kompiuteruli maTematika

ïæéîŽãèâåŽ åâëî控 1. ïæéîŽãèââIJæ ᎠéŽåæ ïìâùæòæçŽùæŽîëàëîù öâïŽãŽèöæ æõë Žôêæöêñèæ, øãâê àãïñîᎠŽàãâàë éŽåâéŽðæçñîæåâëîæŽ ïŽçéŽëá éçŽùîŽá, éŽàîŽé Žé áîëï áŽïŽIJñåâIJæï åãŽèïŽäîæïæåûŽîéëæóéêâIJŽ àŽîçãâñèæ ïæúêâèââIJæ. Žéæï êŽùãèŽá øãâê ãæûõâIJå æïâåæ ïŽûõæïæùêâIJâIJæï Žôûâîæå, îëàëîæù Žîæï ïæéîŽãèâ. éæñýâáŽãŽá æéæïŽ, îëééŽïŽèŽ öâæúèâIJŽ éëàãâøãâêëï éŽîðæãŽá, âï çâåáâIJŽ æéæïŽåãæï, îëé éçæåýãâèéŽàŽŽîøæëï ìŽîŽàîŽòæï IJëèëï éëõãŽêæèæ éŽàŽèæåâIJæ.ïæéîŽãèâ { âï Žîæï àŽîçãâñèæ àŽêïýãŽãâIJñèæ ëIJæâóðâIJæï âîåëIJèæëIJŽ,ïŽáŽù êâIJæïéæâîæ ëIJæâóðæïåãæï öâæúèâIJŽ áŽãŽáàæêëå, éæâçñåãêâIJŽ åñ ŽîŽâï ëIJæâóðæ éëùâéñè ïæéîŽãèâï.ïæéîŽãèâ, îëéâèæù âéëîøæèâIJŽ éýëèëá Žïâå öâäôñáãŽï, öâæúèâIJŽ öâæùŽãáâïåæåóéæï êâIJæïéæâîæ ýŽïæŽåæï ëIJæâóðâIJï. éŽàŽèæåŽá:• ïæéîŽãèâ èëêáëêæï õãâèŽ éæûæïóãâöŽ ïŽáàñîæïŽ;• ïæéîŽãèâ éŽîùýâêŽ òâýïŽùéâèâIJæïŽ;• ïæéîŽãèâ êŽðñîŽèñîæ îæùýãâIJæïŽ 1, 2, 3, 4, . . . .• ïæéîŽãèâ ïìâùæŽèñîæ ïŽIJâüáæ éëûõëIJæèëIJâIJæïŽåãæï ïŽüæîë ïæéIJëèëâIJæïŽ;• ïæéîŽãèâ çëêçîâðñèæ çëéìæñðâîæï ëìâîŽùæâIJæï çëáâIJæïŽ.ïŽIJëèëëá, øãâêåãæï ïŽæêðâîâïë æóêâIJŽ öâéáâàæ ïæéîŽãèââIJæ:• ïæéîŽãèâ æáâêðæòæçŽðëîâIJæïŽ, îëéèâIJæù àãýãáâIJŽ àŽîçãâñè ìîëàîŽéŽöæ;• ïæéîŽãèâ ëìâîŽùæâIJæïŽ æéŽãâ ìîëàîŽéŽöæ;• ïæéîŽãèâ ëìâîŽùæâIJæïŽ, îëéèâIJæù öâæúèâIJŽ öâïîñèáâï éëùâéñèææêïðîñóùææï öâéáâà æéŽãâ ìîëàîŽéŽöæ.ŽéŽïåŽê, âï ïæéîŽãèââIJæ ïŽçéŽëá îåñèæŽ çãèâãæïŽåãæï. ñéâðâïŽá àŽéëãæõâêâIJåäëàæâîå ŽIJïðîŽóðñè ïæéîŽãèâï. æïâåâIJï, îëàëîæùŽŽ îæùýãåŽïæéîŽãèââIJæ.ïæéîŽãèââIJï, øãâñèâIJîæã ŽôêæöêŽãâê áæáæ èŽåæêñîæ ŽïëâIJæå: A, B, . . .ᎠøŽæûâîâIJŽ ëîæáŽê âîåæ ýâîýæå. åñ ïæéîŽãèâ öâæùŽãï îŽéáâêæéâ âèâéâêðï,éŽöæê, ñIJîŽèëá, ãûâîå éæï õãâèŽ âèâéâêðï. éŽàŽèæåŽá, åñ A-ï àŽêãïŽäôãîŽãåîëàëîù 6-ïŽ áŽ 10-ïöëîæï éçŽùîŽá éëåŽãïâIJñè éåâè îæùýãåŽïæéîŽãèâï, éŽöæê âï öâæúèâIJŽ øŽãûâîëå öâéáâàêŽæîŽá:A = {7, 8, 9}5

6ᎠûŽãæçæåýŽãå Žïâ:\A Žîæï ïæéîŽãèâ, îëéâèæù öâæùŽãï 7,8,9-ï".Žó \=" ïæéIJëèë àŽéëæõâêâIJŽ àŽîçãâñèæ Žäîæå: A ñáîæï ïæéîŽãèâï . . . öâéáâàöæàŽéëãæõâêâIJå àŽéëåóéŽï: \ñáîæï åñ ŽîŽ A . . .". Žéæðëé ñêᎠöâéëãæðŽêëåŽé òŽóðæï ïŽéŽîåèæŽêëIJæï áŽáàâêæï ìîëùâáñîŽ. ïý㎠ïæðõãâIJæå,ïæéîŽãèâ öâæúèâIJŽ áŽãŽýŽïæŽåëå àŽîçãâñèæ åãæïâIJâIJæå. ŽéàãŽîŽá, A ïæéîŽãèâöâæúèâIJŽ àŽêãïŽäôãîëå îëàëîùA = { x : x Žîæï éåâèæ îæùýãæ, 6 < x < 10 }ᎠûŽãæçæåýëå Žïâ:\A Žîæï õãâèŽ æïâåæ x-ïŽ, îëé . . .".A ïæéîŽãèâï âçñåãêæï éýëèëá æï âèâéâêðâIJæ, îëéèâIJæù Žîæï 6-äâ éâðæᎠ10-äâ êŽçèâIJæ éåâèæ îæùýãâIJæ. â. æ. 7, 8 Ꭰ9. ŽéîæàŽá, àãŽóãï 7,8,9îëàëîù ûæêŽå.ïæéîŽãèââIJï ýöæîŽá àŽêæýæèŽãâê, îëàëîù \âèâéâêðâIJæï ŽîŽáŽèŽàâIJñèâîåëIJèæëIJŽï", åñéùŽ äëàþâî ïŽüæîëŽ ýŽäæ àŽãñïãŽå, îëé, éŽàŽèæåŽá,{7, 8, 9}, = {8, 9, 7} = {9, 7, 8}, . . .âèâéâðâIJæï áŽèŽàâIJŽäâ ŽîŽãæåŽî ìæîëIJŽï Žî ãŽõâêâIJå. Žéæðëé ŽîŽïûëîææóêâIJëᎠáŽàãâö㎠îŽæéâ àŽêïŽäôãîñèæ áŽèŽàâIJŽ.êâIJæïéæâîæ éëùâéñèæ ëIJæóðæïŽåãæï öâæúèâIJŽ àŽêãïŽäôãîëå âçñåãêæïåñ ŽîŽ æàæ A ïæéîŽãèâï. çâîúëá, åñ îæùýãæ âçñåãêæï ïæéîŽãèâï, éŽöæêãæðõãæå, îëé \âï îæùýãæ Žîæï ïæéîŽãèæï âèâéâêðæ". éŽàŽèæåŽá, åñ 7 ŽîæïA ïæéîŽãèæï âèâéâêðæ, éŽöæê âï òŽóðæ øŽæûâîâIJŽ öâéáâàêŽæîŽá:7 ∈ A.ýëèë òŽóðæ \6 Žî Žîæï A-ï âèâéâêðæ", øŽæûâîâIJŽ Žïâ:6 ∉ A.\∈" ïæéIJëèë ûŽîéëáàâIJŽ IJâîúêñèæ Žïë ε-áŽê. ñŽîõëòŽ ŽôæêæöêâIJŽ ∉ïæéIJëèëåæ. ëìâîŽùææï (Žê ëìâîŽùæñèæ ïæéIJëèëï) ñŽîõëòæï Žïâåæ ŽôêæöãêŽäëàŽáŽá éæôâIJñèæŽ éŽåâéŽðæçŽöæ ᎠýöæîŽá æóêâIJŽ àŽéëõâêâIJñèæ öâéáâàöæ.Žôãêæöêëå, îëé áæáæ õñîŽáôâIJŽ ñêᎠéæâóùâï ïæéîŽãèâåŽ ïìâùæòæçŽùæ-Žï. åæåëâñèæ ïæéîŽãèæïŽåãæï ñêᎠøŽæûâîëï éæïæ ïìâùæòæçŽùæŽ. ïæéîŽãèæïûŽîéëóéêæï ìîëùâïæ öâæúèâIJŽ àîúâèáâIJëáâï ñïŽïîñèëá áæáýŽêï. öâáâàŽáãôâIJñèëIJå ïæéîŽãèâï, îëéâèæù öââïŽIJŽéâIJŽ àŽêïŽäôãîŽï. êŽûæèëIJæåæ ïìâùæòæçŽùæŽæé öâéåýãâãŽöæ öâæúèâIJŽ æõëï ïŽïŽîàâIJèë, îëùŽ àŽîçãâñèæ ŽîŽîæï âçñåãêæï åñ ŽîŽ éëùâéñèæ âèâéâêðæ ïæéîŽãèâï. ŽóŽéáâ àãýãáâIJëáŽïæéIJëèëâIJæ: { . .}, {. . .}, ∈ Ꭰ∉. éŽåæ àŽéëõâêâIJŽ ïŽçéŽëá éŽîðæãŽá àãâøãâêâIJŽ,éŽàîŽé éëæåýëãï àŽîçãâñè øãâãâIJï, îëéèâIJæù æèñïðîæîâIJñèæ æóêâIJŽöâéáâà éŽàŽèæåâIJäâ

7é Ž à Ž è æ å æ 1.1. ïæéîŽãèâåŽ éëõãŽêæèæ àŽêïŽäôãîâIJâIJæáŽê îëéâèæŽïûëîæ:A = {1, 2, 3},B = {5, 6, 6, 7},C = {x : x ∉ A},D = {A, C},E = { x : x = 1 Žê x = {y} Ꭰy ∈ E } ,F = { ïæéîŽãèââIJæ, îëéèâIJæù Žî ûŽîéëŽáàâêï åŽãæï åŽãæï âèâéâêðï } == { x : x ïæéîŽãè⎠Ꭰx ∉ x } .A ïæéîŽãèæï ûâãîåŽ îŽëáâêëIJŽ ŽáãæèŽá àŽéëæåãèâIJŽ ᎠïæéîŽãèæïâèâéâêðâIJæ Žî éâëîáâIJŽ, éŽöæê àŽêïŽäôãîâIJŽ ïûëîæŽ.B ïæéîŽãèâ àŽéëæõñîâIJŽ Žàîâåãâ ïûëîŽá, éýëèëá æéæï àŽéëçèâIJæå,îëé îæùýãæ 6 àãýãáâIJŽ ëîþâî. øãâê öâàãæúèæŽ öâãŽéëûéëå âçñåãêæï åñŽîŽ âèâéâêðæ ïæéîŽãèâï. ŽéàãŽîŽá, âï ñéêæöãêâèëãŽêâïæ éëåýëãêŽ ïæéîŽãèæïàŽêïŽäôãîâIJŽöæ öâïîñèâIJñèæŽ. éŽöŽïŽáŽéâ, öâàãæúèæŽ âï ûŽûâîŽ àŽêãæýæèëåîëàëîù ïûëîæ Ꭰ{5, 6, 7}-æï âçãæãŽèâêðñîæ. ŽéŽïåŽê, Žé ïæðñŽùæŽöæûŽîéëæóéêâIJŽ öâéáâàæ ìîëIJèâéâIJæ. åñ àŽêãæýæèŽãå B-ï ìæîãâè àŽêïŽäôãîâ-IJŽï ᎠŽéëãŽàáâIJå ïæéîŽãèæáŽê âîå-âîå 6-ï, éŽöæê ùýŽáæŽ, àãâóêâIJŽ 6 ∈ BᎠ6 ∉ B. ûŽîéëæóéêâIJŽ ûæꎎôéáâàëIJŽ. Žéæðëé ïæéîŽãèâåŽ àŽêïŽäôãîâIJŽöæïæéIJëèëâIJæï àŽéâëîâIJŽï àŽêãæýæèŽãå, îëàëîù âîåæ ᎠæéŽãâ ïæéIJëèëïéëýïâêæâIJŽï, ýëèë éæï áñIJèæîâIJŽï, îëàëîù ñõñîŽáôâIJëIJŽï. àŽéâëîâIJñèæïæéIJëèëâIJæï ŽéëîâIJŽ ûŽîéëóéêæï öâéáàëéæ éŽåâéŽðæçñîæ éïþâèëIJæï ïŽòñúãâèï.A ïæéîŽãèâ öâæùŽãï îæùýãâIJï, ŽéŽê öâæúèâIJŽ àŽéëæûãæëï àŽñàâIJîëIJŽ,îŽáàŽê îæùýãâIJæ Žî ŽîïâIJëIJï, ñòîë äñïðŽá, øãâê ãæõâêâIJå îæùýãåŽ ïæéIJëèëâIJï.âï ïæéIJëèëâIJæ æûëáâIJŽ æïâãâ, îëàëîù îæùýãâIJæ. Žéæðëé B ŽîæïïŽýâèâIJæï ïæéîŽãèâ ᎠïŽýâèâIJï, øãâñèâIJîæã, ãæõâêâIJå, îŽåŽ ûŽîéëãæáàæêëåæï ëIJæâóðâIJæ (âèâéâêðâIJæ), îëéèâIJäâù ãñåæåâIJå. àŽéëåãèâIJöæ ïŽýâèâIJïŽóãï àŽêïŽçñåîâIJñèæ éêæöãêâèëIJŽ, àŽêïŽçñåîâIJæå ìîëàîŽéñèæ âêâIJæïïâéŽêðæçæï öâïûŽãèŽöæ (ìîëàîŽéæï Žîïæï). Žó Žî Žîæï Žé ìîëIJèâéâIJæï áâðŽèñîæàŽêýæèãæï Žáàæèæ.Žãæôëå, éŽàŽèæåŽá, ïæéîŽãèâX = { \ìŽïçŽèæï öâïŽãŽèæ",\ïðîñóðñîñèæ éëêŽùâéâIJæï ïŽòñúãèâIJæ", \ìŽïçŽèæï öâïŽãŽèæ" } .âï Žîæï ëîæ ûæàêæï áŽïŽýâèâIJæï ïæéîŽãèâ, îëéèæï âîåæ âèâéâêðæ ñõñîŽáôâIJëIJæïàŽéë ëîþâî Žîæï ûŽûâîæèæ, Žê âï Žîæï ïŽéæ ûæàêæï ïæéîŽãèâ, îëéâèåŽàŽêëîï âîåæ Ꭰæàæãâ áŽïŽýâèâIJŽ Žóãï. åñ âï Žïâ, éŽöæê ëîæ ûæàêæ\ìŽïçŽèæï öâïŽãŽèæ" îŽæéâ ýâîýæå ñêᎠàŽãõëå. éëùâéñèæ æêòëîéŽùææå ŽîöâæúèâIJŽ àŽãŽîçãæëå ïûëîæ ìŽïñýæ, Žéæðëé, Žïâå öâéåýãâãŽöæ òîåýæèŽáñêᎠãæõëå.

8C ïæéîŽãèæï àŽêïŽäôãîâIJŽù, æïâãâ îëàëîù A-ï, ïûëîæŽ, îŽáàŽê åñx ∈ A, éŽöæê x ∉ C Ꭰåñ x ∉ A, éŽöæê x ∈ C. C ïæéîŽãèâ úŽèæŽê áæáæŽ: æïöâæùŽãï \õãâèŽòâîï" 1, 2 Ꭰ3 îæùýãæï àŽîáŽ. ŽôêæöãêŽ \õãâèŽòâîï" àŽéëõëòæèæŽáŽ, îëàëîù éŽèâ ãêŽýŽãå, éŽåâéŽðæçñîæ åãŽèïŽäîæïæå \ïŽöæöæŽ".îŽáàŽê A ᎠC àŽêïŽäôãîâIJâIJæ ïæéîŽãèââIJï ûŽîéëŽáàâêï, ŽóâáŽê ãôâ-IJñèëIJëå, îëé D-ï àŽêïŽäôãîâIJŽù, Žàîâåãâ, ïûëîæŽ. öâãêæöêëå, îëé âïïæéîŽãèâåŽ çèŽïæ (ŽîŽòâîæ ŽîŽïûëîæ ŽéŽöæ Žî Žîæï!) æïâåæŽ, îëé éŽï Žóãïéýëèëá ëîæ âèâéâêðæ, çâîúëá, 1 ∉ D, åñêáŽù 1 ∈ A ᎠA ∈ D. âï ŽáãæèŽáöâæúèâIJŽ öâãŽéëûéëå, îŽáàŽê 1 ≠ A, 1 ≠ C Ꭰéýëèëá A ᎠCûŽîéëŽáàâêï D ïæéîŽãèæï âèâéâêðâIJï.E ïæéîŽãèâ Žîæï îâçñîïñèŽá àŽêïŽäôãîñèæ ïæéîŽãèæï ìæîãâèæ éŽàŽèæåæ.æàæ àŽêæïŽäôãîâIJŽ (êŽûæèëIJîæã) åŽãæïæãâ ðâîéæêâIJöæ. éæïæ ŽàâIJæï ìîëùâïæñïŽïîñèëá àîúâèáâIJŽ, Žéæðëé âèâéâêðâIJæï àŽêïŽäôãîæïåãæï ñêᎠàãóëêáâïûâïæ. éŽå øŽûâîŽï ãâî öâãúèâIJå ùýŽáŽá. öâãêæöêëå, îëé E Žî àŽêæïŽäôãîâIJŽéåèæŽêŽá E-ï ðâîéæêâIJöæ, øãâê ñêᎠãæùëáâå ïæéîŽãèâäâ îŽæéâ,îŽù Žî Žîæï áŽéëçæáâIJñèæ àŽêïŽäôãîâIJŽäâ. Žé öâéåýãâãŽöæ âï Žîæï æï îëé1 ∈ E. àãŽóãï:1 ∈ E, Žéæðëé {1} ∈ E,{1} ∈ E, Žéæðëé {{1}} ∈ E,{{1}} ∈ E, Žéæðëé {{{1}}} ∈ E ᎠŽ. ö.åñéùŽ ŽàâIJæï ìîëùâïæ öâéëïŽäôãîñèæ Žî Žîæï, ãæîâIJå îŽ êâIJæïéæâî âèâéâêðïᎠåñ àãŽóãï ïŽçéŽë áîë, öâæúèâIJŽ àŽêãïŽäôãîëå öâáæï åñ ŽîŽ âïâèâéâêðæ E ïæéîŽãèâöæ.ŽýèŽ àŽáŽãæáâå F ïæéîŽãèâäâ. âï ïŽçéŽëá îåñèæ ŽéëùŽêŽŽ. îëé áŽãæêŽýëå,åñ îŽðëé Žî öâæúèâIJŽ ŽîïâIJëIJáâï F , áŽïŽûõæïöæ áŽãñöãŽå éæïæŽîïâIJëIJŽ, ýëèë öâéáâà ãŽøãâêëå, îëé ŽîïâIJëIJï æïâåæ âèâéâêðæ (æàæ Žôãêæöêëåy-æå), îëéèæïåãæïŽù Žî öâàãæúèæŽ àŽêãïŽäôãîëå y ∈ F åñ y ∉ F .ïŽâîåëá, \ñýâîýñèæ " éŽàŽèæåæï àŽéëçãèâãŽ, îëéâèäâù öâàãâúèâIJŽ èëàæçñîæêŽçèæï øãâêâIJŽ, Žáãæèæ Žî Žîæï. éŽàîŽé, éëùâéñè öâéåýãâãŽöæ öâàãæúèæŽàŽéëãæõâêëå åãæå F ïæéîŽãèâ. ïŽçæåýæï àŽïŽîçãâãŽá Žôãêæöêëå âïïæéîŽãèâ G-æå. åñ, îëàëîù áŽãñöãæå, F ïŽúæâIJâèæ ïæéîŽãèâŽ, éŽöæê ŽêG ∈ F , Žê G ∉ F . àŽêãæýæèëå ëîæ öâïŽúèë öâéåýãâãŽ: Ž) G ∈ F , éŽöæê GŽçéŽõëòæèâIJï ìæîëIJŽï: G ∉ G, áŽ, éŽöŽïŽáŽéâ, G ∉ F .IJ) G ∉ F , âï êæöêŽãï, îëé G Žî ŽçéŽõëòæèâIJï F ïæéîŽãèâöæ öâïãèæïìæîëIJŽï, â. æ. G ∈ G, éŽöŽïŽáŽéâ, G ∈ F .éŽöŽïŽáŽéâ, õãâèŽ öâéåýãâãŽöæ éæãáæãŽîå ûæꎎôéáâàëIJŽéáâ. Žéæðëé FŽî öâæúèâIJŽ ŽîïâIJëIJáâï. ïŽ æóêŽ áŽöãâIJñèæ öâùáëéŽ? ïæéîŽãèâåŽ ïæéîŽãèââIJæ,öâïŽúèâIJâèæŽ, ŽéëæýïêâIJŽ, ñïŽïîñèëá áæáæ ïæéîŽãèââIJæù (éŽàŽèæåŽá,äâéëå àŽêýæèñèæ E ïæéîŽãèâ), Žàîâåãâ ŽéëæýïêâIJŽ, éŽàîŽé \õãâèŽ ïæéîŽãèâåŽïæéîŽãèââIJäâ" éñöŽëIJŽ Žî öâæúèâIJŽ ïæéîŽãèâåŽ øãâñèâIJîæãæ åâëîææå.âï éëæåýëãï ïý㎠ïŽýæï éŽåâéŽðæçŽï. ïæéîŽãèâåŽ åâëîææï âï ŽêëéŽèæŽ

9ùêëIJæèæŽ îëàëîù îŽïâèæï ìŽîŽáëóïæ. åñ ñçãâ àãŽóã H ïæéîŽãèâ, éŽöæêöâàãæúèæŽ àŽêãïŽäôãîëå I.ŽéàãŽîŽá, øãâê àŽéëãæçãèâãå éýëèëá æé ïæéîŽãèââIJï, îëéèâIJæù öâæúèâIJŽùýŽáŽá øŽæûâîëï Žê Žæàëï çŽîàŽá àŽêïŽäôãîñèæ ìîëùâïâIJâæå. ŽéæðëéïæéîŽãèââIJæ Žîù æïâåæ ðîæãæŽèñîæŽ, îëàëîù âï áŽïŽûõæïöæ öâæúèâIJŽàãâøãâêëï. ŽéŽïåŽê, åñ äâéëå éëõãŽêæè ûâïâIJï éæãõãâIJæå, éŽåäâ éñöŽëIJŽ ŽîæóêâIJŽ àŽêïŽçñåîâIJñèŽá îåñèæ. öââùŽáâå áŽéëñçæáâãèŽá àŽŽçâåëå öâéáâàæïŽãŽîþæöë:ï Ž ã Ž î þ æ ö ë 1.1.1. àŽêæýæèâå ïæéîŽãèæï æêðâîìîâðŽùææï ëîæ öâïŽúèë ãŽîæŽêðæ éŽæêù{ïéæðæ, ïéæðæ, IJîŽñêæ}.àŽêïŽäôãîâå åæåëâñèæ éŽåàŽêæ æéáâêŽá ùŽèïŽýŽá, îŽéáâêŽáŽù âï öâïŽúèâIJâèæŽ.2. àŽêæýæèâå öâéáâàæ ëåýæ ïæéîŽãèâ:Ž) {1, 2, 3, 4};IJ) {I, II, III, IV, V};à) { 1, âîåæ, one, uno, ein } ;á) { 5, V, ýñåæ, ˛ve } .àŽŽîçãæâå Žé ïæéîŽãèââIJæï øŽûâîŽ îëàëî öâæúèâIJŽ àŽéŽîðæãáâï ᎠéŽåöëîæï îëéèâIJæŽ âçãæãŽèâêðñîæ. àŽêæýæèâå õãâèŽ öâïŽúèë æêðâîìîâðŽùæŽ,îëéèâIJæù ŽçéŽõëòæèâIJï äâéëå Žôûâîæè ãŽîŽñáâIJï.3. öâŽéëûéâå öâéáâàæ éðçæùâIJñèâIJâIJæï ïŽéŽîåèæŽêëIJŽ:\âï éðçæùâIJñèâIJŽ ŽîŽïûëîæŽ" Ꭰ\éâ éŽðõñŽîŽ ãŽî"éðçæùâIJñèâIJŽöæ îëéâèæ ïæðõãâIJæ éëæåýëãï ïŽçñåîæã (â. æ. éŽåâéŽðæçñîŽáäñïð) àŽêïŽäôãîâIJŽï. æéæïŽåãæï, îëé ìŽïñýâIJæ àŽãýŽáëå éŽåâéŽðæçñîŽáéçŽùîæ?4. ãåóãŽå, X Žîæï {1, 2} ïæéîŽãèâ, ýëèë Y { { x : x ∈ y+z; y, z ∈ X }ïæéîŽãèâ. àŽêïŽäôãîâå ùýŽáæ ïŽýæå Y ïæéîŽãèâ. îëàëîæŽ âï ïæéîŽãèââIJæ{y : y = x + z; x, z ∈ X}áŽ{y : x = y + z; x, z ∈ X}.5. áŽãñöãŽå, îëé x Žîæï âèâéâêðæ ᎠŽîŽ ïæéîŽãèâ. éŽöæê y ∉ x êâ-IJæïéæâîæ y-åãæï ᎠŽóâáŽê àŽéëéáæêŽîâëIJï, îëé x ∉ x. öâæúèâIJŽ åñ ŽîŽàŽãŽéŽîðæãëå { x, {x}, {{x}} } ïæéîŽãèâ? îŽ öâæúèâIJŽ æåóãŽï { x, y, {x, y} }ïæéîŽãèæï éæéŽîå?6. ãåóãŽå, A Žîæï õãâèŽ éðâè îæùýãåŽ ïæéîŽãèâ. Žôûâîâå ïæðõãâIJæåöâéáâàæ ïæéîŽãèâ:X = { x : x ∈ A Ꭰx = 1 Žê (x − 2) ∈ X } .

10§ 2. ñéŽîðæãâïæ ëìâîŽùæâIJæ ïæéîŽãèââIJäâîëàëîù 1.1 éŽàŽèæåöæ \éùáŽîæ ïæéîŽãèæï" àŽêýæèãæïŽï ãêŽýâå, ïæéîŽãèââIJæïàŽêïŽäôãîæïŽï ïŽüæîëŽ àŽéëãæøæêëå ïæòîåýæèâ. éæñýâáŽãŽáŽéæïŽ, éëùâéñèæ ïæéîŽãèââIJæáŽê éŽîðæãæ ûâïâIJæå ŽýŽèæ ïæéîŽãèââIJæï Žàâ-IJæïŽï öâæúèâIJŽ ñéðçæãêâñèëá éæãæôëå éîŽãŽèæ ïŽæêðâîâïë éŽàŽèæåæ.éëàãæŽêâIJæå éëãæõãŽêå æé òëîéŽèñî ûâïâIJï, îëéèâIJïŽù ñêᎠŽçéŽõëòæèâIJáâïïæéîŽãèââIJäâ àŽêïŽäôãîñèæ ëìâîŽùæâIJæ, ýëèë ŽýèŽ öâéëãæðŽêåäëàæâîåæ ŽôêæöãêŽ. áŽãæûõëå ñéŽîðæãâïæ ëìâîŽùæâIJæáŽê.à Ž ê ï Ž ä ô ã î â IJ Ž . ãåóãŽå, éëùâéñèæŽ A ᎠB ïæéîŽãèâ. AᎠB ïæéîŽãèââIJæï åŽêŽçãâåŽ âûëáâIJŽ õãâèŽ æé âèâéâêðæï ïæéîŽãèâï,îëéâèæù âçñåãêæï îëàëîù A-ï, Žïâãâ B-ï ᎠŽôæêæöêâIJŽ öâéáâàêŽæîŽá:A ∩ B; ŽéàãŽîŽá,A ∩ B = { x : x ∈ A Ꭰx ∈ B } .ŽêŽèëàæñîŽá, A ᎠB ïæéîŽãèââIJæï àŽâîåæŽêâIJŽ ŽôæêæöêâIJŽ A ∪ BïæéIJëèëåæ ᎠàŽêæïŽäôãîâIJŽ öâéáâàêŽæîŽá:A ∪ B = { x : x ∈ A Žê x ∈ B } .åŽêŽçãâåæïŽ áŽ àŽâîåæŽêâIJæï àŽêïŽäôãîâIJâIJæ àŽéëéáæêŽîâëIJï öâïŽIJŽéæïŽá,\áŽ" Ꭰ\Žê" ïæðõãâIJæáŽê, Žéæðëé îëàëîù öâáâàæ àãŽóãï:A ∪ B = B ∪ A,A ∩ B = B ∩ AᎠîŽù, ŽèIJŽå, êŽçèâIJŽá ùýŽáæŽA ∪ A = A, A ∩ A = A.âï æàæãâëIJâIJæ éêæöãêâèëãŽêæŽ. öâéáàëéæ éŽåâéŽðæçñîæ éïþâèëIJâIJæáŽê àŽéëøêáâIJŽ,îëé äëàþâî ïŽüæîëŽ A ∪ A (öâïŽIJŽéæïŽá, A ∩ A) áŽãæõãŽêëå A-äâ,Žê ìæîæóæå, A àŽãŽòŽîåëãëå A ∪ A (öâïŽIJŽéæïŽá, A ∩ A)-éáâ.à Ž ê ï Ž ä ô ã î â IJ Ž . A ᎠB ïæéîŽãèââIJæï ïýãŽëIJŽ (îëéâèïŽù,Žàîâåãâ, âûëáâIJŽ B-ï áŽéŽðâIJŽ A-éáâ) øŽæûâîâIJŽ A \ B ïŽýæå áŽàŽêæïŽäôãîâIJŽ åŽêŽòŽîáëIJæåA \ B = { x : x ∈ A Ꭰx ∉ B } .Žéæðëé, åñ A = {1, 2, 3} ᎠB = {2, 3, 4}, éŽöæê A\B = {1} ᎠB \A = {4}.öâéáâàæ àŽêïŽäôãîâIJŽ øŽîåñèæŽ ïæïîñèæïŽåãæï. éæñýâáŽãŽá æéæïŽ, îëééŽï ñöñŽèëá æöãæŽåŽá àŽéëãæõâêâIJå, óãâéëå ãêŽýŽãå, îëé Žé ëìâîŽðëîïáæáæ éêæöãêâèëIJŽ Žóãï éŽêóŽêñî ŽîæåéâðæçŽöæ.à Ž ê ï Ž ä ô ã î â IJ Ž . A ᎠB âèâéâêðâIJæï ïæéâðîæñèæ ïýãŽëIJŽŽêñ A△B àŽêæïŽäôãîâIJŽ, îëàëîùA△B = (A ∪ B) \ (A ∩ B).

11öâïŽúèâIJâèæŽ ∪, ∩, \, △ ŽôêæöãêâIJéŽ éçæåýãâèæ ᎎIJêæŽ, Žê öâæúèâIJŽâàëêëï, îëé æïæêæ æéáâêŽá âèâéâêðŽîñèæŽ, îëé éŽå ŽîŽãæåŽîæ ìîŽóðæçñèæàŽéëõâêâIJŽ Žî Žóãå. öâéáâàæ éŽàŽèæåâIJæ éëàãâýéŽîâIJŽ Žéæï àŽîçãâãŽöæ.é Ž à Ž è æ å æ 2.1. ãæàñèæïýéëå, îëé àãŽóãï P ᎠQ ëîæ ìîëàîŽéŽ.A Žîæï P ìîëàîŽéæï éëêŽùâéåŽ õãâèŽ éêæöãêâèëIJæï ïæéîŽãèâ, ýëèë B { QìîëàîŽéæï ïæéîŽãèâ, éŽöæê A ∩ B æóêâIJŽ éëêŽùâéåŽ õãâèŽ éêæöãêâèëIJæï ïæéîŽãèâ,îëéâèæù éæïŽûãáëéæŽ P ᎠQ-åãæï; A∪B éëêŽùâéåŽ õãâèŽ éêæöãêâèëIJæïïæéîŽãèâŽ, îëéâèæù éæïŽûãáëéæŽ P -åãæï Žê Q-åãæï (âîå-âîåæïåãæïéŽæêù), A \ B Žîæï õãâèŽ æé éëêŽùâéåŽ ïæéîŽãèâ, îëéâèæù éæïŽûãáëéæŽ P -åãæï, éŽàîŽé ŽîŽ Q-åãæï. B \ A éæïŽûãáëéæŽ Q-åãæï ᎠŽîŽ P -åãæï, A△BŽîæï õãâèŽ æé éëêŽùâéåŽ ïæéîŽãèâ, îëéâèæù éæïŽûãáëéæŽ éýëèëá âîåæ, ŽêP Žê Q, ìîëàîŽéæïåãæï.öâáàëéæ éïþâèëIJæï ûæê éëïŽýâîýâIJâèæ æóêâIJŽ àŽêãïŽäôãîëå ëîæ ïìâùæŽèñîæïæéîŽãèâ. ìæîãâèæ éŽåàŽêæŽ ùŽîæâèæ ïæéîŽãèâ.à Ž ê ï Ž ä ô ã î â IJ Ž . ùŽîæâèæ ïæéîŽãèâ (ŽôæêæöêâIJŽ ∅-æå),îëéâèïŽù åãæïâIJæå (???????)x ∉ ∅, êâIJæïéæâîæ x-åãæï.éâëîâ ïæéîŽãèâï, îëéèæï àŽêïŽäôãîŽù ŽéëùŽêŽä⎠áŽéëçæáâIJñèæ, ñêæãâîïŽèñîæïæéîŽãèâ âûëáâIJŽ.à Ž ê ï Ž ä ô ã î â IJ Ž . ñêæãâîïŽèñîæ ïæéîŽãèâ (ŽôæêæöêâIJŽ E-æå)Žîæï éëùâéñè ŽéëùŽêŽöæ àŽêïŽýæèãâèæ õãâèŽ âèâéâêðæ ïæéîŽãèâ.à Ž ê ï Ž ä ô ã î â IJ Ž . ëîæ A ᎠB ïæéîŽãèâ Žî åŽêŽæçãâåâIJŽ,åñ A ∩ B = ∅.à Ž ê ï Ž ä ô ã î â IJ Ž . õëãâè öâéåýãâãŽöæ, îëáâïŽù éëùâéñèæŽ E ñêæãâîïŽèñîæïæéîŽãèâ, àŽêãïŽäôãîëå A ïæéîŽãèæï áŽéŽðâIJŽ (ŽôæêæöêâIJŽA ′ -æå), îëàëîùA ′ = E \ A = {x : x ∉ A}.∅, E ᎠA ′ ïæéîŽãèââIJæï àŽêïŽäôãîâIJâIJæáŽê àŽéëéáæêŽîâëIJïæàæãâëIJâIJæï ïŽéŽîåèæŽêëIJŽ.A ∪ A ′ = E,A ∩ A ′ = ∅öâéáâàöæ êŽøãâêâIJæ æóêâIJŽ, îëé éëùâéñèæŽ E-åãæï âï æàæãâëIJâIJæ ïŽçéŽîæïæŽ,îŽåŽ ùŽèïŽýŽá àŽêãïŽäôãîëå A ′ .é Ž à Ž è æ å æ 2.2. ãåóãŽå,E = {1, 2, 3, 4}, A = {1, 3, 4}, B = {2, 3}, C = {1, 4}.àŽêïŽäôãîâIJæáŽê ŽáãæèŽá ãæìëãæå, éŽàŽèæåŽá, A ′ , B ∩ C, C \ A ᎠŽ. ö.ŽéŽïåŽê, öâæúèâIJŽ áŽàãüæîáâï ñòîë îåñèæ àŽéëïŽýñèâIJâIJæï àŽéëçãèâãŽ,îëéâèæù öâæùŽãï ëî Žê éâð ëìâîŽùæŽï. îŽáàŽêŽù Žé öâéåýãâãŽöæ æïéâIJŽ

12ïŽçæåýæ, îëàëî àŽêãïŽäôãîëå îæàæ, îëéèæåŽù ïæéîŽãèââIJäâ ñêᎠéëãŽýáæêëåâèâéâêðŽîñèæ ëìâîŽùæâIJæ, Žéæðëé àŽéëãæõâêâIJå òîøýæèâIJï. òîøýæèâIJöæøŽïéñèæ åæåëâñèæ àŽéëïŽýñèâIJŽ ñêᎠöâïîñèáâï éŽêŽé, ãæáîâ éæïæöâáâàæ æóêâIJŽ àŽéëõâêâIJñèæ ïý㎠àŽéëåãèâIJöæ. éŽàŽèæåŽá, (A ∩ B) ′ àŽéëïŽýñèâIJŽöæ.åŽêŽçãâåŽ àŽéëæåãèâIJŽ ñòîë Žáîâ, ãæáîâ áŽéŽðâIJŽ.âï öâåŽêýéâIJŽ, ùýŽáæŽ, ïŽçéŽîæïæŽ.ŽéŽïåŽê, æéæïŽåãæï, îëé åŽãæáŽê ŽãæùæèëåòîøýæèâIJæï áæáæ îŽëáâêëIJŽ, Žî éëãæåýëãå òîøýæèâIJï éŽöæê, îëùŽàãæêᎠöâãŽïîñèëå áŽéŽðâIJæï ëìâîŽùæŽ ïæéîŽãèââIJäâ êâIJæïéæâîæ{∪, ∩, \, △} ëìâîŽùææï ûæê. Žéæðëé A∩B ′ êæöêŽãï A∩(B ′ ) ᎠŽ. ö. éŽöŽïŽáŽéâ,A ∩ B ′ = A ∩ {1, 4} = {1, 4},(A ∩ B) ′ {3} ′ = {1, 2, 4},(B \ A) ∪ C{2} ∪ C = {1, 2, 4}.éçæåýãâèï öâæúèâIJŽ àŽñçãæîáâï, éïþâèëIJŽ îŽðëé Žîæï ŽàâIJñèæ ïæéîŽãèæïᎠŽîŽ îæùýãæï ùêâIJŽäâ. éŽîåèŽù, ŽóŽéáâ øãâê îæùýãâIJï ãæçãèâãáæåéýëèëá îëàëîù ïæéîŽãèâåŽ âèâéâêðâIJï. âï éýëèëá æéæïåãæï çâåáâIJëáŽ,îëé éçæåýãâèæ àŽùêëIJᎠæé ëIJæâóðâIJï, îëéèâIJåŽêŽù éŽï éëñýáâIJŽ éñöŽëIJŽ.ïŽóéâ æéŽöæŽ, îëé ŽîïâIJëIJï ñòîë îåñèæ ïæéîŽãèââIJæ, ãæáîâ îæùýãåŽ ïæéîŽãèâŽ.øãâê öâàãæúèæŽ îæùýãâIJæ éæãæôëå ïæéîŽãèââIJæáŽê ᎠŽîŽ ìæîæóæå.ŽéŽïåŽê, öâéáàëéæ åâëîææï éîŽãŽèæ áŽéŽðâIJâIJæïŽåãæï ñêᎠàŽãŽçâåëå äñïðæéðçæùâIJŽ äëàæâîåæ ïìâùæŽèñî îæùýãåŽ ïæéîŽãèâäâ. æéæïŽåãæï, îëéñäîñêãâèãõëå ïŽòñúãâèæ, îëéèæï ïŽöñŽèâIJæåŽù ŽàâIJñèæ æóêâIJŽ Žïâåæ ïæéîŽãèââIJæ,àŽêãïŽäôãîëå éåâèæ áŽáâIJæåæ îæùýãâIJæï (êŽðñîŽèñîæ îæùýãâIJæï)N ïæéîŽãèâ:N = {1, 2, 3, . . .}.N ïæéîŽãèæï äñïðæ àŽêïŽäôãîŽ \+" Ꭰ\−" Žîæåéâðæçñè ëìâîŽùæâIJåŽêéëùâéñèæ æóêâIJŽ óãâéëå. ŽéŽïåŽê, Žé åŽãöæ ãæãŽîŽñáâIJå, îëé éçæåýãâèæïåãæïùêëIJæèæŽ N ïæéîŽãèæï äëàæâîåæ åãæïâIJŽ. ŽêŽèëàæñîŽá, Z-ï,îëàëîù õãâèŽ éåâè îæùýãåŽ ïæéîŽãèâï:Z = { . . . , −2, −1, 0, 1, 2, . . . } .îŽ åóéŽ ñêáŽ, N ᎠZ ïæéîŽãèââIJæ Žî öâæúèâIJŽ øŽæûâîëï ùýŽáŽá. ŽéŽïåŽê,àŽéëïŽýñèâIJŽ \. . ." ñêᎠàŽãæàëå îëàëîù \ᎠŽïâ öâéáâà".ŽýèŽ àŽêãæýæèëå ïæéîŽãèâA = { 1, 2, 3, . . . , n } = { x : x ∈ N, 1 ≤ x ≤ n } .éŽï àŽŽøêæŽ n âèâéâêðæ. ãæðõãæå, îëé Žé ïæéîŽãèæï ïæéúèŽãîâ (Žê äëéŽ,êëîéŽ, ïæàîúâ) Žîæï n. âï ŽôæêæöêâIJŽ Žïâ:|A| = card(A) = n.öâáâàöæ êâIJæïéæâî B ïæéîŽãèâï, îëéâèïŽù âèâéâêðâIJæï æàæãâ îŽëáâêë-IJŽ àŽŽøêæŽ, îŽù A ïæéîŽãèâï, âóêâIJŽ æàæãâ ïæéúèŽãîâ áŽ, ùýŽáæŽ, ïŽüæîëŽî æóêâIJŽ Žé âèâéâêðâIJæï øŽéëåãèŽ. éùæîâ ïæéîŽãèââIJæïŽåãæï ïŽçéŽëá ŽáãæèæŽâèâéâêðâIJæï øŽéëåãèŽ, éŽàîŽé ïý㎠ïæéîŽãèââIJæïåãæï (éŽàŽèæåŽá, N

13ïæéîŽãèæïŽåãæï) âï öâæúèâIJŽ æõëï öâñúèâIJâèæ. óãâéëå éëùâéñèæ æóêâIJŽ âèâéâêðâIJæïîŽëáâêëIJæï àŽéëåãèæï, éŽàîŽé, ŽéŽãâ áîëï, ŽîŽòëîéŽèñîæ ûâïæ.à Ž ê ï Ž ä ô ã î â IJ Ž . ãæðõãæå, îëé X ïæéîŽãèâ ïŽïîñèæŽ, åñX = ∅ Žê åñ îëéâèæéâ n ∈ N êŽðñîŽèñîæ îæùýãæïåãæï ŽîïâIJëIJï æïâåæ{1, 2, . . . , n} ïæéîŽãèâ, îëéâèïŽù Žóãï âèâéâêðâIJæï æàæãâ îŽëáâêëIJŽ, îŽùX ïæéîŽãèâï. åñ X ≠ ∅ ᎠŽîŽêŽæîæ n Žî éëæúâIJêâIJŽ, éŽöæê X ïæéîŽãèâïâûëáâIJŽ ñïŽïîñèë.ŽýèŽ, îëùŽ öâéëãæðŽêâå îŽéáâêæéâ àŽêïŽäôãîâIJŽ, öâæúèâIJŽ øŽéëãŽõŽèæ-IJëå îŽéáâêæéâ ïŽãŽîþæöë. ïæéîŽãèââIJæ ŽáãæèŽá øŽïŽûâîæ îëé àŽãýŽáëå,âèâéâðâIJæïŽåãæï çãèŽã àŽéëãæõâêâIJå îæùýãâIJï ᎠŽïëâIJï, éŽàîŽé àãâéŽýïëãîâIJŽ,îëé æàæãâ ëìâîŽùæâIJæ öâæúèâIJŽ àŽéëãæõâêëå êâIJæïéæâîæ ïæéîŽãèââIJæïåãæï.ï Ž ã Ž î þ æ ö ë 2.1.1. ãåóãŽå,E = {1, 2, 3, 4}, X = {1, 5}, Y = {1, 2, 4}, Z = {2, 5}.ãæìëãëå ïæéîŽãèââIJæ:Ž) X ∩ Y ′ ; IJ) (X ∩ Z) ∪ Y ′ ; à) X ∪ (Y ∩ Z); á) (X ∪ Y ) ∩ (X ∪ Z);â) (X ∪Y ) ′ ; ã) X ′ ∩Y ′ ; ä) (X ∩Y ) ′ ; å) (X ∪Y )∪Z; æ) X ∪(Y ∪Z); ç) X \Z;è) (X \ Z) ∪ (Y \ Z).2. ãåóãŽå,E = {a, b, c, d, e, f}, A = {a, b, c}, B = {f, e, c, a}, c = {d, e, f}.ãæìëãëå ïæéîŽãèââIJæ:Ž) A \ C; IJ) B \ C; à) C \ B; á) A \ B; â) A ′ ∪ B; ã) B ∩ A ′ ; ä) A ∩ C;å) C ∩ A; æ) C△A.3. éëùâéñèæŽ A ᎠB ëîæ æïâåæ êâIJæïéæâîæ ïæéîŽãèâ, îëé A ∩ B = ∅.îŽï ûŽîéëŽáàâêï A \ B ᎠB \ A ïæéîŽãèââIJæ?4. éëùâéñèæŽ C ᎠD ëîæ æïâåæ êâIJæïéæâîæ ïæéîŽãèâ, îëé C ∩ D ′ = ∅.îŽ öâæúèâIJŽ æåóãŽï C ∩ D ᎠC ∪ D ïæéîŽãèââIJäâ?5. éëùâéñèæŽ êâIJæïéæâîæ X ïæéîŽãèâ. ãæìëãëå ïæéîŽãèââIJæ:Ž) X ∩ X ′ ; IJ) X ∪ X ′ ; à) X \ X ′ .6. öâéáâàæ éðçæùâIJñèëIJâIJæáŽê îëéâèæŽ ïŽéŽîåèæŽêæ? ãæìëãëå ïæéîŽãèââIJæ:Ž) 0 ∈ ∅; IJ) ∅ = {0}; à) |{}| = 1; á) {{∅}} ∈ {{{∅}}}; â) |{{∅}}| = 2.âï \éäŽçãîñèæ" çæåý㎎. öâæúèâIJŽ éëàãâøãâêëï éŽîðæãŽá ùŽîæâèæ ïæéîŽãèâᎠéæïæ åãæïâIJâIJæ Žî Žîæï ïŽçéŽëá éêæöãêâèëãŽêæ.7. ãåóãŽå, M ᎠN Žîæï ëîæ çëéìæñðâîæ òæóïæîâIJñèæ ìîëàîŽéâIJæå.ãåóãŽå, A Žîæï éëêŽùâéåŽ éêæöãêâèëIJâIJæï ïæéîŽãèâ, îëéâèæù éæïŽûãáëéæŽM çëéìæñðâîæïŽåãæï áŽ, åñ x ∈ A ᎠM éŽêóŽêŽ éñöŽëIJï x öâïŽïãèâèæïæðõãæå, éŽöæê M øâîáâIJŽ Ꭰæúèâ㎠öâáâàï. ŽêŽèëàæñîŽá, ãåóãŽå, B Žîæï

14éëêŽùâéåŽ éêæöãêâèëIJâIJæï ïæéîŽãèâ, îëéèâIJæù æûãâãï N çëéìæñðâîæï àŽøâîâIJŽïᎠöâáâàæï éëùâéŽï. åñ A ïæéîŽãèæï êâIJæïéæâîæ âèâéâêðæ éæïŽûãáëéæŽM ᎠN çëéìæñðâîæïŽåãæï, îŽ öâàãæúèæŽ ãåóãŽå B ′ ïæéîŽãèâäâ? ŽãýïêŽåâï ïæðñŽùæŽ ïæéIJëèëâIJæï ïŽöñŽèâIJæå ᎠàŽêãéŽîðëå Žé æêòëîéŽùææï ñïŽîàâIJèëIJŽ.8. ïæéîŽãèâåŽ ðâîéæêâIJæå ŽãýïêŽå 2.1 éŽàŽèæåæ îŽðëé Žîæï ïûëîæ.9. àŽâîåæŽêâIJæï ëìâîŽùææï àŽêïŽäôãîæï áîëï Žôêæöêñèæ æõë, îëé ãæõâêâIJáæå\Žê"-ï. îëàëîù öâæúèâIJŽ ïæéîŽãèâåŽ ðâîéæêâIJöæ àŽéëãïŽýëå àŽéëéîæùýŽãæ\Žê".10. àŽéëåãèâIJöæ ýöæîŽá ŽýŽèæ ïæéîŽãèââIJæï öâïŽóéêâèŽá àŽéëæõâêâIJŽŽîæåéâðæçñèæ ëIJæâóðâIJæ. Žïâ, åñ A ᎠB îæùýãæåæ ïæéîŽãèââIJæŽ, éŽöæêA + B = { x : x = a + b, a ∈ A, b ∈ B } .ŽêŽèëàæñîŽá àŽéæïŽäôãîâIJŽ +, −, / ëìâîŽùæâIJæ îæùýãåŽ ïæéîŽãèââIJï öëîæï.ãæìëãëå öâéáâàæ ïæéîŽãèââIJæ:Ž) {1, 2}+{1, 3}; IJ) {1, 2}∪{1, 3}; à) {1, 2}∗{1, 3}; á) {1, 2}∩{1, 3}; â){1, 2}−{1, 3}; ã) {1, 2}\{1, 3}; ä) {2, 4}/{2}; å) {2, 4}\{2}; æ) {2, 4}−{2}.11. æéæïŽåãæï, îëé öâãúèëå ïæéîŽãèââIJäâ ëìâîŽùæâIJæï ðâóêæçæï àŽéëõâêâIJŽçëêçîâðñè ŽéëùŽêŽöæ, îŽôŽù âðŽìäâ ïŽüæîë æóêâIJŽ \ŽîŽéŽåâéŽðæçñîæ"éðçæùâIJŽ àŽáŽãæõãŽêëå éŽåâéŽðæçñîöæ. øãâñèâIJîæã (éýëèëá ŽîŽ õëãâèåãæï),Žéæå ŽôûâîŽ ýáâIJŽ ñòîë çëéìŽóðñîæ. ŽéŽïåŽê, éŽåâéŽðæçñîæ àŽéëïŽýñèâIJŽõëãâèåãæï æóêâIJŽ éŽåâéŽðæçñîŽá éçŽùîæ. éŽöæê, îëáâïŽù ïŽûõæïæàŽéëïŽýñèâIJŽ öâæúèâIJŽ Žïâåæ æõëï (ïŽáŽù âï ýáâIJŽ, ïŽüæîëŽ ãæìëãëå çèæŽ ðŽãáŽìæîãâè òëîéñèæîâIJŽï). ŽýèŽ ïŽüæîëŽ:Ž) öâãâùŽáëå øŽéëãŽõŽèæIJëå öâéáâàæ éðçæùâIJñèëIJâIJæ ïæéîŽãèâåŽ âêŽäâ:• éëùâéñèæŽ A, B ᎠC ïæéîŽãèââIJæ. àŽêãïŽäôãîëå ïæéîŽãèâ, îëéâèæùåŽãæï åŽãöæ öâæùŽãï Žé ïæéîŽãèââIJæáŽê éýëèëá ëîï.• ûæêŽ ŽéëùŽêŽ ŽéëãýïêŽå æé ìæîëIJæå, îëé A, B ᎠC ïæéîŽãèââIJæñîåæâîåŽîŽåŽêŽéçãâåæŽ.• éëùâéñèæŽ V , W , Y , X ᎠZ ïæéîŽãèââIJæ. àŽêãïŽäôãîëå ïæéîŽãèâ,îëéâèæù öâæùŽãï V , W , X ᎠY ïæéîŽãèââIJæáŽê ëîï éŽæêùᎠŽî öâæùŽãï Z-ï.IJ) ŽêŽèàæñîŽá Žôãûâîëå ïæðõãâIJæå öâéáâàæ ïæéîŽãèââIJæ(J ∩ (K ∪ L)) ′∪ (H \ L),(P ∪ R ∪ Q) \ ( P ∩ (Q \ R) ) ,((E \ F ) ∪ (F \ E)) ′∪ G.§ 3. ãâêæï áæŽàîŽéâIJæýöæîŽá ïŽïîàâIJèëŽ àãóëêáâï ïæéîŽãèâåŽ àâëéâðîæñèæ ûŽîéëáàâêŽ.Žïâåæ ûŽîéëáàâêâIJæ ãâî öâùãèæï áŽéðçæùâIJâIJï, éŽàîŽé, öâæúèâIJŽ ïŽïŽîàâ-IJèë æõëï æéæïŽåãæï, îëé ïûîŽòŽá ᎠéŽîðæãŽá áŽãîûéñêáâå ïŽéŽîåèæŽêæŽ

15åñ ŽîŽ çëêçîâðñèæ éðçæùâIJñèâIJŽ. áæŽàîŽéâIJï, îëéèâIJïŽù øãâê àŽéëãæõâêâIJå,ãâêæï áæŽàîŽéâIJæ âûëáâIJŽ (æêàèæïâèæ éŽåâéŽðæçëïæï þëê ãâêæïŽïŽýâèæï éæýâáãæå) ᎠŽæàâIJŽ æïâ, îëàëîù âï óãâéëå Žîæï Žôûâîæèæ.ìæîãâè îæàöæ áŽãýŽäëå áæáæ E éŽîåçñåýâáæ (êŽý. 3.1). öâéáâà éŽîåçñåýâáæïöæàêæå øŽãýŽäëå ûîââIJæ (Žê ïý㎠öâïŽIJŽéæïæ öâçîñèæ ûæîæ), îëéûŽîéëãæáàæêëå ïæéîŽãèââIJæ. æïæêæ ñêᎠæçãâåâIJëáêâê ŽéëùŽêŽöæ éëåýëãêæèõãâèŽäâ ñòîë äëàŽá öâéåýãâãŽöæ áŽ, öâïŽIJŽéæïŽá, ñêᎠæõëï Žôêæöêñèæ(êŽý. 3.2). ûâîðæèâIJæ, îëéèâIJæù éáâIJŽîâëIJï áæŽàîŽéæï ïýãŽáŽïý㎠Žîæï öæàêæåöâæúèâIJŽ àŽêãæýæèëå, îëàëîù öâïŽIJŽéæï ïæéîŽãèâåŽ âèâéâêðâIJæ. åñïæéîŽãèââIJöæ âèâéâêðâIJæï îæùýãæ éùæîâŽ, éŽöæê ùŽèçâñèæ âèâéâêðæ öâæúèâIJŽøŽæûâîëï öâïŽIJŽéæïæ ŽîââIJæï öæàêæå, îâëàëîù âï êŽøãâêâIJæŽ 3.1 éŽàŽèæåöæ.êŽý. 3.1êŽý. 3.2êŽý. 3.3é Ž à Ž è æ å æ 3.1. ãåóãŽå, E = {b, c, d, e}, A = {b, c, d}, A = {c, c}.öâïŽIJŽéæïæ áæŽàîŽéŽ àŽéëïŽýñèæŽ 3.3 êŽýŽääâ. Žé ïñîŽåæå ïîñèŽá ŽîæïæèñïðîæîâIJñèæ 3.1 éŽàŽèæåæ. åñ, éŽàŽèæåŽá, A ∈ E, éŽöæê àŽñàâIJŽîæŽ,îæïæ àŽéëïŽý㎠öâæúèâIJëᎠáæŽàîŽéŽäâ. æé öâéåýãâãâIJöæ, îëáâïŽù àŽéëæõâêâIJŽïæéîŽãèâåŽ ñòîë îåñèæ çëêïðîñóùæâIJæ, ñêᎠãâîæáëå áæŽàîŽéâIJæïïŽýæå éŽå àŽéëïŽýãŽï.

16îëáâïŽù àãŽóãï, öâïŽIJŽéæïŽá, ŽàâIJñèæ áæŽàîŽéŽ, éæïæ àŽîçãâñèæ ŽîâöâæúèâIJŽ áŽãöðîæýëå ŽýŽèûŽîéëóéêæè ïæéîŽãèâåŽ àŽêïŽïŽäôãîŽãŽá.é Ž à Ž è æ å æ 3.2. æéæïŽåãæï, îëé ûŽîéëãŽáàæêëå A ∪ (B ′ ∩ C) ïæéîŽãèâ,áŽãæûõëå 3.4 êŽýŽääâ êŽøãâêâIJæ äëàŽáæ áæŽàîŽéæå. B ′ áŽãöðîæýëåâîåæ éæéŽîåñèâIJæå áæŽàëêŽèñîæ ýŽäâIJæåêŽý. 3.4êŽý. 3.5ýëèë C { ïý㎠éæéŽîåñèâIJæï áæŽàëêŽèñîæ ýŽäâIJæå (êŽý. 3.5). ëîéŽàŽááŽöðîæýñèæ òŽîåëIJæ ûŽîéëŽáàâêï B ′ ∩ C ïæéîŽãèâï.áæŽàîŽéæï ŽýŽè Žïèäâ âï Žîâ áŽãöðîæýëå ßëîæäëêðŽèñîæ ýŽäâIJæå,ýëèë A { ãâîðæçŽèñîæå. 3.6 êŽýŽääâ éåèæŽêŽá áŽöðîæýñèæ Žîâ ûŽîéëŽáàâêïA ∪ (B ′ ∩ C) ïæéîŽãèâï. åñ ùŽèçâñè öâéåýãâãŽöæ àŽêïŽýæèãâèïæéîŽãèââIJäâ àãŽóãï áŽéŽðâIJæåæ æêòëîéŽýæŽ, æï öâæúèâIJŽ àŽéëãæõâêëå ãâêæïáæŽàîŽéæï àŽïŽéŽîðæãâIJèŽá.êŽý. 3.6é Ž à Ž è æ å æ 3.3. ãåóãŽå, A ∩ B = ∅. âï öââïIJŽéâIJŽ 3.7 êŽýŽäæïáæŽàîŽéŽï.öâãêæöêëå, îëé ýöæî öâéåýãâãŽöæ ïæéîŽãèââIJæ öâæùŽãï ïŽçéŽëá IJâãîâèâéâêðï áŽ, éŽöŽïŽáŽéâ, âï âèâéâêðâIJæ Žî öâæúèâIJŽ ûŽîéëãŽáàæêëå ùŽèçâ.

17êŽý. 3.7Žéæðëé ñòîë éëïŽýâîýâIJâèæŽ Žé öâéåýãâãŽöæ ãæèŽìŽîŽçëå åæåëâñè ïæéîŽãèâäâ,îëàëîù éåâèäâ ᎠŽî ãŽýïâêëå ùŽèçâñèæ âèâéâêðâIJæ.ï Ž ã Ž î þ æ ö ë 3.1.1. áŽýŽäâå áæŽàîŽéŽ, îëéèæåŽù æèñïðîæîâIJñèæ æóêâIJŽ 2.1 ïŽãŽîþæöëïìæîãâè ŽéëùŽêŽöæ àŽêïŽýæèãâèæ ïæéîŽãèââIJæ.2. îëàëîù öâæúèâIJŽ ûŽîéëãŽáàæêëå ãâêæï áæŽàîŽéâIJæï ïŽöñŽèâIJæå öâéáâàæïæéîŽãèââIJæ:ïŽáŽù{A, {A}}, {{a}, {b}}, {X, Y, Z},X = { x : x = 1 Žê (x − 2) ∈ X } ,Y = { x : x = 3 Žê (x − 3) ∈ Y } ,Z = { x : x = 2 Žê (x − 2) ∈ Z } .§ 4. óãâïæéîŽãèââIJæåŽêŽçãâåæï, àŽâîåæŽêâIJæï, ïýãŽëIJæï ᎠáŽéŽðâIJæï ëìâîŽùæâIJæ àãŽúèâãïŽýŽèæ ïæéîŽãèââIJæï òëîéæîâIJæï ïŽöñŽèâIJŽï. éŽàîŽé, îëàëîù ûâïæ, Žî öâàãæúèæŽãåóãŽå, åñ îëàëî öââòŽîáâIJŽ âîåæ ïæéîŽãèâ ïýãâIJï. éŽàŽèæåŽá,ãåóãŽå, éëùâéñèæŽ ëîæ X ᎠY ïæéîŽãèâ. X ∩ Y åŽêŽçãâåŽ àŽîçãâñèæŽäîæå \êŽçèâIJæ" (Žê ñçæáñîâï öâéåýãâãŽöæ Žî ŽôâéŽðâIJŽ) X ïæéîŽãèâäâ.éŽîåèŽù, X ∩Y ïæéîŽãèæï õãâèŽ âèâéâêðæ, Žàîâåãâ, éæâçñåãêâIJŽ X ïæéîŽãèâï.Žé áŽçãæîãâIJæáŽê öâæúèâIJŽ òëîéŽèñîŽá àŽêãïŽäôãîëå ïæéîŽãèâåŽ áŽïýãŽáŽïý㎠àŽéëïŽýñèâIJâIJæï ðëèëIJŽ æéŽãâ ïæéîŽãèæïŽåãæï. Žé àŽêïŽäôãîâ-IJâIJæï ïŽöñŽèâIJæå öâàãæúèæŽ, Žàîâåãâ, áŽãûâîëå æé éêæöãêâèëãŽêæ òŽóðâIJæïöâïŽòâîæïæ èëàæçñîæ áŽïŽIJñåâIJâIJæ, îëéèâIJæù ïæéîŽãèââIJï öââýâIJŽ.à Ž ê ï Ž ä ô ã î â IJ Ž . ãåóãŽå, A ᎠB ïæéîŽãèââIJæ æïâåæŽ, îëéåñ x ∈ A, éŽöæê x ∈ B. éŽöæê ŽéëIJëIJâê, îëé A Žîæï B-ï óãâïæéîŽãèâ ᎎôêæöêŽãâê öâéáâàêŽæîŽá: A ⊆ B. öâïŽIJŽéæïæ ãâêæï áæŽàîŽéŽ àŽéëïŽýñèæŽ 4.1êŽýŽääâ. ŽéŽïåŽê, åñ ŽîïâIJëIJï B-ï æïâåæ âèâéâêðæ, îëéâèæù Žî âçñåãêæïA-ï, éŽöæê A-ï ñûëáâIJâê B-ï ïŽçñåîæã óãâïæéîŽãèâï ᎠøŽæûâîâIJŽ A ⊂B ïŽýæå. âï êæöêŽãï, îëé îŽôŽù Žäîæå B ñòîë áæáæŽ ãæáîâ A, éŽàîŽé,îëàëîù óãâéëå ãêŽýŽãå, ŽïâåéŽ ðâîéæêéŽ öâæúèâIJŽ áŽàãŽIJêæëï. éŽöŽïŽáŽéâ,

18Žé ðâîéæêæï àŽéëõâêâIJæï ïæòîåýæèâ ñêᎠàŽéëãæøæêëå. âï åŽêŽòŽîáëIJâIJæöâúèâIJŽ, Žàîâåãâ, øŽæûâîëï öâIJîñêâIJñèæ îæàæåéŽöæê ŽéIJëIJâê, îëé B éëæùŽãï A-ï.B ⊃ A ᎠB ⊇ A.êŽý. 4.1ùýŽáæŽ, îëé êâIJæïéæâîæ A ïæéîŽãèæïåãæï ïŽéŽîåèæŽêæŽ öâéáâàæ ïŽéæåŽêŽòŽîáëIJŽ:∅ ⊆ A, A ⊆ A, A ⊆ E.ŽóâáŽê éâëîâ åŽêŽòŽîáëIJŽ Žîæï àŽêïŽçñåîâIJæå éêæöãêâèëãŽêæ. Žéë-IJëIJâê, îëé A ᎠB ïæéîŽãèââIJæ âçãæãŽèâêðñîæŽ (øŽæûâîâIJŽ A = B),åñA ⊆ B, B ⊆ A.âï êæöêŽãï, îëé A-ï õãâèŽ âèâéâêðæ Žîæï B-ï âèâéâêðâIJæ, ýëèë B-ï õãâèŽâèâéâêðæ { A-ï âèâéâêðæ.ïæéîãèâåŽ âçãæãŽèâêðñîëIJæï àŽêïŽäôãîâIJæï àŽéëõâêâIJæå áŽãŽéðçæùëåäëàæâîåæ ïæéîŽãèæï æáâêðñîëIJŽ, îæïåãæïŽù àŽãŽîøæëå ýñåæ éŽàŽèæåæ.âï éŽàŽèæåâIJæ ûŽîéëŽáàâêï áŽéðçæùâIJâIJæï éŽàŽèæåâIJï. Žéæðëé ãŽèáâIJñèæãŽîå éëçèâá àŽêãæýæèëå ïŽçæåýæ æéæï öâïŽýâIJ, åñ îŽðëé ñêᎠãæäîñêëåáŽéðçæùâIJâIJäâ çëéìæñðâîñè éâùêæâîâIJŽöæ. éçŽùîŽá îëé ãåóãŽå, îëàëîùâï øãâñèâIJîæã çâåáâIJŽ, øãâê îŽôŽù ñêᎠáŽãŽéðçæùëå éýëèëá âîåþâî.öâéáâà àŽéëãŽèå æóæáŽê, îëé æêòëîéŽùææï âï êŽûæèæ ïŽéŽîåèæŽêæŽ áŽ,éŽöŽïŽáŽéâ, öâàãæúèæŽ àŽêãæýæèëå îëàëîù èŽóðæ. ŽéŽïåŽê, îëàëîù àŽéëåãèæåæéâùêæâîâIJæï ñéîŽãèâï öâéåýãâãŽöæ ýáâIJŽ, éâåëáæ, îëéèæåŽù öâáâàâIJïãôâIJñèëIJå, æïâãâ éêæöãêâèëãŽêæŽ, îëàëîù åãæå öâáâàæ. áŽéðçæùâ-IJâIJæï ŽêŽèæäæï öâéáâà êŽåâèæ ýáâIJŽ àŽéëåóéñèæ ãŽîŽñáâIJæ Ꭰæï öâáâàâIJæ,îëéèâIJåŽêŽù Žé ãŽîŽñáâIJï éæãõŽãŽîå. Žàîâåãâ, àŽïŽàâIJæ ýáâIJŽ áŽïççãêâ-IJæï àŽéëðŽêæï ìîëùâïâIJæ, îëéèâIJæù öâæúèâIJŽ àŽéëãæõâêëå ïý㎠ŽéëùŽêâIJæïŽéëýïêæïŽï.öâæúèâIJŽ, Žàîâåãâ, öâãêæöêëå, îëé åâëîâéâIJæï áŽéðçæùâIJŽ ŽõŽèæIJâIJïŽãðëéŽðñîŽá àŽáŽïŽûõãâðæ õãâèŽ ŽéëùŽêæï ïŽòñúãâèï. éŽàîŽé ŽéŽï àŽêãæýæèŽãåéëàãæŽêâIJæå. àŽêãæýæèëå îŽéáâêæéâ éŽàŽèæåæ.4.1 éŽàŽèæåöæ ñöñŽèëá éðçæùáâIJŽ öâéáâàæ ëîæ åŽêŽòŽîáëIJæï ïŽéŽîåèæŽêëIJŽ:Ž) A ∩ (B ∪ C) ⊆ (A ∩ B) ∪ (A ∩ C);

19IJ) (A ∩ B) ∪ (A ∩ C) ⊆ A ∩ (B ∪ C).åæåëâñè Žé áŽéðçæùâIJŽåŽàŽê öâáàâIJŽ öâéáâàæ ïŽýæï éðçæùâIJñèâIJâIJæïàŽê:\åñ P , éŽöæê Q".(åñ P éŽîåâIJñèæŽ, éŽöæê éŽîåâIJñèæŽ Q). éëýâîýâIJñèëIJæïŽåãæï âï éðçæùâ-IJñèâIJŽ øŽãûâîëå, îëàëîù \P =⇒ Q" ᎠûŽãæçæåýŽãå:\P -áŽê àŽéëéáæêŽîâëIJï Q".ŽéîæàŽá, åñ àãŽóãï P 0 , P 1 , . . . , P n æïâåæ éæéáâãîëIJŽ, îëé P 0 =⇒ P 1 ,P 1 =⇒ P 2 , . . . , P n−1 =⇒ P n (P 0 -áŽê àŽéëéáæêŽîâëIJï P 1 , P 1 -áŽê àŽéëéáæêŽîâëIJïP 2 , . . . , P n−1 -áŽê àŽéëéáæêŽîâëIJï P n ), éŽöæê àãŽóãï ìæîáŽìæîæéðçæùâIJñèâIJŽ P 0 =⇒ P n .é Ž à Ž è æ å æ 4.1. A, B ᎠC ïæéîŽãèââIJæïåãæï áŽãŽéðçæùëå, îëéá Ž é ð ç æ ù â IJ Ž .A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).x ∈ A ∩ (B ∪ C) =⇒ x ∈ A Ꭰx ∈ (B ∪ C) =⇒(Žïâåæ áŽþàñòâIJŽ ŽñùæèâIJâèæŽ, îŽáàŽê òîøýæèâIJæ êæöêŽãï, îëé àŽâîåæŽêâ-IJŽ ñêᎠàŽéëãåãŽèëå åŽêŽçãâåæï ûæê)=⇒ x ∈ A Ꭰ(x ∈ B Žê x ∈ C) =⇒=⇒ (x ∈ A Ꭰx ∈ B) Žê (x ∈ A Ꭰx ∈ C) =⇒=⇒ (x ∈ A ∩ B) Žê (x ∈ A ∩ C) =⇒ x ∈ (A ∩ B) ∪ (A ∩ C).ŽéîæàŽá,A ∩ (B ∪ C) ⊆ (A ∩ B) ∪ (A ∩ C).ŽýèŽ áŽãŽéðçæùëå öâIJîñêâIJñèæ øŽîåãŽ:éŽöŽïŽáŽéâ,Žéæðëéx ∈ (A ∩ B) ∪ (A ∩ C) =⇒ (x ∈ A ∩ B) Žê (x ∈ A ∩ C) =⇒=⇒ (x ∈ A Ꭰx ∈ B) Žê (x ∈ A Ꭰx ∈ C) =⇒=⇒ x ∈ A Ꭰ(x ∈ B Žê x ∈ C) =⇒ x ∈ A Ꭰx ∈ B ∪ C =⇒=⇒ x ∈ A ∩ (B ∪ C).(A ∩ B) ∪ (A ∩ C) ⊆ A ∩ (B ∪ C).A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).Žé çâîúë öâéåýãâãŽöæ áŽéðçæùâIJæï éâëîâ êŽûæèæ äñïðŽá âéåýãâ㎠ìæîãâèïáŽ, éŽöŽïŽáŽéâ, öâæúèâIJŽ øŽãûâîëåx ∈ A ∩ (B ∪ C) ⇐⇒ x ∈ A Ꭰx ∈ B ∪ C ᎠŽ. ö.Žó \⇐⇒" ïæéIJëèë ŽôêæöêŽãï, îëé P =⇒ Q ᎠQ =⇒ P . âï öâæúèâ-IJŽ ûŽãæçæåýëå Žïâ \éŽöæê Ꭰéýëèëá éŽöæê" ᎠŽôêæöêŽãï ëîæ P ᎠQéðçæùâIJñèâIJâIJæï âçãæãŽèâêðñîëIJŽï.

20éŽàîŽé õëãâèåãæï Žî Žîæï Žïâ Žáãæèæ Žîàñéâêðæï Ꭰöâáâàæï öâóùâãŽ.äëàŽáŽá áŽéðçæùâIJŽ ñêᎠøŽðŽîáâï ëîæãâ éæéŽîåñèâIJæå ùŽèç-ùŽèçâ. öâãêæöêëå,îëé âçãæãŽèâêðñîëIJŽ öâæúèâIJŽ ŽáãæèŽá éæãæôëå ãâêæï öâïŽIJŽéæïæáæŽàîŽéæáŽê.åñéùŽ, õëãâèåãæï Žî Žîæï öâïŽúèâIJâèæ æïâåæ áæŽàîŽéæïáŽýŽäãŽ, îëéèæï éæéŽîå öâæúèâIJŽ ãæõëå áŽîûéñêâIJñèæ, îëé æàæ æéáââêŽáìŽïñýëIJï éëåýëãêâIJï, îŽéáâêŽáŽù ïŽüæîëŽ. Žéæðëé ŽñùæèâIJâèæŽ ñöñŽèëáŽéðçæùâIJŽ. âï áŽéðçæùâIJŽ áŽéëçæáâIJñèæŽ \áŽ" Ꭰ\Žê"-ï éêæöãêâèëIJâIJïöëîæï öæêŽàŽê ñîåæâîåçŽãöæîäâ. öâéáàëéöæ, îëùŽ àŽêïŽäôãîñèæ æóêâIJŽäàæâîåæ ŽèàâIJîñèæ ïðîñóðñîâIJæ, ãŽøãâêâIJï, îëé âï Žî ûŽîéëŽáàâêï ïæîåñèâï.4.2{4.5 éŽàŽèæåâIJöæ Žïâãâ ãæõâêâIJå ìæîáŽìæî áŽéðçæùâIJâIJï, éŽàîŽé âïáŽéðçæùâIJâIJæ øŽûâîæèæŽ îŽéáâêæéâ ïý㎠ïŽýæå.é Ž à Ž è æ å æ 4.2. éëùâéñèæŽ E ïæéîŽãèæï éæéŽîå êâIJæïéæâîæ A(A ⊆ E) ïæéîŽãèæï áŽéŽðâIJŽ âîåŽáâîåæŽ.á Ž é ð ç æ ù â IJ Ž . ãæàñèæïýéëå, îëé ŽîïâIJëIJï ëîæ B ᎠC ïæéîŽãèâ,îëéâèåŽàŽê åæåëâñèæ éŽåàŽêæ ŽçéŽõëòæèâIJï A ïæéîŽãèæï áŽéŽðâIJæïéëåýëãêŽï, â. æ.éŽöæêŽéæðëéB ∩ A = C ∩ A = ∅ ᎠB ∪ A = C ∪ A = E.B = B ∩ E = B ∩ (C ∪ A) = (B ∩ C) ∪ (B ∩ A) == (B ∩ C) ∪ ∅ = B ∩ C,x ∈ B =⇒ x ∈ B Ꭰx ∈ C =⇒ B ⊆ B ∩ C =⇒ B ⊆ B ᎠB ⊆ C.éŽàîŽé ãæùæå, îëé B ⊆ B. Žéæðëé ŽóâáŽê àŽéëéáæêŽîâëIJï, îëéB ⊆ C.ŽêŽèëàæñîŽá (öâãñùãèæå îŽ îëèâIJï B-ï ᎠC-ï), éæãæôâIJå:C ⊆ B.ïŽæáŽêŽù B = C, â. æ. B = C = A ′ ᎠA ′ âîåŽáâîåæŽ.äâéëå éëõãŽêæèæ éŽàŽèæåæ åŽãæï åŽãöæ öâæùŽãï úæîæåŽá éŽåâéŽðæçñîéæáàëéŽï, îëéâèæù àŽéëæõâêâIJŽ âîåŽáâîåëIJæï áŽïŽéðçæùâIJèŽá. áŽïŽûõæïöææàñèæïýéâIJŽ, îëé ŽîïâIJëIJï ëîæ Žïâåæ ëIJæâóðæ, ýëèë öâéáâà éðçæùáâIJŽ,îëé æïæêæ âîåéŽêâåï âéåýãâãŽ.öâéáâà éŽàŽèæåöæ çãèŽã éæãéŽîåŽãå ãŽîŽñáï \Žê"-æï öâïŽýâIJ, îŽåŽ öâïŽúèâIJèëIJŽàãóëêáâï A∪B∪C àŽéëïŽýñèâIJŽ áŽãûâîëå îëàëîù A∪(B∪C)Žê (A ∪ B) ∪ C, îëùŽ âï ñòîë éëïŽýâîýâIJâèæ æóêâIJŽ ïŽüæîë àŽîáŽóéêâIJæïåãæï.é Ž à Ž è æ å æ 4.3. éëùâéñèæŽ A, B ᎠC ïæéîŽãèââIJæ æïâåæ, îëéA ∪ B ∪ C = E

21ᎠA, B ᎠC ûõãæè-ûõãæèŽá Žî ðŽêŽæçãâåâIJŽ, éŽöæêá Ž é ð ç æ ù â IJ Ž .A ′ = B ∪ C, B ′ = A ∪ C, C ′ = A ∪ C.A ∪ B ∪ C = A ∪ (B ∪ C) = E,A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) = ∅ ∪ ∅ = ∅.éŽöŽïŽáŽéâ, B ∪ C ŽçéŽõëòæèâIJï A ′ -æï ìæîëIJŽï, A ′ âîåŽáâîåæŽ. ŽéæðëéA ′ = B ∪ C. ŽêŽèëàæñîŽá ðŽîáâIJŽ áŽéðçæùâIJŽ B ′ ᎠC ′ -åãæï.é Ž à Ž è æ å æ 4.4. êâIJæïéæâîæ X ᎠY ïæéîŽãèââIJæïåãæï ïŽéŽîåèæŽêæŽåŽêŽòŽîáëIJŽ(X ∩ Y ) ′ = (X ∩ Y ′ ) ∪ (X ′ ∩ Y ) ∪ (X ′ ∩ Y ′ ).á Ž é ð ç æ ù â IJ Ž . ãæàñèæïýéëå, îëé 4.3 éŽàŽèæåæáŽê àãŽóãï A, BᎠC ïæéîŽãèââIJæ áŽC = D ∪ E ᎠD ∩ E = ∅.(4.2 êŽýŽääâ àŽéëïŽýñèæŽ ãâêæï áæŽàîŽéŽ, ïŽáŽù E ïæéîŽãèâ áŽõëòæèæŽöâïŽIJŽéæïŽá). éŽöæê A, B, D ᎠE ïæéîŽãèââIJæ ñîåæâîåŽîŽåŽêŽéçãâåæŽ áŽñòîë éâðæù,ŽéæðëéA ∪ B ∪ D ∪ E = E.A ′ = B ∪ C,A ′ = B ∪ D ∪ E.êŽý. 4.2ŽýèŽ ŽáãæèŽá öâæúèâIJŽ ãŽøãâêëå, îëé åñ ãæàñèæïýéâIJåA = X ∩ Y, B = X ∩ Y ′ , D = X ′ ∩ Y ᎠE = X ′ ∩ Y ′ ,éŽöæê éëåýëãêæèæ ìæîëIJâIJæ öâïîñèáâIJŽ ᎠŽéæðëé ïŽüæîë öâáâàæ éŽöæêãâàŽéëéáæêŽîâëIJï ûæêŽ éïþâèëIJâIJæáŽê.é Ž à Ž è æ å æ 4.5. êâIJæïéæâîæ X ᎠY ïæéîŽãèââIJæïåãæï éŽîåâIJñèæŽåŽêŽòŽîáëIJŽ(X ∩ Y ) ′ = X ′ ∪ Y ′ .

22á Ž é ð ç æ ù â IJ Ž .x ∈ (X ∩ Y ) ′ ⇐⇒ x ∈ (X ∩ Y ′ ) ∪ (X ′ ∩ Y ) ∪ (X ′ ∩ Y ′ ) ⇐⇒⇐⇒ x ∈ (X ∩ Y ′ ) ∪ (X ′ ∩ Y ′ ) ∪ (X ′ ∩ Y ) ∪ (X ′ ∩ Y ′ ) ⇐⇒(âîåæ ûâãîæ àŽéâëîâIJñèæŽ. âï öâïŽúèâIJâèæŽ, îŽáàŽê A = A ∪ A êâIJæïéæâîæA-åãæï)⇐⇒ x ∈ ( (X ∩ Y ′ ) ∪ (X ′ ∩ Y ′ ) ) ∪ ( (X ′ ∩ Y ) ∪ (X ′ ∩ Y ′ ) ) ⇐⇒⇐⇒ x ∈ ( (X ∪ X ′ ) ∩ Y ′) ∪ ( X ′ ∩ (Y ∪ Y ′ ) ) ⇐⇒⇐⇒ x ∈ (E ∩ Y ′ ) ∪ (X ′ ∩ E) ⇐⇒ x ∈ Y ′ ∪ X ′ ⇐⇒ x ∈ X ′ ∪ Y ′ ,ŽéîæàŽá, (X ∩ Y ) ′ = X ′ ∪ Y ′ .4.5 éŽàŽèæåöæ éæôâIJñè Ꭰéæï éïàŽãï öâáâàï (æý. 4.5 ïŽãŽîþæöë) áâéëîàŽêæï çŽêëêâIJæ âûëáâIJŽ. âï çŽêëêâIJæ éêæöãêâèëãŽê îëèï ŽïîñèâIJïéŽðâéŽðæçñî èëàæçŽöæ.4.1{4.5 éŽàŽèæåâIJæï åŽêéæéáâãîëIJŽ àãæøãâêâIJï, îëàëî öâæúèâIJŽ àŽêãŽãæåŽîëåéŽåâéŽðæçñîæ åâëîæŽ éŽîðæãæ åâëîâéâIJæï áŽéðçæùâIJâIJæï àŽéëõâêâIJæåᎠàŽéëãæõãŽêëå æïâåæ éêæöãêâèëãŽêæ öâáâàâIJæ, îëàëîæù áâ éëîàŽêæïçŽêëêâIJæŽ. ãæáîâ Žé åŽãæï áŽïçãêæå êŽûæèäâ àŽáŽãŽèå, öâãâùŽáëåàŽáŽãûâîëå 4.2{4.5 éŽàŽèæåâIJöæ àŽêýæèñèæ ŽéëùŽêâIJæï áŽéðçæùâIJâIJæ òëîéŽèñîæïŽýæå, îëàëîù âï 4.1 éŽàŽèæåöæ àŽãŽçâåâå. éëàãæŽêâIJæå öâéëãæðŽêåïŽüæîë ðâîéæêëèëàæŽï, îëéâèæù ïŽöñŽèâIJŽï éëàãùâéï àŽéëãæõâêëåñòîë éëçèâ ŽôêæöãêâIJæ. áŽïŽûõæïöæ öâéëãæðŽêå ëî àŽêïŽäôãîâIJŽï.à Ž ê ï Ž ä ô ã î â IJ Ž . ŽéIJëIJâê, îëé A ᎠB ïæéîŽãèââIJæ ŽîŽâçãæãŽèâêðñîæŽ,åñ æïæêæ Žî Žîæï âçãæãŽèâêðñîæ. âï åãæïâIJŽ æéæï ðëèòŽïæŽ,îëé âîå-âîåæ ïæéîŽãèâ A \ B Žê B \ A ŽîŽùŽîæâèæŽ.à Ž ê ï Ž ä ô ã î â IJ Ž . éëùâéñèæ X ïæéîŽãèæï õãâèŽ óãâïæéîŽãèâåŽïæéîŽãèâï X ïæéîŽãèæï ýŽîæïýæ âûëáâIJŽ ᎠŽôæêæöêâIJŽ P(X)-æå. (äëàæâîåæ Žãðëîæ æõâêâIJï 2 x ŽôêæöãêŽï. Žéæï éæäâäæ àŽïŽàâIJæ àŽýáâIJŽéëàãæŽêâIJæå, îëùŽ àŽãŽîøâãå îŽéáâêæéâ éŽàŽèæåï.) òëîéŽèñîŽáP(X) = {Y : Y ⊆ X}.çâîúëá, öâãêæöêëå, îëé îŽáàŽê ∅ ⊆ X ᎠX ⊆ X, Žéæðëé∅ ∈ P(X), X ∈ P(X).é Ž à Ž è æ å æ 4.6. ãåóãŽå, A = {1, 2, 3}, éŽöæêP(A) = { ∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3} } .Žé ìŽîŽàîŽòï áŽãŽïîñèâIJå áŽéðçæùâIJæï ëîæ æîæIJæ éâåëáæï àŽêýæèãæå.ìæîãâèæ éâåëáæ Žîæï ïŽûæꎎôéáâàëï áŽöãâIJæï éâåëáæ. àŽãæýïâêëåîŽïâèæï ìŽîŽáëóïæ (1.1. éŽàŽèæåæ):F ∈ F =⇒ F ∉ F ᎠF ∉ F =⇒ F ∈ F.

23åñ ŽôãêæöêŽãå F ∈ F éðçæùâIJñèâIJâIJï P -æå, éŽöæê éæãæôâIJå:P éŽîåâIJñèæŽ =⇒ P éùáŽîæŽ áŽ P éùáŽîæŽ =⇒ P éŽîåâIJñèæŽ.éŽåâéŽðæçæï ïŽòñúãâèï ûŽîéëŽáàâêï ãŽîŽñáæ æéæï öâïŽýâIJ, îëé Žî öâæúèâIJŽŽîïâIJëIJáâï éðçæùâIJñèâIJŽ, îëéâèæù Žîæï üâöéŽîæðæ Ꭰîëéâèæù ŽîæïéùáŽîæ (â. æ. èëàæçñîæ ïæïðâéŽ ñêᎠæõëï öæꎎîïæŽêæ. øãâê ñŽîãõëòå îŽïâèæïïæéîŽãèâï æéæðëé, îëé éæïæ àŽêïŽäôãîâIJŽ ñïŽòñúãèëŽ àŽêïŽýæèãâèæŽäîæå.)ãåóãŽå, àãŽóãï àŽéëêŽåóãŽéåŽ P 1 , P 2 , . . . , P n âîåëIJèæëIJŽ ᎠàãæêáŽáŽãŽéðçæùëå (P 1 üâöéŽîæðæŽ áŽ P 2 üâöéŽîæðæŽ áŽ . . . ᎠP n üâöéŽîæðæŽ)=⇒ Q üâöéŽîæðæŽ, Žê, ñòîë éŽîðæãŽá,(P 1 ᎠP 2 Ꭰ. . . ᎠP n ) =⇒ Q.åñ áŽãñöãâIJå àŽéëêŽåóãŽéï (P 1 ᎠP 2 Ꭰ. . . ᎠP n ᎠŽîŽ Q), â. æ.P 1 , P 2 , . . . , P n üâöéŽîæðæŽ, ýëèë { Q éùáŽîæ ᎠŽóâáŽê àŽéëãæðŽêëå îŽæéâP éðçæùâIJñèâIJŽ, îëéâèæù âîåáîëñèŽá üâöéŽîæðæù Žîæï ᎠéùáŽîæ, éŽöæêèëàæçñîæ ïæïðâéŽ, áŽòñúêâIJñèæ (P 1 ᎠP 2 Ꭰ. . . ᎠP n ᎠŽîŽ Q) àŽéëêŽåóãŽéäâ,áŽñöãâIJâèæŽ. éŽöæê öâàãæúèæŽ áŽãŽïçãêŽå, îëé(P 1 ᎠP 2 Ꭰ. . . ᎠP n ) =⇒ Q.îŽáàŽê (ŽîŽ Q) áŽöãâIJŽï éæãõŽãŽîå ûæꎎôéáâàëIJŽéáâ. äâéëå êŽåóãŽéæïïŽæèñïðîŽùæëá àŽêãæýæèëå éŽàŽèæåæ.é Ž à Ž è æ å æ 4.7. áŽãŽéðçæùëå, îëé êâIJæïéæâîæ A ᎠB ïæéîŽãèââIJæïåãæïŽáàæèæ Žóãï åŽêŽòŽîáëIJŽïA ⊆ B ⇐⇒ B ′ ⊆ A ′ .á Ž é ð ç æ ù â IJ Ž . áŽãñöãŽå, îëé E, ∅ ᎠŽ. ö. ïæéîŽãèââIJï åãæïâIJâIJæ(â. æ. àŽêïŽäôãîâIJâIJæ) öâïîñèâIJñèæŽ áŽ, îëé A ⊂ B ᎠB ′ ⊈ A ′ (äâéëåŽôûâîæèæ ïæðñŽùææï ðâîéæêâIJöæ Q Žîæï B ′ ⊆ A ′ ). éŽöæêA ⊆ B =⇒ åñ x ∈ A, éŽöæê x ∈ B;B ′ ⊆ A =⇒ ŽîïâIJëIJï îŽôŽù æïâåæ y âèâéâêðæ, îëéy ∈ B ′ Ꭰy ∉ A ′ .(∗)-áŽê àŽéëéáæêŽîâëIJï åŽêŽòŽîáëIJŽy ∈ A =⇒ y ∈ B =⇒ y ∈ B ′áŽy ∈ B =⇒ y ∈ B ′ ∩ B = ∅(ûæꎎôéáâàëIJŽ),éŽöŽïŽáŽéâ, éæôâIJñèæ B ′ ⊈ A ′ éùáŽîæŽ, Žéæðëé B ′ ⊆ A ′ . ŽêŽèëàæñîŽáöâæúèâIJŽ ãŽøãâêëå, îëééŽöŽïŽáŽéâ, àãŽóãï:B ′ ⊆ A ′ =⇒ A ⊂ B.A ⊆ B ⇐⇒ B ′ ⊆ A ′ .âï éŽàŽèæåæ Žïâãâ ãîùâèáâIJŽ æîæIJæ áŽéðçæùâIJæï éâëîâ éâåëáäâù.(∗)

24é Ž à Ž è æ å æ 4.8. ãåóãŽå, P Žîæï àŽéëêŽåóãŽéæ \áôâï ýñåöŽIJŽåæŽ",ýëèë Q { \áôâï çãæîæï áôâŽ". éŽöæê (P =⇒ Q) ŽôêæöêŽãï \åñ áôâïýñåöŽIJŽåæŽ, éŽöæê âï çãæîæï áôâŽ", ýëèë (ŽîŽ Q =⇒ ŽîŽ P ) æóêâIJŽ \åñáôâï Žî Žîæï çãæîæï áôâ, éŽöæê âï áôâ Žî Žîæï ýñåöŽIJŽåæ".âï ëîæ àŽéëêŽåóãŽéæ âçãæãŽèâêðñîæŽ (â. æ. æïæêæ âîåáîëñèŽá Žê üâöéŽîæð掎ê éùáŽîæŽ).Žé åãŽèïŽäîæïæå ãâêæï áæŽàîŽéâIJæ öâæúèâIJŽ àŽéëãæõâêëå ïæðñŽùææïàŽïŽîçãâãŽá, éŽàîŽé õãâèŽ öâáâàæ àŽéëðŽêæèæ ñêᎠæõëï ŽéëùŽêŽöæ éëùâéñèæãŽîŽñáâIJæáŽê. ñêᎠàãŽýïëãáâï, îëé åñ îŽæéâ éðçæùâIJñèâIJŽ üâöéŽîæðæŽ,ñêᎠöâàãâúèëï éæïæ áŽéðçæùâIJŽ. åñ éðçæùâIJñèâIJŽ éŽîåèŽù ùýŽáæŽ,éŽöæê ŽáãæèŽá áŽãŽéðçæùâIJå éŽï, åñ ŽîŽ, éŽöæê, îëàëîù øŽêï, âï ŽîùæïâùýŽáæŽ, îëàëîù àãâàëêŽ áŽ ïîñèæŽá öâïŽúèâIJâèæŽ, éùáŽîæù æõëï.ï Ž ã Ž î þ æ ö ë 4.1.1. áŽãŽéðçæùëå, îëéA ∩ (B ∩ C) = (A ∩ B) ∩ C.2. ãåóãŽå, éëùâéñèæŽ A, B, C: C ⊆ B ïæéîŽãèââIJæ. áŽãŽéðçæùëå, îëéŽ) A∩C ⊆ A∩B; IJ) A∪C ⊆ A∪B; à) A\B ⊆ A\C; á) C \A ⊆ B \A;â) B ′ \ A ⊆ C ′ \ A.3. áŽãŽéðçæùëå, îëé åñ A ⊆ B, éŽöæê P(A) ⊆ P(B).4. Žøãâêâå öâéáâàæ ðëèëIJæï éŽîåâIJñèëIJŽ:5. áŽãŽéðçæùëå, îëéA ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).(A ∪ B) ′ = A ′ ∩ B ′ .(éæåæåâIJŽ: Žøãâêâå, îëé (A∪B)∪(A ′ ∩B ′ ) = E Ꭰ(A∪B)∩(A ′ ∩B ′ ) = ∅.)6. ᎎéðçæùâå öâéáâàæ éðçæùâIJñèâIJâIJæï âçãæãŽèâêðñîëIJŽ, â. æ. îëé åæåëâñèæáŽêàŽéëéáæêŽîâëIJï éâëîâŽ) A ∪ B = E; IJ) A ′ ⊆ B; à) A ′ ∩ B ′ = ∅.7. öâéáâàæ éðçæùâIJñèâIJâIJæáŽê îëéâèæŽ éŽîåâIJñèæ:Ž) 0 ∈ ∅; IJ) {∅} ⊆ ∅; à) ∅ ⊆ {∅}; á) ∅ ⊆ E; â) {∅} ⊆ {{∅}}?Žé çæåýãæï ìŽïñýâIJæ öâŽáŽîâå 2.1(6) ïŽãŽîþæöëï ìŽïñýâIJï. ∈ Ꭰ⊆ ïæéIJëèëâIJïöëîæï ŽîïâIJëIJï çŽãöæîæ, éŽàîŽé æïæêæ âîåæáŽæàæãâ Žî Žîæï.8. ãŽøãâêëå, îëé ïŽïîñèæ A ïæéîŽãèæïåãæï |2 A | = 2 |A| . (éæåæåâIJŽ:Žéëûâîâå A = {a 1 , a 2 , . . . , a n } ïæéîŽãèâ ᎠàŽêæýæèâå éæïæ óãâïæéîŽãèââIJæ.)§ 5. ïæéîŽãèâåŽ êŽéîŽãèæŽóŽéáâ øãâê úæîæåŽáŽá ŽîïâIJñèæ ïæéîŽãèââIJæáŽê ñòîë éùæîâ äëéæï ïæéîŽãèââIJæïŽàâIJŽï ãæýæèŽãáæå, ŽýèŽ àŽêãæýæèŽãå áæáæ ïæéîŽãèââIJæï ŽàâIJæï

25âîå-âîå õãâèŽäâ ñòîë äëàŽá ýâîýï. ïŽæèñïðîŽùæëá àŽêãæýæèëå üŽáîŽçæïáŽòæï ñþîâáâIJäâ àŽêåŽãïâIJñèæ ïæéîŽãèâ (ïñî. 1.10).êŽý. 5.1àŽêãæýæèëå ïãâðâIJæï ïæéîŽãèâ, îëéèâIJïŽù ŽôãêæöêŽãå a, b, . . . , h-æå (éŽîùýêæáŽêéŽîþãêæã), ïðîæóëêâIJæï ïæéîŽãèâ, îëéèâIJïŽù ŽôãêæöêŽãå 1, 2, . . . , 8-æå (óãâéëáŽê äâéëå). éŽöŽïŽáŽéâ, åæåëâñèæ ñþîŽ ùŽèïŽýŽá öâæúèâIJŽ æõëïéëùâéñèæ ëîæ ïæéIJëèëåæ: âîåæ F = {a, b, . . . , h} ïæéîŽãèæáŽê, éâëîâR = {1, 2, . . . , 8} ïæéîŽãèæáŽê. éŽàŽèæåŽá, a1, b3, h5, e3 ᎠŽ. ö. ŽéàãŽîŽá,ïãâðâIJæï F ïæéîŽãèæïŽ áŽ ïðîæóëêâIJæï R ïæéîŽãèæïŽàŽê öâãóéâêæå áŽòæïõãâèŽ ñþîæï ïæéîŽãèâ.âï éŽàŽèæåæ åŽãæï åŽãöæ öâæùŽãï ŽýŽè æáââIJï, îëéèâIJæù ïæéîŽãèâåŽêŽéîŽãèæï ŽàâIJæïŽï àŽéëæõâêâIJŽ. ŽéŽïåŽê, æéæïŽåãæï, îëé öâàãâúèëï àŽêïŽýæèãâèæïæðñŽùææï àŽêäëàŽáâIJŽ, ñêᎠãæõëå ùëðŽ ñòîë éâðŽá äñïðâIJæ.à Ž ê ï Ž ä ô ã î â IJ Ž . Žôãêæöêëå n-âèâéâêðæŽêæ x 1 , x 2 , . . . , x n éæéáâãîëIJŽ(x 1 , x 2 , . . . , x n )-æå. Žó éîàãŽèæ òîøýæèâIJæ àŽéëõâêâIJñèæŽ æéæïŽåãæï,îëé éæãñåæåëå æé îæàäâ, îëéèæåŽù øŽûâîæèæŽ âèâéâêðâIJæ. éŽàŽèæåŽá, åñx − 1 ≠ x 2 , éŽöæê (x 2 , x 1 , . . . , x n ) Žî âéåýãâ㎠(x 1 , x 2 , . . . , x n ) éæéáâãîëIJŽï.ãñûëáëå n ïæàîúæï êŽçîâIJæ. 2 ïæàîúæï êŽçîâIJï ãñûëáëå ûõãæèæ.ãåóãŽå, éëùâéñèæŽ A 1 , A 2 , . . . , A n n ïæéîŽãèâ. õãâèŽ (x 1 , x 2 , . . . , x n )êŽçîâIJæï æïâå ïæéîŽãèâï, ïŽáŽù x 1 ∈ A 1 , x 2 ∈ A 2 , . . . , x n ∈ A n -ï A 1 , A 2 , . . . , A nïæéîŽãèâåŽ ìæîáŽìæîæ êŽéîŽãèæ âûëáâIJŽ. ᎠŽôæêæöêâIJŽ A 1 ×A 2 ×· · ·×A n ïæéIJëèëåæ. ïý㎠ŽôêæöãêâIJæï àŽéëõâêâIJæï Žé êŽéîŽãèï øŽãûâîå ñòîëéëçèâá.é Ž à Ž è æ å æ 5.1. ãåóãŽå, X = {0, 1}, Y = {x, y}, z = {0, 1, 2},éŽöæêX × Y = { (0, x), (0, y), (1, x), (1, y) } ,Y × X = { (x, 0), (x, 1), (y, 0), (y, 1) } .

26ŽéàãŽîŽá, X × Y ≠ Y × X.éŽàŽèæåŽá,X × Z = { (0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2) }(0, 1) Ꭰ(1, 0) X × Z ïæéîŽãèæï ïýãŽáŽïý㎠âèâéâêðâIJæŽ. üŽáîŽçæï áŽòæïéŽàŽèæåäâ F ᎠR ïæéîŽãèââIJæ Žî æçãâåâIJæŽê. (e, 3) ∈ F ×R, ýëèë (3, e) ∉F × R (∈ R × F ), Žéæðëé 3e øŽêŽûâîæ ñêᎠñŽîãõëå, îëàëîù ŽîŽïûëîæüŽáîŽçæï áŽòæïåãæï.ýöæîŽá àŽéëãæõâêâIJå ìæîáŽìæî êŽéîŽãèï âîåêŽæîæ ïæéîŽãèââIJæïåãæï.Žé öâéåýãâãŽöæ ñòîë éëïŽýýâîýâIJâèæ æóêâIJŽ A × A × · · · × A øŽãûâîëå A nïŽýæå.ï Ž ã Ž î þ æ ö ë 5.1.1. ãåóãŽå, X = {a, b, c} ᎠY = {a, b, e, f}. æìëãâå X × Y ᎠY 2 .2. ᎎéðçæùâå, îëé A ⊆ X ᎠB ⊆ Y , A × B ⊆ X × Y .3. ᎎéðçæùâå, îëé A × (B ∩ C) = (A × B) ∩ (A × C).4. ᎎéðçæùâå, îëé êâIJæïéæâîæ ŽîŽùŽîæâèæ A ᎠB ïæéîŽãèââIJæïåãæïïîñèáâIJŽŽ) ∅ × A = ∅; IJ) E × A ≠ A; à) A ⊆ A × A; á) |A × {x}| = |A|; â)A × B = B × A, éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ A = B.

éæéŽîåâIJâIJæàŽéëåãèâIJöæ ýöæîŽá ïŽüæîëŽ Žãæîøæëå ïæéîŽãèâåŽ æïâåæ âèâéâêðâ-IJæ, îëéèâIJæù ŽçéŽõëòæèâIJï îŽæéâ \éæéŽîåâIJŽï". âï ùêâIJŽ ïŽçéŽëá äëàŽáæŽ.Žéæðëé æàæ òŽîåëá àŽéëïŽõâêâIJâèæŽ. IJñêâIJîæãæŽ, éæéŽîåâIJæï öâïŽIJŽéæïæ öâîøâãæïŽïéæïæ ŽîàñéâêðâIJæ âîåæéâëîâïåŽê öâæúèâIJŽ áŽçŽãöæîâIJñèæ æõëïïŽçéŽëá éŽîðæãŽá. ŽñùæèâIJâèæ ŽîŽ îæï æïæêæ áŽçŽãöæîâIJñèæ æõëï îŽôŽùéŽîðæãæ Žê ùýŽáæ òëîéñèæå, åñéùŽ, æïâå ïæðñŽùæâIJöæ, îëùŽ éëåýëãêæèæŽîŽæéâ àŽéëåãèâIJæï øŽðŽîâIJŽ. äëàþâî öâæúèâIJŽ ãæìëãëå éæéŽîåâIJæïéëýâîýâIJñèæ ŽôûâîŽ.ãæáîâ Žé ïŽçæåýï éŽåâéŽðæçñîæ ìëäæùææáŽê éæãñáàâIJæå, àŽêãæýæèëåîŽéáâêæéâæáâŽ, îëéèâIJæù ûŽîéëæóéêâIJŽ öâéáâàæ éŽîðæãæ ïæðñŽùææï àŽêýæèãæáŽê(îëéèâIJïŽù, Žàîâåãâ, éæãõŽãŽîå éæéŽîåâIJæï ùêâIJæï öâéëðŽêŽéáâ).ãåóãŽå, îŽôŽù àãŽóãï ìîëàîŽéâIJæï P ïæéîŽãèâ. éëêŽùâéåŽ éêæöãêâèëIJâIJæïD ïŽïîñèæ ïæéîŽãèâ ᎠöâáâàâIJæï R ïæéîŽãèâ. åñ D-áŽê Žãæîøâãå çëêçîâðñèéêæöãêâèëIJŽï, æàæ öâæúèâIJŽ àŽéëãæõâêëå P -áŽê ŽîâIJñè äëàæâîåìîëàîŽéŽöæ ᎠP -áŽê ŽôâIJñèæ åæåëâñèæ ìîëàîŽéæïåãæï ŽîâIJëIJï éêæöãêâèëIJŽåŽâîåëIJèæIJŽ D ïæéîŽãèæáŽê, îëéèâIJæù Žé ìîëàîŽéŽöæ àŽéëæõâêâ-IJŽ. ŽéàãŽîŽá, àãŽóãï öâïŽIJŽéæïëIJŽ éëêŽùâéåŽ éêæöãêâèëIJâIJïŽ áŽ ìîëàîŽéâIJïöëîæï. éŽöŽïŽáŽéâ, D × P ïæéîŽãèâöæ ŽîïâIJëIJï âèâéâêðâIJæ, îëéèâIJæù ïŽæêðâîâïëŽàŽîçãâñèæ åãŽèïŽäîæïæå. ŽêŽèëàæñîŽá, åñ àŽêãæýæèŽãå îŽæéâp ∈ P ìîëàîŽéŽï, éŽöæê æï D-áŽê ŽîâIJñè éëêŽùâéåŽ öâïŽIJŽéæï éêæöãêâèë-IJâIJï ᎎçŽãöæîâIJï R ïæéîŽãèæáŽê ŽôâIJñè öâáâàâIJåŽê. öâæúèâIJŽ àŽêãæýæèëåéëêŽùâéâIJæ, îëéèâIJïŽù P éæßõŽãå àŽøâîâIJŽéáâ Žê àŽêãæýæèëå öâáâàâIJæ, îëéèâIJæùŽî öâæúèâIJŽ éæãæôëå P -áŽê. éŽöŽïŽáŽéâ, øãâê éæãáæãŽîå (D × R)ïæéîŽãèæï óãâïæéîŽãèââIJåŽê D-áŽê R-çâê àŽáŽéñöŽãâIJæï áîëï ûŽîéëæóéêâ-IJŽ îŽôŽù ŽïëùæŽùæâIJæ, îëéâèIJæù öâæúèâIJŽ ïŽïŽîàâIJèë Žôéëøêáâï ðâîéæêëèëàææïáŽéŽýïëãîâIJæïåãæï.àŽáŽãæáâå òëîéŽèñî àŽêýæèãâIJäâ.§ 1. úæîæåŽáæ ùêâIJâIJæà Ž ê ï Ž ä ô ã î â IJ Ž . A 1 , A 2 , . . . , A n ïæéîŽãèââIJäâ àŽêïŽäôãîñèæ n-ŽáàæèæŽêæ R éæéŽîåâIJŽ âûëáâIJŽ A 1 ×A 2 ×· · ·×A n ìæîáŽìæîæ êŽéîŽãèæïóãâïæéîŽãèâï.ŽéîæàŽá, x 1 , x 2 , . . . , x n âèâéâêðâIJæ áŽçŽãöæîâIJñèæ R éæéŽîåâIJæå éŽöæêᎠéýëèëá éŽöæê, îëùŽ (x 1 , x 2 , . . . , x n ), ïŽáŽù (x 1 , x 2 , . . . , x n ) áŽèŽàâIJñèæíàáîð-æŽ.27

28æòîë ýöæîŽá àãýãáâIJŽ éæéŽîåâIJŽ, îëùŽ n = 2. Žé öâéåýãâãŽöæ éŽï âûëáâIJŽIJæêŽîñèæ éæéŽîåâIJŽ. ŽéîæàŽá, A ᎠB ïæéîŽãèââIJï öëîæï IJæêŽîñèæéæéŽîåâIJŽ Žîæï A×B ìæîáŽìæîæ êŽéîŽãèæï óãâïæéîŽãèâ. åñ âï ïæéîŽãèââIJæâçãæãŽèâêðñîæŽ (ãåóãŽå, A-ï ðëèæŽ), éŽöæê ãæðõãæå, îëé A 2 -ï óãâïæéîŽãèâàŽêïŽäôãîŽãï éæéŽîåâIJŽï A ïæéîŽãèâäâ.éæéŽîåâIJâIJæ Žî ûŽîéëŽáàâêï îŽæéâ ŽýŽèï. öâæúèâIJŽ ŽãŽàëå éæéŽîåâIJâIJæ,îëéèâIJæù ñâüãâèŽá êŽùêëIJæ æóêâIJŽ éçæåýãâèæïåãæï. àŽêãæýæèëå öâéáâàæéŽàŽèæåâIJæ:é Ž à Ž è æ å æ 1.1. ãåóãŽå, A = {1, 2, 3, 4, 5, 6, 7, 8} ᎠR ={(x, y) : x, y ∈ A; x Žîæï y-æï þâîŽáæ}. éŽöæê R éæéŽîåâIJŽ öâæúèâIJŽ øŽãûâîëåùýŽáŽá:{R = (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (7, 1), (8, 1), (2, 2), (4, 2),}(6, 2), (8, 2), (3, 3), (6, 3), (4, 4), (8, 4), (5, 5), (6, 6), (7, 7), (8, 8) .é Ž à Ž è æ å æ 1.2. ãåóãŽå, A = {a, b, c, d}. àŽêãæýæèëå éæïæ õãâèŽóãâïæéîŽãèâåŽ ïæéîŽãèâ P(A). éŽïäâ àŽêãïŽäôãîëå IJæêŽîñèæ éæéŽîåâIJŽR = {(X, Y ) : X, Y ∈ P(A), X ⊂ Y }.é Ž à Ž è æ å æ 1.3. àŽêãæýæèëå üŽáîŽçæï áŽòŽ. ãåóãŽå, F ={a, b, c, d, e, f, g, h}, R = {1, 2, 3, 4, 5, 6, 7, 8} áŽ, ãåóãŽå,S = F × R.ŽéîæàŽá, S Žîæï üŽáîŽçæï áŽòæï ñþîâáâIJæï ïæéîŽãèâ. ñþîâáâIJæ ŽôæêæöêâIJŽ(x, y) ûõãæèâIJæå, ïŽáŽù x ∈ F , y ∈ R. S ïæéîŽãèâäâ àŽêãïŽäôãîëåC IJæêŽîñèæ éæéŽîåâIJŽ: (s, t) ∈ C éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ s Ꭰt ŽîæïS ïæéîŽãèæï âèâéâêðâIJæ Ꭰâðèï öâñúèæŽ âîåæ ïãèæå àŽæŽîëï s-áŽê t-éáâùŽîæâè áŽòŽäâ. (àŽãæýïâêëå, îëé âðèï öâñúèæŽ éëúîŽëIJŽ ãâîðæçŽèñîæᎠßëîæäëêðŽèñîæ éæéŽîåñèâIJæå.) Žéæðëé æùãèâIJŽ Žê ßëîæäëêðŽèñîæŽê ãâîðæçŽèñîæ çëëîáæêŽðæ ᎠŽîŽ ëîæãâ âîåáîëñèŽá, â. æ. C ∈ S × SáŽC ={ ((fs, r s , (t f , r t ) ) : ( f s = f t Ꭰr s ≠ r t)Žê(fs ≠ f t Ꭰr s = r t) } .à Ž ê ï Ž ä ô ã î â IJ Ž . êâIJæïéæâîæ A ïæéîŽãèæïåãæï àŽêãïŽäôãîëåJ A æàæãñîæ éæéŽîåâIJŽ ᎠU A ñêæãâîïŽèñîæ éæéŽîåâIJŽ öâéáâàêŽæîŽá:J A = { (a, a) : a ∈ A } , U A = { (a, b) : a ∈ A, b ∈ A } .ŽéîæàŽá, U A = A 2 . îŽáàŽê ∅ ⊆ A 2 , Žéæðëé ∅ ùŽîæâèæ ïæéîŽãèâù ûŽîéë-Žáàâêï IJæêŽîñèæ éæéŽîåâIJŽï ᎠéŽï âûëáâIJŽ ùŽîæâèæ éæéŽîåâIJŽ.à Ž ê ï Ž ä ô ã î â IJ Ž . A ᎠB ïæéîŽãèââIJï öëîæï õëãâè R IJæêŽîñèéæéŽîåâIJŽï áŽãñçŽãöæîëå ëîæ ïæéîŽãèâ { D(R) àŽêïŽäôãîæï Žîâ ᎠR(R)

29éêæöãêâèëIJŽåŽ Žîâ. æïæêæ àŽêæïŽäôãîâIJŽ öâéáâàêŽæîŽáD(R) = {x : (x, y) ∈ R}, R(R) = {y : (x, y) ∈ R}.é Ž à Ž è æ å æ 1.3. äâéëå àŽêýæèñè 1.1 éŽàŽèæåöæD(R) = A, R(R) = A.é Ž à Ž è æ å æ 1.4. áŽãñöãŽå, îëé àãŽóãï îŽôŽù ìîëàîŽéŽ, îëéâèæùA = {1, 2, 3, 4, 5} ïæéîŽãèæáŽê çæåýñèëIJï ëî x Ꭰy îæùýãï Ꭰåñ x < y,IJâüáŽãï æïâå z îæùýãï (Žïâãâ A ïæéîŽãèæáŽê), ïŽáŽù x ≤ z < y. ìîëàîŽéŽøâîáâIJŽ A-áŽê õãâèŽ îæùýãæï ûŽçæåýãæï öâéáâà. âï ŽéëùŽêŽ àŽêïŽäôãîŽãïöâéáâà P ⊆ A 2 × A éæéŽîåâIJâIJï:{P = ((1, 2, 1), ((1, 3), 1), ((1, 3), 2), ((1, 4), 1), ((1, 4), 2),éŽöæê((1, 4), 3), ((1, 5), 1), ((1, 5), 2), ((1, 5), 3), ((1, 5), 4),((2, 3), 2), ((2, 4), 2), ((2, 4), 3), ((3, 4), 3), ((3, 5), 3), ((3, 5), 4),}((4, 5), 4), ((2, 5), 2), ((2, 5), 3), ((2, 5), 4) ,{D(P ) = (1, 2), (1, 3), (1, 4), (1, 5), (2, 4), (2, 3), (2, 5),}(3, 4), (3, 5), (4, 5) ⊂ A 2 ,R(P ) = {1, 2, 3, 4} ⊂ A.åæåëâñèæ éæéŽîåâIJŽ ïæéîŽãèâï ûŽîéëŽáàâêï ᎠöâàãæúèæŽ áæáæ èŽåæêñîæŽïëâIJæå Žôãêæöêëå, Žàîâåãâ ŽîïâIJëIJï éæéŽîåâIJâIJæï IJâîúêñèæ ìŽðŽîŽŽïëâIJæå Žôêæöãêæï ìîŽóðæçŽù. éŽàŽèæåŽá, ρ, σ, τ. ýöæîŽá æõâêâIJâê öâéáâàŽôêæöãêâIJï:Ž) (a, b) ∈ ρ, â. æ. (a, b) Žîæï ρ-öæ.IJ) aρb, â. æ. a áŽçŽãöæîâIJñèæŽ b-åŽê ρ éæéŽîåâIJæå.à) b ∈ ρ(a).éëùâéñèæ IJæêŽîñèæ éæéŽîåâIJæáŽê öâæúèâIJŽ àŽéëãæõâêëå ïý㎠éæéŽîåâ-IJâIJæù.à Ž ê ï Ž ä ô ã î â IJ Ž . ãåóãŽå, R Žîæï IJæêŽîñèæ éæéŽîåâIJŽ.àŽêãïŽäôãëå R −1 öâIJîñêâIJñèæ éæéŽîåâIJŽ. öâéáâàêŽæîŽá:R −1 = { (x, y) : (y, x) ∈ R } .ŽéîæàŽá, R −1 âèâéâêðâIJæï æàæãâ ûõãæèï ŽçŽãöæîâIJï, éŽàîé ïý㎠áŽèŽàâIJæå.éŽöŽïŽáŽéâ, åñ R ⊆ A × B, éŽöæêR −1 ⊆ B × A, D(R −1 ) = R(R), R(R −1 ) = D(R).

30à Ž ê ï Ž ä ô ã î â IJ Ž . ãåóãŽå, R Žîæï A ïæéîŽãèâäâ àŽêïŽäôãîñèæîŽæéâ éæéŽîåâIJŽ. R éæéŽîåâIJæï áŽéŽðâIJŽ ãñûëáëå æïâå R ′ éæéŽîåâIJŽï, îëéâèæùàŽêïŽäôãîñèæ öâéáâàêŽæîŽá: xR ′ y éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ xŽî Žîæï R éæéŽîåâIJŽöæ y-åŽê. ùýŽáæŽ, ïæéîŽãèæï âêŽäâ àãâóêâIJŽ R ′ = A 2 \R.ñòîë éëýâîýâIJñèæ æóêâIJŽ D R ᎠR R ŽôêæöãêâIJæï àŽéëõâêâIJŽ D(R) áŽR(R) ŽôêæöãêâIJæï êŽùãèŽá.ï Ž ã Ž î þ æ ö ë 1.1.1. ãåóãŽå, X = {6, 7, 8, 9, 10}, ρ = {(x, y) : x, y ∈ X, x < y}. Žéëûâîâåρ-ï Ꭰρ −1 -ï õãâèŽ âèâéâêðæ.2. ãåóãŽå, A = {a, b, c} ᎠE = P(A). Žéëûâîâå E-äâ àŽêïŽäôãîñèæ ⊂Ꭰ⊆ éæéŽîåâIJâIJæï õãâèŽ âèâéâêðæ.3. A = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 } . R ŽîæïA-äâ àŽêïŽäôãîñèæ IJæêŽîñèæ éæéŽîåâIJŽ:R = { (x, y) : x, y ∈ A, x 2 = y } .Žôûâîâå R ïæéîŽãèâ. îŽéáâê âèâéâêðï öâæùŽãï F −1 , R ′ ?4. ãåóãŽå, B = {1, 2, 3, 4, 5, 6} ïæéîŽãèâäâ àŽêïŽäôãîñèæŽ ρ éæéŽîåâIJŽ:ρ = { (x, y) : x, y ∈ B Ꭰx + y = 7 } .Žôûâîâå ρ, îëàëîù ïæéîŽãèâ. îŽ çŽãöæîæŽ ρ Ꭰρ −1 éæéŽîåâIJâIJï öëîæï?5. ãåóãŽå, E = Z 2 Ꭰρ = {(x, y) : x < y}. Žôûâîâå ρ-ï áŽéŽðâIJŽ ρ ′(\êŽçèâIJæŽ" éæéŽîåâIJæï àŽéëõâêâIJæï àŽîâöâ).6. ãåóãŽå, σ = ρ = {(x, y) : x ⊂ y}. öâæúèâIJŽ åñ ŽîŽ σ ′ Žôãûâîëåæàæãâ ýâîýæå, îëàëîù ρ ′ ûæêŽ ŽéëùŽêŽöæ? ᎎïŽIJñåâå.7. ãåóãŽå, óñøŽäâ Žîæï øãâñèâIJîæãæ ûâïæå áŽêëéîæèæ 30 ïŽýèæ: çâêðæêëéîâIJæ âîå éýŽîâï, èñûæ êëéîâIJæ éâëîâ éýŽîâï. ãåóãŽå, h n ŽôêæöêŽãïn-êëéîæŽê ïŽýèöæ éùýëãîâIJ éëïŽýèâï. Žé óñøæï éëïŽýèâëIJæï ïæêîŽãèâäâàŽêãïŽäôãîëå N éæéŽîåâIJŽ: h i Žîæï h j -åŽê N éæéŽîåâIJâIJæ, åñ æïæêæ éâäëIJèâIJæŽîæŽê. ïæéIJëèëâIJæï ïŽöñŽèâIJæå Žôûâîâå N éæéŽîåâIJŽ.8. àŽêãæýæèëå üŽáîŽçæï áŽòæï ñþîâáâIJæï S ïæéîŽãèâäâ éŽïäâ àŽêãïŽäôãîëåK éæéŽîåâIJŽ: x ñþîŽ Žîæï y ñþîŽïåŽê K éæéŽîåâIJŽöæ, åñ ùýâêïöâñúèæŽ àŽáŽãæáâï x-áŽê y-äâ âîåæ ïãèæå. Žôûâîâå K éæéŽîåâIJŽ ïæéIJëèëâIJæïïŽöñŽèâIJæå.9. ãåóãå, G Žîæï éæéŽîåâIJŽ S-äâ àŽêïŽäôãîñèæ éæéŽîåâIJŽ: x éæéŽîåâIJŽöæŽy-åŽê éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ x Žîæï åâåîæ ìŽæçæï ïŽûõæïæìëäæùæŽ, ýëèë y Žîæï ñþîŽ, îëéâèæù ìŽæçæï ìæîãâèæ ïãèŽ éåŽãîáâIJŽ.Žôûâîâå G, D(G) ᎠR(G) ïæéîŽãèââIJæ.§ 2. àîŽòæçñèæ ûŽîéëáàâêâIJæŽéëùŽêæï Žéëýïêæï áîëï ìæîãâè âðŽìäâ ýöæîŽá éëïŽýâîýâIJâèæŽ áŽýŽäãŽæéæïŽåãæï, îëé ñòîë êŽåèŽá áŽãæêŽýëå ŽéëùŽêæï õãâèŽ çëéìëêâêðæ. âï

31àŽêïŽçñåîâIJæå éëïŽýâîýâIJâèæŽ éæéŽîåâIJâIJæï Žôûâîæïåãæï. Žé ìŽîŽàîŽòöæàŽêãæýæèŽãå éæéŽîåâIJâIJæï Žôûâîæï àîŽòæçñè éâåëáâIJï.àŽêãæýæèëå X = {a, b, c, d, e} ïæéîŽãèâï ᎠJ X , U X ᎠR éæéŽîåâIJâIJæ:R = { (a, b), (a, c), (b, d), (c, e), (e, b) } .áŽãýŽäëå ëîæ, ñîåæâîåéŽîåëIJñèæ ôâîúæ: OX { ßëîæäëêðŽèñîæ,OY { ãâîðæçŽèñîæ. åæåëâñè éŽåàŽêäâ ûâîðæèâIJæå Žôãêæöêëå X ïæéîŽãèæïâèâéâêðâIJæ.êŽý. 2.1ïæéîŽãèââIJæ, îëéâèIJæù öââïŽIJŽéâIJŽ J X , U X ᎠR-ï àŽéëïŽýñèæŽ 2.1(IJ),2.1(à), 2.1(á) êŽýŽäâIJäâ.éæéŽîåâIJæï àîŽòæçñèæ ûŽîéëáàâêæï éâëîâ éâåëáæ éáàëéŽîâëIJï öâéáâàöæ:ûâîðæèâIJæï êŽùãèŽá æïîâIJæå öâãŽâîåëå x ∈ D Ꭰy ∈ R. âèâéâêðâIJæ(êŽý. 2.2(Ž), 2.3(IJ), 2.3(à), 2.3(á))àŽêïŽäôãîæï D X Žîâ ᎠéêæöãêâèëIJŽåŽ R X Žîâ öâæúèâIJŽ ûŽîéëãŽáàæêëåìŽîŽèâèñî ôâîúâIJäâ, îëàëîù 2.3 êŽýŽää⎠êŽøãâêâIJæ:

32êŽý. 2.2êŽý. 2.3ŽîïâIJëIJï Žïâåæ éâåëáæù, îëùŽ X ïæéîŽãèâ éëùâéñèæŽ ûâîðæèâIJæå,ýëèë éŽïäâ àŽêïŽäôãîñèæ éæéŽîåâIJŽ êŽøãâêâIJæŽ æïîâIJæå. 2.4 êŽýŽääâ éëùâéñèæŽJ X , U X ᎠR éæéŽîåâIJâIJæ ŽéàãŽîæ àîŽòæçñèæ éâåëáæåa•• be •• c•dI xa • c • • e• b a•• d • b e • •dêŽý. 2.4• c

33ï Ž ã Ž î þ æ ö ë 2.1.1. áŽýŽäâå áæŽàîŽéŽ, îëéâèæù ûŽîéëŽáàâêï ïŽãŽîþæöë 2.1(1)-öæ àŽêýæèñèρ éæéŽîåâIJŽï.2. áŽýŽäâå áæŽàîŽéŽ, îëéâèæù ûŽîéëŽáàâêï ïŽãŽîþæöë 2.1(3)-öæ àŽêýæèñèéæéŽîåâIJŽï.3. áŽýŽäâå áæŽàîŽéŽ, îëéâèæù ûŽîéëŽáàâêï ïŽãŽîþæöë 2.1(5)-öæ àŽêýæèñèN éæéŽîåâIJŽï, åñ óñøŽäâ Žîæï Žåæ ïŽýèæ.§ 3. éæéŽîåâIJâIJæï åãæïâIJâIJæùýŽáæŽ, îëé äëàŽáæ éæéŽîåâIJâIJæ, îëéèâIJæù ïæéîŽãèâåŽ ìæîáŽìæîæêŽéîŽãèæï éýëèëá óãâïæéîŽãèââIJæŽ, Žî Žîæï àŽêïŽçñåîâIJæå ïŽæêðâîâïë.éŽàîŽé, îëùŽ éæéŽîåâIJâIJæ ŽçéŽõëòæèâIJï äëàæâîå áŽéŽðâIJæå ìæîëIJâIJï, éŽåäâöâæúèâIJŽ ñòîë éâðæ îŽéæï åóéŽ. Žé ìŽîŽàîŽòöæ àŽêãæýæèŽãå äëàæâîå æéúæîæðŽá åãæïâIJâIJï, îëéèâIJæù öâæúèâIJŽ ßóëêáâï éæéŽîåâIJŽï. ŽéIJëIJâê, îëéåãæïâIJŽï Žáàæèæ Žóãï éŽöæê, åñ öâïîñèâIJñèæŽ öâïŽIJŽéæïæ ìæîëIJŽ.à Ž ê ï Ž ä ô ã î â IJ Ž . ãåóãŽå, ρ Žîæï A ïæéîŽãèâäâ àŽêïŽäôãîñèæéæéŽîåâIJŽ. éŽöæêŽ) ρ îâòèâóïñîæŽ, åñ xρx, êâIJæïéæâîæ x ∈ A;IJ) ρ ïæéâðîæñèæŽ, åñ xρy éæéŽîåâIJŽ æûãâãï yρx éæéŽîåâIJŽï.à) ρ ðîŽêäæðñèæŽ, åñ xρy Ꭰyρz éæéŽîåâIJâIJæ æûãâãï xρz éæéŽîåâIJŽï.á) ρ ŽêðæïæéâðîæñèæŽ, åñ xρy Ꭰyρx éæéŽîåâIJâIJæ æûãâãï x = yðëèëIJŽï.öâéëðŽêæèæ ùêâIJâIJæ àŽêãéŽîðëå éŽàŽèæåâIJäâé Ž à Ž è æ å æ 3.1.1. ρ = { (x, y) : x, y ∈ N ᎠàŽéõëòæŽ y-æï } , éŽöæê ρ éæéŽîåâIJŽŽ) îâòèâóïñîæŽ, îŽáàŽê êâIJæïéæâîæ êŽðñîŽèñîæ x îæùýãæ åŽãæï åŽãæïàŽéõëòæŽ, â. æ. xρx, êâIJæïéæâîæ x ∈ N;IJ) ïæéâðîæñèæ Žî Žîæï, îŽáàŽê 3 àŽéõëòæŽ 6-æï, éŽàîŽé 6 Žî Žîæï3-æï àŽéõëòæ, â. æ. xρy ≠⇒ yρx.à) ðîŽêäæðñèæŽ, îŽáàŽê åñ y/x ∈ N Ꭰz/y ∈ N, éŽöæêz( y) ( z)y = · ∈ N.x yâ. æ. xρy, yρz =⇒ xρz.á) ŽêðæïæéâðîæñèæŽ, îŽáàŽê åñ y x ∈ N Ꭰx y∈ N, éŽöæê x = y, â. æ.(xρy Ꭰyρx) =⇒ x = y2. σ = { (x, y) : x, y ∈ N, x ≤ y } , éŽöæê σ

34Ž) îâòèâóïñîæŽ, îŽáàŽê êâIJæïéæâîæ x-åãæï x ≤ x, â. æ. xσx, êâIJæïéæâîæx ∈ N.IJ) ŽîŽïæéâðîæñèæŽ, îŽáàŽê åñ 5 ≤ 7, éŽàîŽé 7 ≰ 5, â. æ. xσy ≠⇒yσx.à) ðîŽêäæðñèæŽ, îŽáàŽê åñ x ≤ y Ꭰy ≤ z, éŽöæê x ≤ z, â. æ.(xσy, yσz) =⇒ xσz.á) ŽêðæïæéâðîæñèæŽ, îŽáàŽê åñ x ≤ y Ꭰy ≤ x, éŽöæê x = y, â. æ.(xσy Ꭰyσx) =⇒ x = y.3. τ = { (x, y) : x, y ∈N\{1} Ꭰx Ꭰy-ï ïŽâîåë àŽéõëòæ Žóãï } , éŽöæê τŽ) îâòèâóïñîæŽ, â. æ. xτx, êâIJæïéæâîæ x ∈ N \ {1}.IJ) ïæéâðîæñèæ, â. æ. xτy = yτx.à) Žî Žîæï ðîŽêäæðñèæ îŽáàŽê 4-ïŽ áŽ 6-æï ïŽâîåë àŽéõëòæŽ 2 ∈N\{1}-æï Ꭰ9-æï { 3 ∈ N\{1}, éŽàîŽé 4-ïŽ áŽ 9-ï ïŽâîåë àŽéõëòæŽî Žóãå.á) Žî Žîæï Žêðæïæéâðîæñèæ: 4σ6, 6σ4, éŽàîŽé 4 ≠ 6.é Ž à Ž è æ å æ 3.2. ãåóãŽå, P Žîæï éåâèæ ýŽèýæï ïæéîŽãèâ. àŽêãïŽäôãîëåëîæ éæéŽîåâIJŽA = { (x, y) : x, y ∈ P, x Žîæï y-æï ûæêŽìŽîæ } .âï éæéŽîåâIJŽ îâòèâóïñîæ Žî Žîæï, ïæéâðîæñèæ Žî Žîæï, ŽêðæïæéâðîæñèæŽ,ðîŽêäæðñèæŽ.S = { (x, y) : x, y ∈ P, Ꭰx Ꭰy ïŽâîåë éöëIJèâIJæ ßõŽãå } .ùýŽáæŽ, îëé A éæéŽîåâIJŽ îâòèâóïñîæŽ, ïæéâðîæñèæŽ, ðîŽêäæðñèæŽ.öâãêæöêëå, îëé ïæéâðîæñèëIJæïŽ áŽ ŽêðæïæéâðîæñèëIJæï åãæïâIJâIJæ ŽîŽîæï ñîåæâîåàŽéëéîæùýŽãæ. êâIJæïéæâîæ X ïæéîŽãèæïŽåãæï J X æàæãñîæ éæéŽîåâIJŽŽîæï ïæéâðîæñèæù ᎠŽêðæïæéâðîæñèæù. öâæúèâIJŽ àãóëêáâï éæéŽîåâIJŽ,îëéâèæù Žîù ïæéâðîæñèæŽ áŽ Žîù Žêðæïæéâðîæñèæ.é Ž à Ž è æ å æ 3.3. ãåóãŽå, P Žîæï éåâèæ ýŽèýæï ïæéîŽãèâ. àŽêãïŽäôãîëåB éæéŽîåâIJŽB = { (x, y) : x, y ∈ P, x Žîæï y-æï úéŽ } .àŽêãæýæèëå ïæðñŽùæŽ, îëáâïŽù ëþŽýöæ Žîæï ëîæ úéŽ. p Ꭰq Ꭰâîåæ áŽ{ r. B éæéŽîåâIJŽ Žî Žîæï ïæéâðîæñèæ, îŽáàŽê pBr éŽàîŽé ŽîŽ rBp. âïéæéŽîåâIJŽ, Žïâãâ, Žî Žîæï Žêðæïæéâðîæñèæ, îŽáàŽê pBq ᎠqBp, éŽàîŽé pᎠq àŽêïýãŽãâIJñèæ (êŽý. 3.1)

35p • • q • rêŽý. 3.1ñòîë äëàŽá ïæðñŽùæŽöæ öâàãæúèæŽ éæéŽîåâIJæï äâéëå àŽêýæèñèæ åãæïâIJâIJæïæêðâîìîâðŽùæŽ áæŽàîŽéâIJæï ŽàâIJæï àäæå:Ž) éæéŽîåâIJŽ îâòèâóïñîæŽ éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ åæåëâñèæçãŽêúæïåãæï (ûâîðæèæïåãæï) áæŽàîŽéŽäâ ŽîïâIJëIJï æïŽîæ, îëéâèæù æûõâIJŽáŽ éåŽãîáâIJŽ Žé çãŽêúäâ.IJ) éæéŽîåâIJŽ ïæéâðîæñèæŽ éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ åæåëâñèæ æïîæïåãæï,îëéâèæù ëî çãŽêúï ŽâîðâIJï, Žàîâåãâ, ŽîïâIJëIJï Žé çãŽêúâIJæï ïŽûæꎎôéáâàëéæéŽîåñèâIJæå öâéŽâîåâIJâèæ æïŽîæ.à) éæéŽîåâIJŽ ðîŽêäæðñèæŽ éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ x Ꭰy çãŽêúâIJæïõëãâèæ ûõãæèæïåãæï, îëéâèIJæù öââîåâIJñèæŽ âîåéŽêâååŽê æïîâIJæï éæéáâãîëIJæå:x-áŽê a 1 -éáâ, a 1 -áŽê a 2 -éáâ, . . ., a n -áŽê y-éáâ, ŽîïâIJëIJï, Žàîâåãâ,x-æï y-åŽê öâéŽâîåâIJâèæ æïŽîæ.á) éæéŽîåâIJŽ ŽêðæïæéâðîæñèæŽ éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ Žî Žîïâ-IJëIJï ñîåæâîåïŽûæꎎôéáâàëæïîâIJæï ûõãæèæå öââîðâIJñèæ ëîæ ïýãŽáŽïýãŽçãŽêúæ.ŽîïâIJëIJï éæéŽîåâIJæï IJâãîæ ïý㎠åãæïâIJŽ, îëéèâIJæù öâæúèâIJëᎠàŽêàãâýæèŽ,éŽàîŽé, äâéëå Žôêæöêñèæ åãæïâIJâIJæ Žîï ñòîæë éêæöãêâèëãŽêæ áŽýöæîŽá æóêâIJŽ öâéáàëéöæ àŽéëõâêâIJñèæ.ï Ž ã Ž î þ æ ö ë 3.1.1. Žîæï åñ ŽîŽ öâéáâàæ éæéŽîåâIJâIJæ îâòèâóïñîæ, ïæéâðîæñèæ, ðîŽêäæðñèæŽê Žêðæïæéâðîæñèæ:Ž) R = { (a, b) : a, b ∈ Z, a − b Žîæï èñûæ } .IJ) A = {(a, b) : a, b ∈ N, a + b Žîæï èñûæ } .à) P éåâèæ ýŽèýæï ïæéîŽãèâŽ,B = { (a, b) : a, b ∈ P a Ꭰb-ï ïŽâîåë ûæêŽìŽîæ ßõŽãå } .á) D = { (a, b) : a, b ∈ P a Žîæï b-ï éöëIJâèæ } .â) M = { (a, b) : a, b ∈ Z, a = 2b } .ã) Q = { (a, b) : a, b ∈ Z, a æõëòŽ b-äâ } .ä) E = { (a, b) : a, b ∈ Z \ {0}, a õëòï b-ï } .å) F = { (a, b) : a, b ∈ N, a õëòï b-ï } .

36æ) L = {a, b, c}, éæéŽîåâIJŽ L-äâ:A = { (a, b), (b, a), (a, c), (c, a), (b, c), (c, a) } .2. öâéáâàæ éðçæùâIJñèâIJŽ Žîæï éùáŽîæ: ïæéâðîæñèæ ᎠðîŽêäæðñèæ éæéŽîåâIJŽŽîæï îâòèâóïñîæ, îŽáàŽê aRb ᎠbRa æûãâãï aRa-ï. æìëãâå öâùáëéŽ.{1, 2, 3} ïæéŽîãèâäâ àŽêïŽäôãîâå éæéŽîåâIJŽ, îëéâèæù æóêâIJŽ ïæéâðîæñèæ,ðîŽêäæðñèæ ᎠŽî æóêâIJŽ îâòèâóïñîæ.3. ãåóãŽå, ρ Žîæï éæéŽîåâIJŽ A ᎠB ïæéîŽãèââIJï öëîæï. åñ a ∈ A,éŽöæê ρ(a)-æå Žôãêæöêëå {b : aρb} ïæéîŽãèâ. ρ(a) Žîæï B ïæéîŽãèæï óãâïæéîŽãèâ.ãåóãŽå, {−4, −3, −2, −1, 0, 1, 2, 3, 4} ïæéîŽãèâäâ àŽêïŽäôãîñèæŽöâéáâàæ éæéŽîåâIJâIJæ:ρ = {(a, b) : a < b}, σ = { (a, b) : b − 1 < a < b + 2 } ,τ = {(a, b) : a 2 ≤ b}, δ = { (a, b) : |a| = |b| } .æìëãâå öâéáâàæ ïæéîŽãèââIJæŽ) ρ(0); IJ) σ(0); à) τ(0); á) ρ(1); â) σ −1 ; ã) τ(−1); ä) δ(0), δ(−2), δ(2).§ 4. áŽõëòŽ ᎠâçãæãŽèâêðëIJæï éæéŽîåâIJŽéîŽãŽè àŽéëåãèæå ŽéëùŽêŽöæ ãæôâIJå áæá ïæéîŽãèââIJï Ꭰãõëòå æïâåêŽæîŽá,îëé õãâèŽ øãâêåãæï ïŽæêðâîâïë ïæðñŽùæŽ öâïŽúèâIJâèæ æõëï àŽéëãæçãèæëåîŽéáâêæéâ ïûëîŽá öâîøâñè éŽàŽèæåäâ. éŽàŽèæåŽá, ìîëàîŽéæîâIJæïâêæï éŽýŽïæŽåâIJèâIJæï ýŽîæïýëIJîæãæ öâòŽïâIJæï éæôâIJæï âîå-âîåæ à䎎îæï Žé âêŽäâ áŽûâîæèæ çëêçîâðñèæ ìîëàîŽéâIJæï àŽêýæèãŽ. éŽàîŽé åæåëâñèæïŽæêðâîâïë âêŽ, æïâåæ éŽôŽèæ áëêæï âêâIJæï øŽåãèæå, îëàëîæùŽŽìŽïçŽèæ ûŽîéëöëIJï ñïŽïîñèëá IJâãî ìîëàîŽéŽï áŽ, éŽöŽïŽáŽéâ, ìîëàîŽéâ-IJæ æïâ ñêᎠöâãŽîøæëå, îëé æïæêæ ïûëîŽá ŽïŽýŽãáâï âêæï ôæîïâIJâIJï ᎠêŽçèëãŽêâIJâIJï.ñòîë çëêçîâðñèâIJæ îëé ãæõëå, ãæàñèæïýéëå, îëé âêŽï ŽóãïïŽéæ úæîæåŽáæ ïŽéŽîåæ ïðîñóðñîŽ, éæïŽûãáëéëIJæï ëåýæ éâåëáæ ᎠéâðæéŽï Žî Žóãï ŽîŽêŽæîæ àŽêïŽçñåîâIJñèæ åãæïâIJâIJæ. øãâê öâàãæúèæŽ éŽàŽèæåæïïŽýæå Žàãâôë öãæáæ ìîëàîŽéŽ, îëéâèåŽàŽê åæåëâñèæ éŽåàŽêæ öâæùŽãï âêæïéýëèëá âîå éŽýŽïæŽåâIJâèï (åñéùŽ, äëàŽáŽá, åæåëâñèæ ìîëàîŽéŽ öâæúèâ-IJŽ æõâêâIJáâï âêæï âîåäâ éâð éŽýŽïæŽåâIJâèï). éŽöæê Žé ìîëàîŽéâIJæï àŽéëçãèâãŽáŽòŽîŽãᎠâêæï åãæïâIJâIJæï ñáæáâï êŽûæèï. éŽåâéŽðæçñîŽá âï öâæúèâIJŽàŽêãïŽäôãîëå öâéáâàêŽæîŽá.à Ž ê ï Ž ä ô ã î â IJ Ž . ãåóãŽå, A ŽîŽùŽîæâèæ ïæéîŽãèâï, A i ,(i = 1, 2, . . . , n, n ∈ N) Žîæï A ïæéîŽãèæï æïâå óãâïæéîŽãèâåŽ âîåëIJèæëIJŽ,îëén⋃A i = A.i=1Žé óãâïæéîŽãèâåŽ âîåëIJèæëIJŽï A ïæéîŽãèæï áŽòŽî㎠âûëáâIJŽ.é Ž à Ž è æ å æ 4.1. {A, B} Žîæï A ∪ B ïæéîŽãèæï áŽòŽîãŽ.

37åñ àŽéëãæõâêâIJå áŽòŽîãæï ùêâIJŽï, öâàãæúèæŽ ñäîñêãâèãõëå, îëé ŽîùâîåæåãæïâIJŽ Žî æõëï àŽéëðëãâIJñèæ, îŽáàŽê åæåëâñèæ âèâéâêðæ âçñåãêæïáŽòŽîãæï âîå óãâïæéîŽãèâ éŽæêù. ŽéŽïåŽê, äëàŽáŽá, öâæúèâIJŽ àãýãáâIJëáâïàŽéâëîâIJæï öâéåýãâãâIJæù. åñ öâéáâàöæ éëãæåýëãå, îëé áŽòŽîãæï âèâéâêðâIJæûõãæè-ûõãæèŽá Žî æçãâåâIJëáâï, éŽöæê, àŽéâëîâIJŽ ŽôŽî æóêâIJŽ, â. æ.åæåëâñèæ âèâéâêðæ éýëèëá âîå óãâïæéîŽãèâöæ öâãŽ. ŽóâáŽê ûŽîéëæóéêâIJŽáŽõëòæï ùêâIJŽ.à Ž ê ï Ž ä ô ã î â IJ Ž . ŽîŽùŽîæâèæ A ïæéîŽãèæï áŽõëòŽ âûëáâ-IJŽ óãâïæéîŽãèâåŽ æïâå P(A) âîåëIJèæëIJŽï, îëéèæïåãæïŽù ïîñèáâIJŽ ëîæìæîëIJŽn⋃1. A i = A, A i ∈ P(A);i=12. A i ∩ A j = ∅, i ≠ j.ŽéàãŽîŽá, A ïæéîŽãèâ áŽõëòæèæŽ æïâåêŽæîŽá, îëé éæïæ åæåëâñèæ âèâéâêðæâçñåãêæï áŽõëòæï éýëèëá âîå óãâïæéîãèâï.é Ž à Ž è æ å æ 4.2. E ñêæãâîïŽèñîæ ïæéîŽãèâï1) {A, A ′ } Žîæï E-æï áŽõëòŽ;IJ) {A ∩ B, A ∩ B ′ , A ′ ∩ B, A ′ ∩ B ′ } Žîæï E-æï áŽõëòŽ;à) {A \ B, A ∩ B, B \ A} Žîæï A ∪ B-æï áŽõëòŽ.A ïæéîŽãèæïn ⋃i=1A i áŽõëòŽ Žé ïæéîŽãèâäâ àŽêïŽäôãîŽãï àŽîçãâñè πéæéŽîåâIJŽï:xπy éŽöæê Ꭰéýèëá éŽöæê, îëùŽ x, y ∈ A k . ùýŽáæŽ, êâIJæïéæâîæ x ∈ AâèâéâêðæïŽåãæï àãâóêâIJŽ xπx. éŽöŽïŽáŽéâ, π éæéŽîåâIJŽ îâòèâóïñîæŽ.ãåóãŽå, xπy Ꭰyπz. ìæîãâèæ éæéŽîåâIJŽ êæöêŽãï x, y ∈ A k , éâëîâ { y, z ∈A k . ŽóâáŽê àŽéëéáæêŽîâëIJï, îëé x, z ∈ A k . îŽù êæöêŽãï xπz éæéŽîåâIJŽï.ŽóâáŽê ãŽïçãêæå, îëé π éæéŽîåâIJŽ ðîŽêäæðñèæŽ.ãåóãŽå, xπy, â. æ. x, y ∈ π, îŽù æàæã⎠y, x ∈ π. âï êæöêŽãï, îëé yπx áŽπ éæéŽîåâIJŽ ïæéâðîæñèæŽ.à Ž ê ï Ž ä ô ã î â IJ Ž . ïæéîŽãèâäâ àŽêïŽäôãîñè IJæêŽîñè éæéŽîåâ-IJŽï âçãæãŽèâêðëIJæï éæéŽîåâIJŽ âûëáâIJŽ, åñ æï îâòèâóïñîæ, ïæéâðîæñèæ áŽðîŽêäæðñèæŽ.é Ž à Ž è æ å æ 4.3. õãâèŽ ïŽéçñåýâáæï ïæéîŽãèâäâ àŽêãïŽäôãîëåéæéŽîåâIJŽ{(x, y) : x Ꭰy ïŽéçñåýâáâIJï Žóãå âîåêŽæîæ òŽîåëIJæ}.ùýŽáæŽ, âï âçãæãŽèâêðëIJæï éæéŽîåâIJŽŽ.à Ž ê ï Ž ä ô ã î â IJ Ž . ãåóãŽå, ρ âçãæãŽèâêðëIJæï éæéŽîåâIJŽŽA ïæéîŽãèâäâ. x ∈ A âèâéâêðæïŽåãæï àŽêãïŽäôãîëå âçãæãŽèâêðëIJëIJæï [x]

38çèŽïæ[x] = {y : xρy}.ŽéàãŽîŽá, [x] Žîæï A-ï õãâèŽ æé âèâéâêðæï ïæéîŽãèâ, îëéèâIJæù ρ-âçãæãŽèâêðñîæŽx-æï æé öâéåýãâãŽöæ, îëùŽ ïæéîŽãèâäâ àŽêæýæèâIJŽ éýëèëáâîåæ âçãæãŽèâêðëIJæï éæéŽîåâIJŽ, öâàãæúèæŽ àŽéëãæõâêëå Žàîâåãâ \≡" (\âçãæãŽèâêðñîæŽ")ŽôêæöãêŽ, Žéæðëé[x] = {y : x ≡ y}.ùŽèçâñè ïìâùæŽèñî öâéåýãâãŽöæ âçãæãŽèâêðëIJæï Žôêæöãêæïåãæï æõâêâ-IJâê \∼" ïæéIJëèëï. îëáâïŽù àŽêïŽäãîñèæŽ âçãæãŽèâêðëIJæï éæéŽîåâIJŽ, éåâèæïæéîŽòãèæï öâéëûéâIJæï êŽùãèŽá öâàãæúèæŽ êâIJæïéæâîŽá Žãæîøæëå ûŽîéëéŽáàâêèâIJæ(åæåë-åæåë âçãæãŽèâêðëIJæï çèŽïæáŽê), îŽù ŽéŽîðæãâIJï àŽéëåãèâIJï.é Ž à Ž è æ å æ 4.4. ãåóãŽå, s ïæòóïæîâIJñèæ êŽðñîŽèñîæ îæùýãæŽ.éåâè îæùýãåŽ Z ïæéîŽãèâäâ àŽêãïŽäôãîëå ρ s éæéŽîåâIJŽ:ρ s = { (x, y) : x − y = ns, ïŽáŽù n ∈ Z } .àŽêãæýæèëå öâéåýãâ㎠s = 10. éŽöæê[1] = {1, 11, 21, −9, −19, . . .},[25] = {35, 75, −105, −15, . . .} ᎠŽ. ö.ïæêŽéáãæèâöæ ŽîïâIJëIJï âçãæãŽèâêðëIJæï éýëèëá Žåæ ïýãŽáŽïý㎠çèŽïæ.0, 1, 2, . . . , 8, 9 îæùýãâIJæ âçñåãêæï ïýãŽáŽïý㎠çèŽïâIJï. Žéæðëé öâàãæúèæŽàŽéëãæõâêëå æïæêæ Žé çèŽïâIJæï ûŽîéëéŽáàâêèâIJŽá. Žé âçãæãŽèâêðëIJæï çèŽïâ-IJæï ïæéîŽãèâ ŽôæêæöêâIJŽ Z 10 -æå ᎠâûëáâIJŽ êŽöååŽ çèŽïâIJæ éëáñèæå 10.ïŽäëàŽáëá, êâIJæïéæâîæ m ∈ N-åãæï, Z m êŽöååŽ çèŽïâIJæ éëáñèæå m, öâæùŽãïm ïýãŽáŽïý㎠çèŽïï:{}Z k = [0], [1], [2], . . . , [m − 1] .àŽêãæýæèëå Z × N ïæéîŽãèâ. (a, b) ûõãæèæ öâæúèâIJŽ àŽêãæýæèëå, îëàëîùa/b ûæèŽáæ. = bZ × N ïæéîŽãèæï âèâéâêðâIJæï Žôêæöãêæï âï ëîæ ýâîýæïýãŽáŽïý㎎, éŽàîŽé æïæêæ \æäëéëîòñèæŽ". öâãêæöêëå, îëé öâïŽúèâIJâè掎ôêæöãêæï âîåæ òëîéæáŽê éâëîâäâ àŽáŽïãèŽ.ŽîïâIJëIJï ïýãŽáŽïý㎠âèâéâêðæ Z × N ïæéîŽãèâöæ, îëéèâIJæù ïŽïñîãâèæŽàŽêãæýæèëå, îëàëîù âîåæ Ꭰæàæãâ, åñéùŽ æïæêæ ïýãŽáŽïýãŽêŽæîŽáøŽæûâîâIJŽ. âï ïæîåñèâ îëé àŽáŽãèŽýëå, Z × N ïæéîŽãèâäâ àŽêãïŽäôãîëåâçãæãŽèâêðëIJæï éæéŽîåâIJŽ öâéáâàêŽæîŽá: (a, b) ≡ (c, d) éŽöæê ᎠéýëèëáéŽöæê, îëùŽ a·d = b·c. Žé éæéŽîåâIJæå àŽêïŽäôãîñè âçãæãŽèâêðëIJæï çèŽïâIJïâûëáâIJŽ îŽùæëêŽèñîæ îæùýãâIJæ, îŽùæëêŽèñî îæùýãåŽ ïæéîŽãèâï ŽôêæöêŽãâêQ ïæéIJëèëåæ. êŽéáãæè îæùýãåŽ ïæéîŽãèâ ŽôæêæöêâIJŽ R ïæéIJëèëåæ.âï îæùýãâIJæ öâæúèâIJŽ ûŽîéëãŽáàæêëå öâéáâàæ òëîéæå:. . . od n . . . d 2 d 1 d 0 δ 1 δ 2 . . . δ n . . . ,

39ïŽáŽù õëãâèæ d i Ꭰδ j âçñåãêæï ïæéîŽãèâï{0, 1, 2, 3, 4, 5, 6, 7, 8, 9} = Z 10 ,d n ≠ 0, àŽîᎠn = 0 öâéåýãâãæïŽ. áŽïŽöãâIJæŽ ñïŽïîñèë ŽîŽìâîæëáñèæŽåûæèŽáæ. êñèæ d n -æï ûæê, øãâñèâIJîæã àŽéëæðëãâIJŽ.A ïæéîŽãèâäâ àŽêïŽäôãîñèæ âçãæãŽèâêðëIJæï ρ éæéŽîåâIJæï æêáâóïæ âûëáâIJŽρ éæéŽîåâIJæå æêáñùæîâIJñè A çèŽïâIJï.ï Ž ã Ž î þ æ ö ë 4.1.1. ᎎéðçæùâå, îëé êâIJæïéæâîæ âçãæãŽèâêðëIJæï éæéŽîåâIJŽ ûŽîéëóéêæïæïâå áŽõëòŽï, îëé êâIJæïéæâîæ x, y ∈ A âèâéâêðâIJæïåãæï ïîñèáâIJŽ Žê [x] =[y] Žê [x] ∩ [y] = ∅.2. ãåóãŽå, A ïŽïîñèæ ïæéîŽãèâŽ. îëàëîæ âçãæãŽèâêðëIJæï éæéŽîåâIJâIJæûŽîéëóéêæï âçãæãŽèâêðëIJæï çèŽïâIJæï ñáæáâï Ꭰñéùæîâï.3. åñ {A 1 , A 2 , . . . , A n } Žîæï A ïæéîŽãèæï áŽõëòŽ ᎠA ïŽïîñèæŽ, Žøãâêâå,îëén∑|A| = |A i |.i=1§ 5. áŽèŽàâIJæï éæéŽîåâIJŽîŽéáâêŽáŽù ðëèëIJæï ùêâIJæáŽê ûŽîéëóæéêâIJŽ âçãæãŽèâêðëIJæï éŽåâéŽðæçñîæùêâIJŽ, äëàæâîåæ ñðëèëIJŽ, Žïâãâ öâæúèâIJŽ àŽéëãæõâêëå, îëàëîù éëáâèæéæéŽîåâIJŽåŽ òŽîåë çèŽïâIJæïåãæï.A ïæéîŽãèâäâ àŽêïŽäôãîñè îâòèâóïñî, ðîŽêäæðñè ᎠŽêðæïæéâðîæñèéæéŽîåâIJâIJæï êŽûæèëIJîæãæ áŽèŽàâIJŽ (Žê áŽèŽàâIJŽ) âûëáâIJŽ. áŽèŽàâIJŽ(Žêñ áŽèŽàâIJæï éæéŽîåâIJŽ) Žîæï ≤ éæéŽîåâIJæï àŽêäëàŽáâIJŽ.åñ àŽêïŽäôãîñèæ ≤ éæéŽîåâIJŽ, éŽöæê öâæúèâIJŽ < éæéŽîåâIJæï àŽêïŽäôãîŽöâéáâàêŽæîŽá:a < b ⇐⇒ a ≤ b Ꭰa ≠ b.ŽêŽèëàæñîŽá, åñ éëùâéñèæŽ < éæéŽîåâIJŽ, éŽöæê ≤ éæéŽîåâIJŽŽ àŽêãïŽäôãîŽãåŽïâ:a ≤ b ⇐⇒ a < b Žê a = b.é Ž à Ž è æ å æ 5.1. ãåóãŽå, éëùâéñèæŽ êâIJæïéæâîæ A ïæéîŽãèâ, éŽöæêP(A) ïæéîŽãèâäâ àŽêïŽäôãîñèæ ⊆ éæéŽîåâIJŽ Žîæï ðîæãæŽèñîæ áŽèŽàâIJæïéæéŽîåâIJŽ (X ⊆ X õãâèŽ X-åãæï, åñ X ⊆ Y ᎠY ⊆ X, éŽöæê X = Y ; åñX ⊆ Y ᎠY ⊆ Z, éŽöæê X ⊆ Z).áŽèŽàâIJæï éæéŽîåâIJŽï âûëáâIJŽ ïîñèæ, åñ êâIJæïéæâîæ x, y ∈ A âèâéâêðâIJæïåãæïïîñèáâIJŽ Žê X ⊆ Y Žê Y ⊆ X (Žê ëîæãâ).é Ž à Ž è æ å æ 5.2. ùýŽáæŽ, îëé éëùâéñèæ ïæéîŽãèæï óãâïæéîŽãèâåŽïæéîŽãèâäâ àŽêïŽäôãîñèæ áŽèŽàâIJŽ Žî Žîæï ïîñèæ, ýëèë îæùýãåŽIJñêâIJîæãæ áŽèŽàâIJŽ R êŽéáãæè ôâîúäâ ïîñèæ.

40öâãêæöêëå, îëé ïîñèŽá áŽèŽàâIJñè ïæéîŽãèâï ûîòæãŽá áŽèŽàâ-IJñèïŽù ñûëáâIJâê.éŽåâéŽðæçæï öâïûŽãèŽöæ ìîëàîâïæï éæôûâãâIJåŽé âîåŽá ùýŽáæ ýáâIJŽ,îëé éŽåâéŽðæçŽ Žî Žîæï ùŽèçâñèèæ æáââIJæï êŽçîâIJæ, ŽîŽéâá âîåæéâëîâïåŽêáŽçŽãöæîâIJñèæ çëêùâìðñŽèñîæ ùêâIJâIJæï âîåëIJèæëIJŽŽ, îëéèâIJæù àŽéëæõâêâIJŽéîŽãŽèæ âîåæéâëîæïàŽê àŽêïýãŽãâIJñèæ ïæðñŽùæâIJæ. ŽóâáŽê àŽéëéáæêŽîâëIJï,îëé åñ úæîæåŽáæ ìîæêùæìæ áŽáàâêæèæ ᎠàŽéëçãèâñèæŽ, æàæ, òŽóðëIJîæãŽá,âîåêŽæîŽá ûõãâðï ìîëIJèâéâIJï õãâèŽ Žé ïýãŽáŽïý㎠öâéåýãâãæïåãæï.öâéâàæ éŽàŽèæåæ ûŽîéëŽáàâêï Žéæï éŽîðæã æèñïðîŽùæŽï.é Ž à Ž è æ å æ 5.3. N ïæéîŽãèâäâ àŽêïŽäôãîñèæ áŽèŽàâIJæï ïŽòñúãâèäâöâàãæúèæŽ òëîéŽèñîŽá éæãæôëå øãâñèâIJîæãæ áŽèŽàâIJæï éæéŽîåâIJŽ Z,Q ᎠR îæùýãåŽ ïæéîŽãèââIJäâ.àŽêãæýæèëå Z éåâè îæùýãåŽ ïæéîŽãèâ.éïþâèëIJæï àŽéŽîðæãâIJæï éæäêæåZ áŽãõëå öâéáâàêŽæîŽá:Z = N ∪ {0} ∪ A, ïŽáŽù A = {−x : x ∈ N}.àŽêãïŽäôãîëå Z ïæéîŽãèâäâ éæéŽîåâIJŽ õãâèŽ öâïŽúèë x Ꭰy âèâéâêðâIJæïàŽêýæèãæå {N ∪ {0} ∪ A} áŽõëòæáŽê.åñ x = y, éŽöæê x ≤ y Ꭰy ≤ x. ãåóãŽå, x ≠ yŽ) åñ x, y ∈ N, éŽöæê Z-öæ áŽèŽàâIJŽ æïâåæãâŽ, îëàëîù N-öæ.IJ) åñ x, y ∈ A, éŽöæê x ≤ y éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ −y ≤ −xN-öæ (éŽà., −7 ≤ −5, îŽáàŽê 5 ≤ 7).à) åñ x = 0 Ꭰy ∈ N, éŽöæê x ≤ y;á) åñ x ∈ A, y = 0, éŽöæê x ≤ y;â) åñ x ∈ A, y ∈ N, éŽöæê x ≤ y.Z ïæéîŽãèâäâ áŽèŽàâIJãæïŽ áŽ éåâè ôâîúäâ øãâñèâIJîæãæ ŽîæåéâðæçñèæëìâîŽùæâIJæï ïŽòñúãâèäâ öâàãæúèæŽ àŽêãïŽäôãîëå áŽèŽàâIJŽ îŽùæëêŽèñîîæùýãåŽ Q ïæéîŽãèâäâ: a/b ≤ c/d éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ a·d ≤ b ·c.Žé éðçæùâIJñèâIJâIJæï öâéëûéâIJŽï áŽãðëãâIJå ïŽãŽîþæöëï ïŽýæå.IJëèëï, àŽêãïŽäôãîëå áŽèŽàâIJæï éæéŽîåâIJŽ êŽéáãæè îæùýãåŽ R ïæéîŽãèâäâ.àŽêãæýæèëå ëîæ áŽáâIJæåæ îæùýãæï ŽåëIJæå ûŽîéëáàâêŽD = · · · od n · · · d 2 d 1 d 0 δ 1 δ 2 · · · ,C = · · · oc m · · · c 2 c 1 c 0 γ 1 γ 2 · · · .åñ d i = c i Ꭰδ i = γ i õãâèŽ i-åãæï, éŽöæê D = C áŽ, éŽöŽïŽáŽéâ, D ≤ CᎠC ≤ D. ûæꎎôéáâà öâéåýãâãŽöæŽ) åñ d n ≠ 0, c n ≠ 0 Ꭰn ≠ m, éŽöæêD ≤ C, åñ n < m ᎠC ≤ D, åñ m < n;IJ) åñ n = m Ꭰd i ≠ c i , éŽàîŽé d j = c j õãâèŽ æïâåæ j-åãæï, îëéi < j ≤ n, éŽöæêD ≤ C, åñ d i < c i ᎠC ≤ D, åñ c i < d i ;

41à) åñ m = n Ꭰd i = c i õãâèŽ i-åãæï, éŽàîŽé δ k ≠ γ k îëéâèæéâ k-åãæïᎠδ j = γ j õãâèŽ j-åãæï, îëéèæïåãæïŽù 0 < j < k, éŽöæêC ≤ D, åñ γ k < δ k ᎠD ≤ C, åñ δ k < γ k .éçæåýãâèï öâñúèæŽ âï öâŽéëûéëï áŽéëñçæáâIJèŽá. ñŽîõëòæåæ îæùýãâIJææïâãâ öâæúèâIJŽ àŽéëãæçãèæëå, îëàëîù Z-öæ.X ïæéîŽãèæï ≤ áŽèŽàâIJæï éæéŽîåâIJŽïåŽê âîåŽá âûëáâIJŽ êŽûæèëIJîæãáŽèŽàâIJñèæ ïæéîŽãèâ (ŽôæêæöêâIJŽ (X, ≤)-æå) êâIJæïéæâî u ∈ (X, ≤) âèâéâêðïâûëáâIJŽ x Ꭰy âèâéâêðâIJæï äâᎠïŽäôãŽîæ, åñ ïîñèáâIJŽ x ≤ uᎠy ≤ v. ŽêŽèëàæñîŽá, v ∈ X âèâéâêðï âûëáâIJŽ x Ꭰy âèâéâêðâIJæïóãâᎠïŽäôãŽîæ, åñ ïîñèáâIJŽ v ≤ x Ꭰv ≤ y. x Ꭰy âèâéâêðâIJæïõãâèŽ äâᎠïŽäôãîæï ïæéîŽãèâ ûŽîéëŽáàâêï X ïæéîŽãèæï óãâïæéîŽãèâï áŽáŽèŽàâIJñèæŽ ≤ éæéŽîåâIJæå. åñ ŽîïâIJëIJï Žé ïæéîŽãèæï âîåŽáâîåæ ñéùæîâïæâèâéâêðæ, â. æ. åñ ŽîïâIJëIJï æïâåæ µ ∈ (X, ≤), îëéèæïåãæïŽù ïîñèáâIJŽ:x ≤ µ, y ≤ µ, µ ≤ u êâIJæïéæâîæ u äâᎠïŽäôãîæïåãæï, éŽöæê µ-ï ñûëáâIJâê xᎠy âèâéâêðâIJæï äñïð äâᎠïŽäôãŽîï (sup). ŽêŽèëàæñîŽá àŽêæéŽîðâ-IJŽ x Ꭰy âèâéâêðâIJæï äñïðæ óãâᎠïŽäôãŽîæ (inf) (â. æ. ñáæáâïæ óãâáŽïŽäôãŽîæ).öâãêæöêëå, îëé êŽéáãæè îæùýãåŽ R ïæéîŽãèâäâ IJñêâIJîæãæ áŽèŽàâIJŽàŽêïŽäôãîŽãï ŽýŽè ïæéîŽãèââIJï. éŽå ñûëáâIJâê æêðâîãŽèâIJï:[a, b] = { x : x ∈ R, a ≤ x ≤ b } .âï Žîæï øŽçâðæèæ æêðâîãŽèæ (éëêŽçãâåæ, ïâàéâêðæ) a-áŽê b-éáâ.]a, b[ = { x : x ∈ R, a < x < b }Žîæï ôæŽ æêðâîãŽèæ a-áŽê b-éáâ. ëîæãâ öâéåýãâãŽöæ a-ï Ꭰb-ï âûëáâIJŽIJëèë ûâîðæèâIJæ.øŽçâðæèæ æêðâîãŽèæ åŽãæïåŽãöæ öâæùŽãï IJëèë ûâîðæèâIJï, ýëèë ôæŽæêðâîãŽèæ Žî öâæùŽãï. êŽýâãîŽáôæŽ æêðâîãŽèâIJæ àŽêæïŽäôãîâIJŽ öâéáâàêŽæîŽá:àŽêãæýæèëå öâéáâàæ æêðâîãŽèâIJæ:ï Ž ã Ž î þ æ ö ë 5.1.[a, b[ = { x : x ∈ R, a ≤ x < b } ,]a, b] = { x : x ∈ R, a < x ≤ } .] − ∞, a] = {x : x ≤ a},] − ∞, a[ = {x : x < a},[a, +∞[ = {x : a ≤ x},]a, +∞[ = {x : a < x},] − ∞, +∞[ = R.1. ãåóãŽå, A êâIJæïéæâîæ ïæéîŽãèâŽ, P(A) éæïæ õãâèŽ óãâïæéîŽãèâåŽ ïæéîŽãèâ,ρ { P() × P(A) ïæéîŽãèâäâ àŽêïŽäôãîñèæ IJæêŽîñèæ éæéŽîåâIJŽŽ,

42(P, Q)ρ(X, Y ) éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ (P △Q) ⊆ (X△Y ). Žîæï åñŽîŽ ρ áŽèŽàâIJŽ éæéŽîåâIJŽ (P △Q = (P ∪ Q) \ (∩Q))?2. ãåóãŽå, A êâIJæïéæâîæ ïæéîŽãèâŽ, ýëèë σ { P(A) × P(A) ïæéîŽãèâäâöâéáâàêŽæîŽá àŽêïŽäôãîñèæ IJæêŽîñèæ éæéŽîåâIJŽ: (P, Q)σ(X, Y ) éŽöæêᎠéýëèëá éŽöæê, îëùŽ P ≤ X ᎠQ ≤ Y . Žîæï åñ ŽîŽ σ áŽèŽàâIJæïéæéŽîåâIJŽ? (Ꭰåñ Žîæï áŽèŽàâIJæï éæéŽîåâIJŽ, Žîæï åñ ŽîŽ ïîñèæ?)3.ãåóãŽå, τ Ꭰπ N 2 ïæéîŽãèâäâ öâéáâàêŽæîŽá àŽêïŽäôãîñèæ IJæêŽîñèæéæéŽîåâIJâIJæŽ: (a, b)τ(c, d) éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ a ≤ c Ꭰb ≤ d;(a, b)π(c, d) éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ a ≤ c Ꭰb ≥ d. Žîæï åñ ŽîŽ τᎠπ IJæêŽîñèæ éæéŽîåâIJâIJâIJæ?4. ãåóãŽå, π Žîæï Q-ï áŽáâIJæå îæùýãâIJäâ öâéáâàêŽæîŽá àŽêïŽäôãîñèæIJæêŽîñèæ éæéŽîåâIJâIJŽ: (a/b)π(c/d) éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ a·d ≤ b·c.Žøãâêâå, îëé π Žîæï ïîñèæ áŽèŽàâIJæï éæéŽîåâIJŽ.§ 6. öâáàâêæèæ éæéŽîåâIJâIJæà Ž ê ï Ž ä ô ã î â IJ Ž . ãåóãŽå, éëùâéñèæŽ A, B ᎠC ïæéîŽãèââIJæ,σ éæéŽîåâIJŽ àŽêïŽäôãîñèæ A ᎠB ïæéîŽãèââIJï öëîæï, ýëèë ρ éæéŽîåâIJŽ{ B ᎠC ïæéîŽãèââIJï öëîæï. A ᎠC ïæéîŽãèââIJï öëîæï àŽêãïŽäôãîëåéæéŽîåâIJŽ, îëéâèæù A-áŽê B-öæ æéëóéâáâIJï σ-ï ïŽöñŽèâIJæå, ýëèë B-áŽêC-öæ ρ-ï ïŽöñŽèâIJæå. Žïâåæ àäæå ŽàâIJñè éæéŽîåâIJŽï âûëáâIJŽ öâáàâêæèæ ᎎôæêæöêâIJŽ (ρ ◦ σ) ïæéIJëèëåæ, â. æ.(ρ ◦ σ)(a) = ρ(σ(a)).êŽý. 6.6ŽéîæàŽá, (x, y) ∈ ρ ◦ σ, åñ B ïæéîŽãèâöæ ŽîïâIJëIJï æïâåæ z âèâéâêðæ,îëéèæïåãæïŽù öâïîñèáâIJŽ (x, z) ∈ σ, (z, y) ∈ ρ, ŽóâáŽê àŽéëéáæêŽîâëIJï,îëé ρ ◦ σ éæéŽîåâIJæï àŽêïŽäôãîæï ŽîâŽD ρ◦σ = σ −1 D ρ .ïæðñŽùææï ïŽæèñïðŽùæëá àŽêãæýæèëå êŽý. 6.6. σ éæéŽîåâIJæï D σ àŽêïŽäôãîæïŽîâ ᎠR σ éêæöãêâèëIJŽåŽ ïæéîŽãèâ áŽöðîæýñèæŽ âîåæ éæéŽîåñèâ-IJæå, ýëèë ρ éæéŽîåñèâIJæï D ρ ᎠR ρ { ïý㎠éæéŽîåñèâIJæå, éŽöŽïŽáŽéâ, A,

43B ᎠC ïæéîŽãèââIJäâ ëîéŽàŽá áŽöðîæýñèæ ïâàéâêðâIJæ ûŽîéëŽáàâêï D ρ◦σ ,D ρ ∩ R σ ᎠR ρ◦σ ŽîââIJï, öâïŽIJŽéæïŽá.ö â ê æ ö ã ê Ž . σ Ꭰρ éæéŽîåâIJâIJæï øŽûâîæáŽê àŽéëéáæêŽîâëIJï,îëé æïæêæ àŽéëæõâêâIJŽ éŽîþãêæáŽê éŽîùýêæã. éŽöŽïŽáŽéâ, (ρ ◦ σ)(a) êæöêŽãï,îëé áŽïŽûõæïöæ ŽæôâIJŽ a Ꭰσ-ï ïŽöñŽèâIJæå àŽîáŽæóéêâIJŽ, ýëèë öâéáâààŽîáŽæóéêâIJŽ ρ-ï ïŽöñŽèâIJæå. ŽèàâIJîŽöæ, äëàþâî, âï øŽæûâîâIJŽ aσρ-ï ïŽýæå.ïý㎠éŽåâéŽðæçñîæ ûæàêâIJæï ûŽçæåýãæï õñîŽáôâIJŽ ñêᎠéæãŽóùæëå æéŽï, åñéæéŽîåâIJâIJæï øŽûâîæï îëàëîæ ûâïæŽ éæôâIJñèæ Žé ûæàêöæ.é Ž à Ž è æ å æ 6.1. ãåóãŽå, êŽðñîŽèñî îæùýãåŽ N ïæéîŽãèâäâàŽêïŽäôãîñèæŽ σ Ꭰρ IJæêŽîñèæ éæéŽîåâIJâIJæ:éŽöæêσ = { (x, y) : x, y ∈ N, y = x + 1 } ,ρ = { (x, y) : x, y ∈ N, x = y 2} ,D σ = {x : x, x + 1 ∈ N} = N,D ρ = {x 2 : x ∈ N},D ρ◦σ = σ −1 D ρ = { x 2 − 1 : x, x 2 − 1 ∈ N } = {3, 8, 15, 24, 35, . . .}.ãåóãŽå, A ïæéîŽãèâäâ àŽêïŽäôãîñèæŽ σ éæéŽîåâIJŽ, öâæúèâIJŽ ŽãŽàëåöâáàâêæèæ σ ◦ σ éæéŽîåâIJŽ. éŽàŽèæåŽá, åñσ = { (x, y) : x, y ∈ N, y = x + 1 } ,σ ◦ σ = { (x, y) : x, y ∈ N, y = x + 2 } ,ρ = { (x, y) : x, y ∈ N, x = y 2} ,ρ ◦ ρ = { (x, y) : x, y ∈ N, x = y 4} .âï éæéŽîåâIJâIJæ öâæúèâIJŽ øŽæûâîëï σ 2 Ꭰρ 2 ïŽýæå (ïŽäëàŽáëá, âï ŽôêæöãêŽïæéîŽãèæïåãæï éåèŽá çŽêëêæâîæ Žî Žîæï, åñéùŽ, öâàãæúèæŽ ŽáãæèŽááŽãŽéçãæáîëå, îŽáàŽê, åñ (x, y) ∈ σ ◦ σ, éŽöæê ((x, z), (z, y)) ∈ σ × σ îŽæéâz-åãæï. Žé áîëï ŽîŽãæåŽîæ àŽñàâIJîëIJŽ Žî ûŽîéëæóéêâIJŽ.åñ àŽéëãæõâêâIJå Žé ŽôêæöãêâIJï, öâãúèâIJå êâIJæïéæâîæ n ∈ N-åãæï àŽêãïŽäôãîëåσ n öâéáâàêŽæîŽá:{}σ n = (x, y) : xσz Ꭰzσ n−1 y îëéâèæéâ z-åãæï .ûæêŽ éŽàŽèæåäâσ n = { (x, y) : x, y ∈ N, y = x + n } ,ρ n = { (x, y) : x, y ∈ N, x = y 2n} .ïŽæêðâîâïëŽ, Žé öâéåýãâãŽöæ, îŽéáâêŽá éæïŽôâIJæŽ ŽêŽèëàæŽ àŽéîŽãèâ-IJŽïåŽê. ãåóãŽå, A Žîæï ïæéîŽãèâ, ýëèë R { éæéŽîåâIJŽ éŽïäâ. J A ◦ R, R áŽR◦J A âçãæãŽèâêðñîæŽ. Žéæðëé J A Žîæï A-äâ àŽêïŽäôãîñèæ æàæãñîæ éæéŽîåâIJŽ,îëéâèïŽù 1-æï ŽêŽèëàæñîæ åãæïâIJŽ Žóãï, â. æ. êŽðñîŽèñîæŽ àŽéîŽãèâIJæïëìâîŽùææï áîëï. öâãêæöêëå, îëé R −1 ◦ R = J A = R ◦ R −1 ðëèëIJæï

44áŽûâîŽ, ïŽäëàŽáëá, õëãâèåãæï Žî öâæúèâIJŽ. Žéæïåãæï ïŽüæîëŽ áŽéŽðâIJæåæìæîëIJâIJæï éëåýëãêŽ.ï Ž ã Ž î þ æ ö ë 6.1.1. ãåóãŽå, R ᎠS éæéŽîåâIJâIJæ àŽêïŽäôãîñèæ éåâè ýŽèýåŽ P ïæéîŽãèâäâ:R = { (x, y) : x, y ∈ P Ꭰx Žîæï y-æï éŽéŽ } ,S = { (x, y) : x, y ∈ P Ꭰx Žîæï y-æï óŽèæöãæèæ } .Žôûâîâå öâéáâàæ éæéŽîåâIJâIJæ: Ž) R 2 ; IJ) S 2 ; à) R ◦ S; á) S ◦ R; â) S ◦ R −1 ;ã) R −1 ◦ S; ä) R −1 ◦ S −1 ; å) S −1 ◦ R, æ) S −1 ◦ S −1 , ç) S −1 ◦ R −1 .2. Z éåâè îæùýãåŽ ïæéîŽãèâäâ éëùâéñèæŽ IJæêŽîñèæ éæéŽîåâIJâIJæ, àŽêïŽäôãîñèæöâéáâàêŽæîŽá:σ = { (x, y) : x, y ∈ Z Ꭰy = 2x } ,ρ = { (x, y) : x, y ∈ Z Ꭰx = 2y + 1 } .Žôûâîâå öâéáâàæ éæéŽîåâIJâIJæ (åñ ŽîïâIJëIJï) Ž) σ ◦ ρ; IJ) ρ ◦ σ; à) σ ◦ ρ −1 ;á) σ −1 ◦ ρ; â) ρ −1 ◦ σ −1 ; ã) ρ 2 ; ä) σ 2 .§ 7. éæéŽîåâIJâIJæï øŽçâðãŽøŽçâðãæï ùêâIJŽ ûŽîéëŽáàâêï òñêáŽéâêðñî éŽåâéŽðæçñî ùêâIJŽï ᎠàŽéëæõâêâIJŽéŽåâéŽðæçæï ñéîŽãèâï áŽîàöæ. Žé ùêâIJæï ïŽæèñïðîŽùæëá àŽêãæýæèëåöâéáâàæ éŽàŽèæåæ:Žãæôëå x 0 ëIJæâóðæ Ꭰp ìîëùâïæ, îëéâèæù ûŽîéëóéêæï ïæéîŽãèâï áŽàŽêïŽäôãîŽãï x 1 , x 2 , . . . , x n , . . . éæéáâãîëIJŽï, ïŽáŽùx 1 ∈ p(x 0 )x 2 ∈ p(x 1 ). . . . . . . . . . . .x n ∈ p(x n−1 ). . . . . . . . . . . . . . .ïæéîŽãèâ, îëéâèæù öâæùŽãï õãâèŽ éæéáâãîëIJæï õãâèŽ âèâéâêðï, îëéâèæùöâæúèâIJŽ àŽéëõãŽêæèæ æõëï p ìîëùâïæï ïŽöñŽèâIJæå, áŽûõâIJñèæ x 0 -áŽê, âûëáâIJŽp ìîëùâïæï øŽçâð㎠x 0 -æï éæéŽîå. Žéæðëé ìîëùâïæï öâáâàæ öâáæïp n (x 0 )-öæ îëéâèæôŽù n-åãæï. éŽàîŽé øãâê Žî ãæùæå ûæêŽïûŽî n-æï éêæöãêâèëIJŽ.ñòîë éâðæù, åñ ŽãæôâIJå êâIJæïéæâî y âèâéâêðï Žé øŽçâðãæáŽê áŽöâãŽïîñèâIJå p ìîëùâïï y-áŽê áŽûõâIJñèæ, ŽîŽòâîï ŽýŽèï Žî éæãæôâIJå. öâáâàïñçãâ åãæå øŽçâð㎠öâæùŽãï. ïæéîŽãèâ Žî öâæúèâIJŽ àŽòŽîåëãáâï Žïâåæàäæå (æàæ øŽçâðæèæŽ).é Ž à Ž è æ å æ 7.1. Žãæôëå s çãŽáîŽðæ, îëéâèæù éëêæöêñèæŽ æïâ, îëàëîù7.1(Ž) êŽýŽääâ êŽøãâêâIJæ ᎠàŽêãïŽäôãîëå r ìîëùâïæ öâéáâàêŽæîŽá:

45S-æï éëùâéñèæ éáâIJŽîâëIJæáŽê r ìîëùâïæ ûŽîéëóéêæï õãâèŽ éáâIJŽîâëIJæï ïæéîŽãèâï,îëéâèæù éææôâIJŽ åæï æïîæï éæéŽîåñèâIJæå éŽîåæ çñåýæå éë-IJîñêâIJæï öâáâàŽá.êŽý. 7.1ŽéàãŽîŽá, r(s) àãŽúèâãï ïñî. 7.1(IJ)-äâ àŽéëïŽýñè çëêòæàñîŽùæŽï. r-æï ëåýþâî àŽéëõâêâIJæï öâáâàŽá øãâê áŽãñIJîñêáâIJæå ïŽûõæï éáàëéŽîâëIJŽï.éŽöŽïŽáŽéâ, øŽçâð㎠éëùâéñè öâéåýãâãŽöæ Žîæï ëåýæ ìëäæùææï ïæéîŽãèâ (æý.êŽý. 7.2).êŽý. 7.2ŽýèŽ àŽêãæýæèëå îŽ éëýáâIJŽ, åñ ìîëùâïï àŽêãïŽäôãîŽãå éæéŽîåâIJâIJæïïŽöñŽèâIJæå (âï õëãâèåãæ öâïŽúèâIJâèæ, æéæðëé îëé öâïŽòâîæïæ éæéŽîåâIJŽöâàãæúèæŽ àŽêãïŽäôãîëå {(x, y) : y ∈ p(x), ïŽáŽù p öâïŽïûŽãèæ ìîëùâïæŽ}ïæéîŽãèââIJæï ïŽöñŽèâIJæå. øŽçâðãæï ŽïŽàâIJŽá ïŽçéŽîæïæŽ A éæéŽîåâIJŽï ßóëêáâïA, A 2 , . . . , A n , . . . öâáàâêæèæ éæéŽîåâIJâIJæ, îëéèâIJæù öâéáâà çëéIJæêæîáâIJŽøãâñèâIJîæãæ åâëîæñè-ïæéîŽãèñîæ àäæå.à Ž ê ï Ž ä ô ã î â IJ Ž . ïæéîŽãèâäâ àŽêïŽäôãîñèæ A éæéŽîåâIJæïðîŽêäæðñèæ øŽçâð㎠(Žê ñIJîŽèëá øŽçâðãŽ) âûëáâIJŽ ñïŽïîñèë àŽâîåæŽêâ-IJŽï∞⋃A n = A ∪ A 2 ∪ A 3 ∪ · · · .n=1A éæéŽîåâIJæï ðîŽêäæðñèæ øŽçâð㎠ŽôæêæöêâIJŽ A + ïæéIJëèëåæ.é Ž à Ž è æ å æ 7.2.

461. ãåóãŽå, êŽðñîŽèñî îæùýãåŽ ïæéîŽãèâäâ àŽêïŽäôãîñèæŽ R éæéŽîåâIJŽR= { (x, y) : y =x+1 } ðëèëIJæå, éŽöæê R + ={(x, y) : x

47é Ž à Ž è æ å æ 7.3. ûæêŽ éŽàŽèæåâIJöæ àŽêïŽäôãîñèæ éæéŽîåâIJâIJæïàŽéëõâêâIJæå éæãæôâIJå:R ∗ = { (x, y) : x ≤ y, x, y ∈ N } ,σ ∗ = { (x, y) : x ≤ y, x, y ∈ Q } ,ρ ∗ = ρ + ∪ {0, 0},N ∗ = N + ,ï Ž ã Ž î þ æ ö ë 7.1.τ ∗ = (Z (1) × Z (1) ) ∪ (Z (2) × Z (2) ).1. àŽéëæõâêâå 6.1(1) ïŽãŽîþæöëöæ éëùâéñèæ R ᎠS éæéŽîåâIJâIJæ ᎎôûâîâå öâéáâàæ øŽçâðãâIJæ: Ž) R + ; IJ ) S + ; à) R ∗ ; á) S ∗ ; â) (S ◦ S −1 ) + ; ã)(R −1 ◦ R) ∗ ; ä) (S 2 ◦ R 2 ) + .2. 6.1(2) ïŽãŽîþæöëöæ éëùâéñèæ σ, ρ éæéŽîåâIJâIJæïŽåãæï Žôûâîâå øŽçâðãâIJæ:Ž) σ + ; IJ) σ ∗ ; à) (ρ −1 ) + ; á) (ρ −1 ) ∗ .

Žîæåéâðæçæï úæîæåŽáæ ùêâIJâIJæ§ 1. \éùæîâ" ïŽïîñèæ Žîæåéâðæ玎îæåéâðæçŽ öâàãæúèæŽ àŽêãæýæèëå, îëàëîù ïæéîŽãèâ ëîæ ëìâîŽùææå,îëéèâIJæù öâçîâIJæïŽ áŽ àŽéîŽãèâIJæï éïàŽãïŽá éëóéâáâIJï. æï öâæúèâIJŽ öâãæïûŽãèëåéîŽãŽèæ ýâîýæå. îëé àŽãŽîçãæëå Žîæåéâðæçñèæ ïæïðâéæï éëåýëãêâIJæ,éæãæôëå çëêïðîóñùæñèæ éæŽýèëâIJŽ, Ꭰéåâèæ îæùýãâIJæ (0, 1, 2, . . .)àŽêãæýæèëå, îëàëîù ïæéIJëèëâIJæ öâéáàëéöæ àŽêãæýæèŽãå éýëèëá ïŽïîñèŽîæåéâðæçŽï, îëéâèöæù àŽéëæõâêâIJŽ éýëèëá îæùýãåŽ ïŽïîñèæ ïæéîŽãèâ.ãæàñèæïýéëå, îëé åñ A ∼ N m , éŽöæê éëàãâåýëãâIJŽ m ïýãŽáŽïý㎠ïæéIJëèë.ŽéŽïåŽê, ïæéIJëèëâIJæï ŽîŽãæåŽîæ çëéIJæêŽùææï ñòèâIJŽ Žî àãŽóãï. åñ éýëèëáŽåëIJæåæ îæùýãâIJæ àŽéëæõâêâIJŽ, éŽöæê m ≤ 10. îŽáàŽê éëùâéñèæ äëéæïõãâèŽ ïæéîŽãèâ IJæâóùæñîæŽ, öâàãæúèæŽ àŽêãæýæèëå éýëèëá N m ïæéîŽãèâ.ñòîë éâðæ åãŽèïŽøæêëâIJæïåãæï àŽêãæýæèëå N 6 ïæéîŽãèâ. Žéæïåãæï ŽñùæèâIJâè掎ãŽàëå öâçîâIJæï ᎠàŽéîŽãèâIJæï ëìâîŽùæŽåŽ ùýîæèâIJæ.öâçîâIJæï ëìâîŽùæŽ öâæùŽãï êâæðîŽèñî âèâéâêðï, îëéâèæù øãâñèâIJîæã,o ïæéIJëèëåæ ŽôæêæöêâIJŽ. éŽàîŽé o ∉ N 6 . Žéæðëé àŽéëãæõâêëå Z 6 ={0, 1, 2, 3, 4, 5} ïæéîŽãèâ. ùýŽáæŽ, Z 6 ∼ N 6 . ŽéŽïåŽê, Žîùâîåæ åãæïâIJŽ ŽîáŽàãâçŽîàâIJŽ. ŽéîæàŽá, åñ ãæïŽîàâIJèâIJå êñèëãŽêæ âèâéâêðæï åãæïâIJæå,àãâóêâIJŽ 1.1 ùýîæèæ.+ 0 1 2 3 4 50 01 12 23 34 45 5ùýîæèæ 1.1.îŽáàŽê öâçîâIJæï ëìâîŽùæŽ çëéñðŽùæñîæŽ, ùýîæèæ ïæéâðîæñèæ ñêáŽæõëï. åñ éëãæåýëãå, îëé õëãâèæ âèâéâêðæïŽåãæï ŽîïâIJëIJáâï éëìæîáŽìæîââèâéâêðæ, éŽöæê êâIJæïéæâîæ âèâéâêðæ åæåëâñè ïðîæóëêïŽ áŽ åæåëâñèïãâðöæ ñêᎠöâãæáâï äñïðŽá âîåýâè. éŽîåèŽù, åña + b = a + c,49

50éŽöæê−a + (a + b) = −a + (a + c),(−a + a) + b = (−a + a) + c,0 + b = 0 + c,b = c.öâãêæöêëå, îëé Žó àŽéëãæõâêâå, Žïâãâ, öâçîâIJæï ëìâîŽùææï ŽïëùæŽùæñîëIJŽ.àŽêãæýæèëå 1.2 ùýîæèæå àŽêïŽäôãîñèæ öâçîâIJæï ïŽéæ ëìâîŽùæŽ: A,B ᎠC. A ëìâîŽùæŽ ŽîŽçëéñðŽùæñîæŽ, îŽáàŽê ýîæèæ Žî Žîæï ïæéâðîæñèæ,B ëìâîŽùæŽöæ áŽîôãâñèæŽ \öâáâàæï âîåŽáâîåëIJæï" çîæðâîæñéâIJæ.C ŽçéŽõëòæèâIJï õãâèŽ ìæîëIJŽï.A 0 1 2 3 4 50 0 1 2 3 4 51 1 2 0 4 5 32 2 0 1 5 3 43 3 5 4 0 2 14 4 3 5 1 0 25 5 4 3 2 1 0B 0 1 2 3 4 50 0 1 2 3 4 51 1 1 2 3 4 52 2 2 2 3 4 53 3 3 3 3 4 54 4 4 4 4 4 55 5 5 5 5 5 5ùýîæèæ 1.2.C 0 1 2 3 4 50 0 1 2 3 4 51 1 5 3 4 2 02 2 3 1 5 0 43 3 4 5 0 1 24 4 2 0 1 5 35 5 0 4 2 3 1îëàëî ŽãŽàëå ëìâîŽùæŽ, îëéâèæù ᎎçéŽõëòæèâIJï äâéëå àŽêýæèñèõãâèŽ åãæïâIJŽï? õãâèŽäâ îåñèŽá öâïŽïîñèâIJâèæŽ ŽïëùæŽùæñîëIJŽ. óãâéëåöâéëåŽãŽäâIJñè ìîëùâáñîŽöæ ŽïëùæŽùæñîëIJŽï àŽéëãæõâêâIJå, îëàëîù Žàâ-IJæï úæîæðŽá êŽIJæþï áŽ, éŽöŽïŽáŽéâ, âï åãæïâIJŽ öâïîñèáâIJŽ ŽãðëéŽðñîŽá.êŽIJæþæ 1. öâçîâIJæï ëìâîŽùææïåãæï o Žîæï êñèëãŽêæ âèâéâêðæ, ŽéæðëéãôâIJñèëIJå 1.3 ùýîæèï+ 0 1 2 3 4 50 0 1 2 3 4 51 12 23 34 45 5ùýîæèæ 1.3.êŽIJæþæ 2. àŽêãïŽäôãîëå ùýîæèæï öâéáâàæ ïðîæóëêæ, îëéâèæù ŽçéŽõëòæèâIJï\öâáâàæï âîåŽáâîåëIJæï" ìæîëIJŽï. ýŽäæ îëé àŽãñïãŽå àŽéëõâêâIJñèðâóêæçŽï, ïìâùæŽèñîŽá ãæîøâãå öâáâàï, îëéâèæù àŽêïýãŽãáâIJŽ øãâñèâIJîæãæïàŽê.Žãæôëå:+ 0 1 2 3 4 51 1 3 0 5 2 4

51îŽáàŽê ëìâîŽùæŽ çëéñðŽùæñîæ ñêᎠæõëï, öâãŽãïëå 1.4 ùýîæèæï öâïŽIJŽéæïæïãâðæ+ 0 1 2 3 4 50 0 1 2 3 4 51 1 3 0 5 2 42 2 03 3 54 4 25 5 4ùýîæèæ 1.4.êŽIJæþæ 3. öâãŽãïëå ùýîæèæï áŽêŽîøâêæ ñþîâIJæ ŽïëùæŽùæñîëIJæï àŽéëõâêâIJæå.åæåëâñè áâðŽèï áŽãŽçãæîáâå áŽûãîæèâIJæå2 + 2 = 2 + (1 + 4) = 2 + 1 = 0 + 4 = 4,2 + 3 = 2 + (1 + 1) = (2 + 1) + 1 = 0 + 1 = 1,2 + 4 = 2 + (1 + 5) = (2 + 1) + 5 = 0 + 5 = 5,2 + 5 = 2 + (1 + 3) = (2 + 1) + 3 = 0 + 3 = 3.Žó àŽéëãæõâêâå 2 + 1 = 0 Ꭰ0 + x = x åŽêŽòŽîáëIJâIJæ. öâéáâà3 + 3 = 3 + (1 + 1) = (3 + 1) + 1 = 5 + 1 = 4,3 + 4 = (1 + 1) + 4 = 1 + (1 + 4) = 1 + 2 = 0,3 + 5 = (1 + 1) + 5 = 1 + (1 + 5) = 1 + 4 = 2ᎠŽ. ö. ŽéàãŽîŽá, (1 + x)-æï éêæöãêâèëIJâIJæï ïŽòæúãâèäâ + ëìâîŽùææïåãæïãôâIJñèëIJå ùýîæèï:+ 0 1 2 3 4 50 01 1 3 0 5 2 42 2 0 4 1 5 33 3 5 1 4 0 24 4 2 5 0 3 15 5 4 3 2 1 0ùýîæèæ 1.5.àŽáŽãæáâå àŽéîŽãèâIJæï ëìâîŽùæŽäâ. àŽéîŽãèâIJæïåãæï êâæðîŽèñîæ âèâéâêðæŽâîåâñèëãŽêæ âèâéâêðæ. öâãåŽêýéáâå, îëé âîåâñèëãŽêæ âèâéâêðæàŽêïýãŽãáâIJŽ 0-àŽê, îŽáàŽê, ûæꎎôéáâà öâéåýãâãŽöæ, êâIJæïéæâîæ x Ꭰy-åãæïàãâóêâIJëáŽx = 0 · x = x · 0, y = 0 + y = y + 0,Žéæðëéx · y = x · (0 + y) = (x · 0) + (x · y) = x + (x · y),â. æ. x êñèëãŽêæ âèâéâêðæŽ: x = 0 âï Žî Žîæï ïûëîæ, Žéæðëé, 0 Žî ŽîæïàŽéîŽãèâIJæïåãæï êâæðîŽèñîæ âèâéâêðæ. âîåâñèëãŽê âèâéâêðŽá Žãæôëå 1.

521.6 ùýîæèæï ïŽòñúãâèäâ öâàãæúèæŽ ŽêŽèëàæñîŽá àŽêãïŽäôãîëå àŽéîŽãèâIJæïëìâîŽùæŽ. ŽéŽïåŽê, Žé öâéåýãâãŽöæ Žî ñêᎠéëãæåýëãëå \öâáâàæïâîåŽáâîåëIJæï" çîæðâîæñéæï öâïîñèâIJŽ. àŽéëãæõâêëå àŽéîŽãèâIJæï ëìâîŽùææïáæïðîæIJñùæñèëIJŽ öâçîâIJæï éæéŽîå.· 0 1 2 3 4 50 01 0 1 2 3 4 52 23 34 45 5ùýîæèæ 1.6.öâãêæöêëå, îëé 1 + 1 = 3. Žéæðëé áæïðîæIJñùæñèëIJæï àŽéë éæãæôâIJå3 · 0 = (1 + 1) · 0 = 1 · 0 + 1 · 0 = 0 + 0 = 0,3 · 2 = (1 + 1) · 2 = 1 · 2 + 1 · 2 = 2 + 2 = 4,3 · 3 = (1 + 1) · 3 = 1 · 3 + 1 · 3 = 3 + 3 = 4,3 · 4 = (1 + 1) · 4 = 1 · 4 + 1 · 4 = 4 + 4 = 3,3 · 5 = (1 + 1) · 5 = 1 · 5 + 1 · 5 = 5 + 5 = 0.ŽýèŽ öâæúèâIJŽ ãæïŽîàâIJèëå 3 + 3 = 4, 3 + 1 = 5, 1 + 4 = 2, 1 + 2 =0 åŽêŽòŽîáëIJâIJæå. åñ æïâ ãæéëóéâáâIJå, îëàëîù ûæêŽå, éæãæôâIJå öâéáâàëìâîŽùæŽï· 0 1 2 3 4 50 0 0 0 0 01 0 1 2 3 4 52 0 2 1 4 3 53 0 3 4 4 3 04 0 4 3 3 4 05 0 5 5 0 0 5ùýîæèæ 1.7.éŽöŽïŽáŽéâ, ùýîæèöæ ïðîæóëêæï åæåóéæï êâIJæïéæâîæ öâîøâãæå áŽûõâ-IJñèæ Ꭰîæàæ éŽîðæãæ öâäôñáãâIJæï áŽáâIJæå éæïŽôâIJæ Žîæåéâðæçñèæ ïæïðâ鎎.ŽýèŽ ïŽçéŽîæïæŽ áŽãŽáàæêëå, îëé éæôâIJñèæ ïæïðâéŽ Žî âûæꎎôéáâàâIJŽûæêŽå àŽéëåóéñè éëïŽäîâIJâIJï. åñ êëîéŽèñî ŽîæåéâðæçŽöæ a+b = cᎠc ∈ Z 6 , éŽöæê ïŽïñîãâèæŽ øãâêï ŽîæåéâðæçŽöæù (a + b)-ï ìŽïñýæ æõëï c.éŽöŽïŽáŽéâ, éæãâáæå ŽîøâãêŽéáâ+ 0 1 2 3 4 51 1 2 3 4 5 ?âèâéâêðæ, îëéâèæù ŽçèæŽ, ñêᎠæõëï 0, îŽáàŽê 6 ∉ Z 6 Ꭰ0 Žîæï Z 6 -æïâîåŽáâîåæ âèâéâêðæ, îëéâèæù Žî Žîæï Žé ïðîæóëêöæ. Žïâåæ öâîøâãæï öâáâàŽáãôâIJñèëIJå 1.8 ùýîæèï. æàæ àŽêïŽäôãîŽãï â. û. ŽîæåéâðæçŽï éëáñèæå

536. (âï ŽîæåéâðæçŽ äñïðŽá æïâ éëóéâáâIJï, îëàëîù øãâñèâIJîæãæ éåâèîæùýãâIJæŽêæŽîæåéâðæçŽ, æé àŽéëêŽçèæïæå, îëé õãâèŽ éåâèæ îæùýãæ öâùãèæèæŽéŽåæ 6-äâ àŽõëòæå éæôâIJñèæ êŽöåæå.)ùýîæèæ 1.8.+ 0 1 2 3 4 50 0 1 2 3 4 51 1 2 3 4 5 02 2 3 4 5 0 13 3 4 5 0 1 24 4 5 0 1 2 35 5 0 1 2 3 4· 0 1 2 3 4 50 0 0 0 0 0 01 0 1 2 3 4 52 0 2 4 0 2 43 0 3 0 3 0 34 0 4 2 0 4 25 0 5 4 3 2 1ï Ž ã Ž î þ æ ö ë 1.1.1. Z 6 -åãæï éæôâIJñèæ \IJñêâIJîæãæ" Žîæåéâðæçæï éïàŽãïŽá ŽãŽàëå ŽêŽèëàæñîæŽîæåéâðæçŽ Z 16 -åãæï {0, 1, . . . , 9, A, B, C, . . . , F } ïæéIJëèëâIJæï àŽéëõâêâIJæå.2. ŽãŽàëå ŽîæåéâðæçŽ Z 6 -åãæï, îëéâèæù öââïŽIJŽéâIJŽ öâéáâà ïðîæóëêï:+ 0 1 2 3 4 52 2 3 4 0 5 13. (1+3)-æï àŽêýæèãæå ãŽøãâêëå, îëé öâéáâà ùýîæèï éæãõŽãŽå ûæꎎôéáâàëIJŽéáâ:+ 0 1 2 3 4 50 0 1 2 3 4 51 1 0 3 4 5 2§ 2. \áæáæ" ïŽïîñèæ ŽîæåéâðæçŽøãâê ñçãâ ŽãŽàâå ŽîæåéâðæçŽ Z 6 -åãæï. àŽãŽòŽîåëãëå âï ïæïðâéŽ. ïŽæèñïðîŽùæëáàŽêãæýæèëå áŽèŽàâIJñèæ ïŽéâñèæ Z 6 -áŽê, Žêñ Z 6 × Z 6 × Z 6 ïæéîŽãèæïâèâéâêðâIJæ. åñ Z 6 áŽèáŽàâIJñèæŽ øãâñèâIJîæãæ ýâîýæå, â. æ. 0

54(0, 1, 0), (0, 1, 1), . . . , (0, 1, 5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(0, 5, 0), . . . . . . . . . . . . , (0, 5, 5)(1, 0, 0), . . . . . . . . . . . . , (1, 0, 5)(5, 5, 0), (5, 5, 1), . . . , (5, 5, 5)ŽéîæàŽá, ŽîïâIJëIJï 6 3 = 216 ïýãŽáŽïý㎠ïŽéâñèæ. äâéëå éëõãŽêæèæ Z 3 6âèâéâêðâIJæï áŽèŽàâIJæïŽï (0,0,5) ñöñŽèëá (0,1,0)-æï ûæê Žîæï, (0,1,5) ñöñ-Žèëá ûæê ñïûîâIJï (0,2,0)-ï ᎠŽ. ö. (0,5,5) ñöñŽèëá ûæê ñïûîâIJï (1,0,0)-ï.éŽöŽïŽáŽéâ, ïŽïñîãâèæ æóêâIJëᎠïîñèáâIJëáâï(0, 0, 5) + 1 = (0, 1, 0).ŽéŽïåŽê, ŽóŽéáâ Žî æõë 1-æï ûŽîéëáàâêŽ Z 3 6-öæ. éæñýâáŽãŽá æéæïŽ, îëé âïáŽèŽàâIJŽ IJñêâIJîæãæŽ, ñêᎠãæäîñêëå æéŽäâ, îëé Žî àŽéëãæõâêëå ŽîùâîåæùêâIJŽ, îëéèæï àŽêïŽäôãîâIJŽù Žî öâéëàãæðŽêæŽ. Žôûâîæï àŽïŽŽáãæèâIJèŽáZ 3 6 ûŽîéëãŽáàæêëå, îëàëîù A 2 ×A 1 ×A 0 ᎠàŽêãæýæèëå (a 2 , a 1 , a 0 )-æï áŽ(b 2 , b 1 , b 0 )-æï þŽéæ. çëéìëêâêðâIJŽá öâçîâIJŽ àãŽúèâãï (a 2 +b 2 , a 1 +b 1 , a 0 +b 0 ),ïŽáŽù öâçîâIJŽ ýáâIJŽ Z 6 .àŽêãæýæèëå éŽàŽèæåæé Ž à Ž è æ å æ 2.1. àŽêãæýæèëå þŽéæ(0, 1, 3) + (4, 2, 1) = (4, 3, 4),îëéâèæù ñòîë åãŽèïŽøæêë æóêâIJŽ, åñ éŽï øŽãûâîå öâéáâàæ ïŽýæååñéùŽ1, 2, 5+ 2, 2, 1= 3, 5, 00, 1, 3+ 4, 2, 1= 4, 3, 40, 0, 5+ 0, 0, 1 (= 1?)= 0, 0, 0 (= 0?)öâçîâIJæï ëìâîŽùææï öâáâàŽá A 0 ïæéîŽãèâ àŽáŽáæï åŽãæï åŽãöæ. éŽàîŽéæéæïŽåãæï, îëé ïŽçéŽëá áæáæ îæùýãâIJæï þŽéï (æïâåï, îëàëîæù 5 +1-æŽ) öââúèëï àŽéëãæáâï A 0 -æï ïŽäôãîâIJæáŽê, ïŽüæîëŽ ãŽûŽîéëëå îŽôŽùéëóéâáâIJŽ A 1 -öæ áŽ, Žïâãâ, öâïŽúèâIJâèæŽ A 2 -öæ (âï êŽøãâêâIJæŽ 2.1 ùýîæèöæ).ùýîæèæ 2.1.+ s 0 1 2 3 4 50 0 1 2 3 4 51 1 2 3 4 5 02 2 3 4 5 0 13 3 4 5 0 1 24 4 5 0 1 2 35 5 0 1 2 3 4+ c 0 1 2 3 4 50 0 0 0 0 0 01 0 0 0 0 0 12 0 0 0 0 1 13 0 0 0 1 1 14 0 0 1 1 1 15 0 1 1 1 1 1

55+ s -æå ŽôêæöêñèæŽ öâçîâIJæï ëìâîŽùæŽ Z 6 -öæ (â. æ. êŽöåæ, îëéâèæù éææôâIJŽþŽéæï 6-äâ àŽõëòæïŽï), ýëèë + c ùýîæèöæ éëùâéñèæŽ àŽêŽõëòæ, îëéâèæùéææîâIJŽ þŽéæï 6-äâ àŽõëòæïŽï). Žãæôëå êâIJæïéæâîæ a Ꭰb îæùýãâIJæ Z 6 -áŽê.éŽöæê éŽåæ þŽéæ (Z-öæ) æóêâIJŽa + b = 6 · (a + c b) + (a + s b).é Ž à Ž è æ å æ 2.1. 4 + 4 = 6 · (4 + c 4) + (4 + s +4) = 6 · 1 + 2 = 8.åñ 0 ≤ x ≤ n − 1 Ꭰ0 ≤ x ≤ n − 1, Žéæðëé0 ≤ x + y ≤ 2n − 2 < 2n − 1 (n = 6 Z 6 − öæ).åñ áŽãñIJîñêáâIJæå (a 2 , a 1 , a 0 )-æïŽ áŽ (b 2 , b 1 , b 0 )-æï öâçîâIJŽï ᎠìŽïñýïŽôãêæöêŽãå (d 2 , d 1 , d 0 )-æå, éæãæôâIJåd 0 = a 0 + s b 0 ,x 0 = a 0 + c b 0 ,d 1 = a 1 + s b 1 + s x 0 ,{1, åñ a 1 + c b 1 = 1x 1 =(a 1 + s b 1 ) + c x 0 , åñ a 1 + c b 1 ≠ 1d 1 = (a 2 + s b 2 ) + s x 1 ,{1, åñ a 2 + c b 2 = 1x 2 =(a 2 + s b 2 ) + c x 1 , åñ a 2 + c b 2 ≠ 1 .îŽáàŽê 0 ≤ a i + b i < 2n − 1 Ꭰx 0 = 0 Žê x i = 1, Žéæðëé àŽáŽïŽðŽêæ öâáâàæ(a i + b i + x i−1 )-áŽê 1-äâ éâðæ ŽîŽïëáâï æóêâIJŽ. æé öâéåýãâãŽï, îëùŽ x 2 > 0,éëàãæŽêâIJæå àŽêãæýæèŽãå.öâãêæöêëå, îëé Z 3 6-æï êñèëãŽêæ âèâéâêðæŽ (0,0,0), ýëèë âîåâñèëãŽêæâèâéâêðæ { (0,0,1).ŽêŽèëàæñîŽá àŽêæïŽäôãîâIJŽ àŽéîŽãèâIJæï ëìâîŽùæŽ.·p 0 1 2 3 4 50 0 0 0 0 0 01 0 1 2 3 4 52 0 2 4 0 2 43 0 3 0 3 0 34 0 4 2 0 4 25 0 5 4 3 2 1·c 0 1 2 3 4 50 0 0 0 0 0 01 0 0 0 0 0 02 0 0 0 1 1 13 0 0 1 1 2 24 0 0 1 2 2 35 0 0 1 2 3 4ùýîæèæ 2.2.·p ùýîæèöæ éëùâéñèæŽ êŽöåæ, îëéâèæù éææôâIJŽ Z 6 -æï âèâéâêðâIJæï êŽéîŽãèæï6-äâ àŽõëòæïŽï (â. æ. + p Žîæï Z 6 -öæ àŽêïŽäôãîñèæ êŽéîŽãèæï ëìâîŽùæŽ),ýëèë + c ùýîæèöæ éëùâéñèæŽ àŽêŽõëòæ, îëéâèæù éææôâIJŽ Z 6 -æïâèâéâêðâIJæï êŽéîŽãèæï 6-äâ àŽõëòæïŽï.ŽóŽéáâ øãâê ãæýæèŽãáæå éýëèëá æé ïæéIJëèëâIJï, îëéèâIJïŽù Žóãï áŽáâ-IJæåæ îæùýãâIJæï ïŽýâ. ùýŽáæŽ, 0, 1, 2, . . . ïæéIJëèëâIJï éëãâóùâå \IJñêâIJîæãŽá"

56áŽ, éŽöŽïŽáŽéâ, éŽåæ æêðâîìîâðŽùæŽ öâæúèâIJŽ éëãŽýáæêëå, îëàëîù ŽîŽñŽîõëòæåæîæùýãâIJæ. Z 3 6-äâ àŽêïŽäôãîñèæ ŽîæåéâðæçŽ éëóéâáâIJï îæùýãâIJäâ0 = (0, 0, 0)-áŽê 215 = (5, 5, 5)-éáâ, îëéèâIJæù éæôâIJñèæŽ (0-áŽê 5-éáâ) éæéáâãîëIJæáŽêZ 6 -öæ. åñ Z 6 -æï êŽùãèŽá ŽãæôâIJå {−3, −2, −1, 0, 1, 2} ïæéîŽãèâï,éæãæôâIJå ñŽîõëòæå îæùýãâIJï, âï ïæéîŽãèâ Žôãêæöêëå Z − 6 -æå. àŽêãæýæèëåZ − 6 ×Z 6 ×Z 6 ïæéîŽãèâ. éŽöæê öâàãæúèæŽ ŽãŽàëå ŽîæåéâðæçŽ îæùýãâIJäâ−108-áŽê 107-éáâ. ïæêŽéáãæèâöæ ŽîæåéâðæçŽ æàæãâŽ, îŽù Z 3 6-öæ, éýëèëá A 2ïæéîŽãèæï 3, 4 Ꭰ5 îæùýãâIJæ öâïŽIJŽéæïŽá æùãèâIJŽ −3, −2 Ꭰ−1-æå. Žéæðëé,éŽàŽèæåŽá, (−2, 4, 2) Z-öæ àŽéëæåãèâIJŽ, îëàëîù(−2 · 36) + (4 · 6) + 2 = −46.Z 6 -ïŽ áŽ Z −16 ïæïðâéâIJï öëîæï IJæâóùæŽ àŽêæïŽäôãîâIJŽ öâéáâàêŽæîŽá3 ↦−→ −34 ↦−→ −25 ↦−→ −1x ↦−→ x áŽêŽîøâê öâéåýãâãâIJöæ.àŽéëåãèâIJæ, îëéèâIJæù öâæùŽãï öâçîâIJŽï ᎠàŽéëçèâIJŽï. ŽýŽè ŽîæåéâðæçŽöæïŽçéŽëá éŽîðæãæŽ áŽ áŽéëçæáâIJñèæŽ ëî æàæãâëIJŽäâ:áŽ(5, 5, 5) + (0, 0, 1) = (0, 0, 0)(a 2 , a 1 , a 0 ) + (b 2 , b 1 , b 0 ) = (5, 5, 5).éŽå Žáàæèæ Žóãï éŽöæê Ꭰéýëèëá éŽöæê, îëùŽa 2 + b 2 = 5, a 1 + b 1 = 5, a 0 + b 0 = 5.ŽéîæàŽá, æéæïŽåãæï, îëé àŽéëãåãŽèëå (a 2 , a 1 , a 0 ) îæùýãæï éëìæîáŽìæîâîæùýãæ (b 1 , b 1 , b 0 ), þâî ñêᎠãæìëãëå éëùâéñèæ îæùýãæï áŽéŽðâIJŽ(5,5,5)-äâ öâéáâà éæãñéŽðëå 1 = (0, 0, 1).é Ž à Ž è æ å æ 2.3. ãæìëãëå (3,4,1)-æï éëìæîáŽìæîâ îæùýãæ.3, 4, 1+ 2, 1, 41= 0, 0, 0(5-éáâ áŽéŽðâIJŽ),â. æ. (2,1,5) Žîæï (3,4,1)-æï éëìæîáŽìæîâ:3, 4, 1+ 2, 1, 5= 0, 0, 0â. æ. −(3, 4, 1) = (2, 1, 5), Žïâãâ −(−3, 4, 1) = (2, 1, 5).

57é Ž à Ž è æ å æ 2.4. àŽéëãåãŽèëå (1, 3, 4)−(2, 1, 5). ãæìëãëå −(2, 1, 5).â. æ. −(2, 1, 5) = (3, 4, 1).öâéáâà éæãñéŽðëå (1,3,4)-ï:2, 1, 5+ 3, 4, 01= 3, 4, 11, 3, 4+ 3, 4, 1= 5, 1, 5Žïâãâ (5.1.5) = (−1, 1, 5) Z − 6 -öæ.öâãŽéëûéëå éæôâIJñèæ öâáâàæ Z-öæ. àãŽóãï: (1, 3, 4) = 1·6 2 +3·6+4 = 58,(2, 1, 5) = 2 · 6 2 + 1 · 6 + 5 = 83, (5, 1, 5) = (−1, 1, 5) = −6 2 + 6 + 5 = −25,58 − 83 = −25.ïŽäëàŽáëá, åñ àŽéëåãèâIJæ ûŽîéëâIJï N m -öæ (Z m ), ñêᎠàŽéëãæõâêëåáŽéŽðâIJŽ (m − 1)-éáâ Ꭰm-éáâ.ö â ê æ ö ã ê Ž . Z n m-æï îæùýãæï øŽûâîæïŽï, øãâñèâIJîæã, àŽéëæðëãâIJŽéúæéââIJæ ᎠõãâèŽ êñèæ, îëéâèæù áàŽï ìæîãâèæ ŽîŽêñèëãŽêæ ùæòîæï éŽîùýêæã.éŽöŽïŽáŽéâ, (1,3,4) øŽæûâîâIJŽ, îëàëîù 134; (0,0,6) îëàëîù 6, (0,0,0)îëàëîù 0.áŽï Ž ã Ž î þ æ ö ë 8.1. Z − m × Z n−1m Žôãêæöêëå, îëàëîù Z n−m , ïŽáŽù{Z − m = − m 2 , −m 2 + 1, . . . , 0, 1, . . . , m }2 − 1 , åñ m èñûæŽ{Z = − m − 1 , . . . , 0, 1, . . . , m − 1 }, åñ m çâêðæŽ.22àŽéëåãŽèâå: Ž) 10 − 7; IJ) (−8) + (−21) öâéáâà ïæïðâéâIJöæZ 4−7 , Z3− 10 , Z5− 5 , . . . , Z2− 12 .(ö â ê æ ö ã ê Ž . îæùýãâIJæ éŽàŽèæåöæ éëùâéñèæŽ Z-äâ. Žéæðëé åŽãæáŽêñêᎠàŽáŽãæõãŽêëå öâïŽIJŽéæï ïæïðâéŽöæ, ýëèë öâéáâà àŽéëãåãŽèëå).§ 3. ëîëIJæåæ ŽîæåéâðæçŽZ n m ᎠZ n−m ïæéîŽãèââIJäâ ñçãâ ŽàâIJñèæ ŽîæåéâðæçæáŽê ŽáãæèŽá öâàãæúèæŽàŽéëãõëå ëîëIJæåæ Žîæåéâðæçæï ïŽòñúãèâIJæ. ŽîïâIJëIJï ëîæ, â. û.ëîëIJæåæ ŽîæåéâðæçŽ. ìæîãâèæ â êæöêæŽêæ ᎠéëáñèæŽêæ òëîéŽ, îëéâèæùàŽêïŽäôãîñèæŽ {−, +} × ′ bZ2 n -äâ. â. æ. Z n 2 -äâ áŽéŽðâIJñèæ êæöêæå,êæöêæï çëáæîâIJŽ, øãâñèâIJîæã, ýáâIJŽ IJæêŽîñèæ òëîéæå: 0 \+"-åãæï Ꭰ1\−"-åãæï. éâëîâ ŽîæåéâðæçŽ (áŽéŽðâIJæï ëîëIJæåæ ŽîæåéâðæçŽ) Žîæï Z n−2 ,,.

58{0, 1} âèâéâêðâIJæå õãâèŽ ìëäæùæŽöæ. ëîëIJæåæ Žîæåéâðæçæï âï ïŽýâ àŽéëæõâêâIJŽçëéìæñðâîâIJæï ñéîŽãèâïëIJŽöæ. Žéæðëé øãâê àŽêãæýæèŽãå éýëèëá Z n−2 -ï. éïþâèëIJŽ ñòîë çëêçîâðñèæ îëé àŽãýŽáëå, àŽêãæýæèëå Z 5−2 ïæéîŽãèâ,æàæ öâæùŽãï 2 5 = 32 âèâéâêðï, îëéèâIJæù éëåŽãïâIJñèæŽ −16-ᎠᎠ+15-ïöëîæï (−16 Žé ïæïðâéŽöæ ûŽîéëáàâIJŽ, îëëàîù (10000) = −2 4 + 0 · 2 3 + 0 ·2 2 +0·2 1 +0·2 0 , ýëèë 15, îëàëîù (01111) = 0·2 4 +1·2 3 +1·2 2 +1·2 1 +1·2 0 ).îæùýãæ −1 ûŽîéëáàâIJŽ, îëàëîù 11111 : −2 4 +1·2 3 +1·2 2 +1·2 1 +1·2 0 , îæùýãæ1 ûŽîéëáàâIJŽ, îëàëîù ãæùæå, 00001 : 0·2 4 +0·2 3 +0·2 2 +0·2 1 +1·2 0 .ëîëIJæå ùæòîï IJæðæ âûëáâIJŽ. æéæïŽåãæï, îëé ãæìëãëå îŽæéâ îæùýãåŽ éëìæîáŽìæîâîæùýãæ Z 5 2-öæ (Žêñ áŽéŽðâIJŽ 2-éáâ), þâî ñêᎠãæìëãëå áŽéŽðâIJŽ11111-éáâ. öâéáâà 1-æï éæéŽðâIJæå éæãæôâIJå áŽéŽðâIJŽï 2-éáâ.é Ž à Ž è æ å æ 3.1. ãæìëãëå { (01011)0 1 0 1 11 0 1 0 0+1= 1 0 1 0 1{ (áŽéŽðâIJŽ 11111-éáâ),(01001) aris 2 3 + 2 1 + 2 0 = 11, xolo 10101 = −2 4 + 2 2 + 2 0 = −11-(1 0 1 1 0)0 1 0 0 1+1= 0 1 0 1 0(10110 Žîæï −2 4 + 2 2 + 2 1 = −10, ýëèë 01010 { 2 3 + 2 1 = 10).ùýŽáæŽ, îæùýãåŽ öâéëïŽäôãîñèëIJŽé öâæúèâIJŽ àŽéëæûãæëï ìîëIJèâéâ-IJæ, îëéâèåŽ ŽùæèâIJŽï ãâî öâãúèâIJå, éŽàîŽé ñêᎠãæùëáâå, îëáæï ŽîæïöâïŽúèâIJâèæ \öâùáëéŽ". áŽéŽðâIJæï òëîéŽ àŽáŽãïâIJæï ìæîëIJæï öâéëûéâIJŽïàŽîçãâñèŽá ŽŽáãæèâIJï, éýëèëá ñòîëïæ êæöêŽáæ IJæðæï àŽéëõâêâIJæï àŽéëZ 5−2 -öæ Žî îŽæëï IJæðæ 24 êëéîæå). âï IJæðæ ŽôêæöêŽãï ûŽîéëáàâêæèæ îæùýãæïêæöŽêï ᎠâûëáâIJŽ êæöŽêæï IJæðæ Žê êæöêæï åŽêîæàæ. ãæáîâ öâãŽéëûéâIJáâå,åñ îëàëîæ éêæöãêâèëIJŽ Žóãï êæöêæï IJæðï, àŽãæýïâêëå, îëé Z n m-öæéŽóïæéŽèñî áŽáâIJæå îæùýãäâ 1-æï éæéŽðâIJŽ àãŽúèâãï éŽóïæéŽèñî ñŽîõëòæåîæùýãï (ñáæáâïæ ñŽîõëòæåæ îæùýãæ Žîæï êñèæïŽàŽê õãâèŽäâ áŽöëîâIJñèæîæùýãæ). ïý㎠ïæðõãâIJæå îëé ãåóãŽå, îæùýãâIJæ éâëîâáâIJŽ ùæçèñîŽá. Z 5−2 -öæ àãŽóãï 3.1 ïñîŽåäâ àŽéëïŽýñèæ ïæðñŽùæŽ.êŽý. 3.1

59îŽ éëýáâIJŽ, åñ öâãçîâIJå ëî x Ꭰy îæùýãï, ïŽáŽù −a ≤ x < a áŽ−a ≤ y < a (Z 5−2 -öæ a = 16). x + y þŽéæ æóêâIJŽ öâéëïŽäôãîñèæ:àŽêãæýæèëå ïŽéæ öâïŽúèë öâéåýãâãŽ:−2a ≤ x + y ≤ 2a − 2 < 2a − 1.(I) åñ −a ≤ x < 0 Ꭰ−a ≤ y < 0, éŽöæê −2a;(II) åñ 0 ≤ x < a Ꭰ0 ≤ y < a (Žêñ 0 ≤ x ≤ a − 1, 0 ≤ y ≤ a − 1),éŽöæê 0 ≤ x + y ≤ 2a − 2 < 2a − 1;(III) åñ −a ≤ x < 0 Ꭰ0 ≤ y < a, éŽöæê −a ≤ x + y < a.öâãêæöêëå, îëé (III) öâéåýãâãŽöæ öâáâàæ Žîæï éëåýëãêæè ïŽäôãîâIJöæáŽ, éŽöŽïŽáŽéâ, õëãâèåãæï \ïûëîæŽ". àŽãæýïâêëå, îëé åñ z ∈ Z áŽ−2a ≤ z < −a, éŽöæê z îæùýãæ ïŽïîñè ŽîæåéâðæçŽöæ ûŽîéëáàâIJŽ, îëàëîùz ′ = z + 2a, 0 ≤ z ′ < a. ŽêŽèëàæñîŽá, åñ a ≤ z < 2a − 1, éŽöæê zûŽîéëæáàæêâIJŽ, îëàëîù z ′′ = z − 2a Ꭰ−a ≤ z ′′ < 0 (â. æ., öâïŽIJŽéæïŽá,2a-ï áŽéŽðâIJæáŽê àŽéëçèâIJæå îæùýãæ éëåŽãïáâIJŽ éëåýëãêæè ïŽäôãîâIJöæ).ŽéîæàŽá, ìŽïñýæ \ŽîŽïûëîæ" æóêâIJŽ åñ æï (I) öâéåýãâãŽöæ áŽáâIJæåæŽ Žê (II)öâéåýãâãŽöæ ñŽîõëòæåæ.âï áŽïçãêâIJæ îëé àŽêãéŽîðëå êæöêæï åŽêîæàæï åãæïâIJâIJæï ðâîéæêâIJöæ,àŽêãæýæèëå ëîæ îæùýãæï ïýãŽáŽïý㎠öâïŽúèâIJèëIJŽ:Ž) ëîæãâ îæùýãæ ñŽîõëòæåæŽ;IJ) ëîæãâ îæùýãæ áŽáâIJæåæŽ;à) îæùýãâIJï Žóãï ïýãŽáŽïý㎠êæöŽêæ.é Ž à Ž è æ å æ 3.2.1.1 0 1 0 1+ 1 1 0 1 01 0 1 1 1 1−11+(−6) = −17 Z{öæ, −17 < −16, Žéæðëé −17+2 · 16 = 15 = 01111 Z 5−2 -öæ;2.1 1 1 0 0+ 1 0 1 1 11 0 0 0 1 1−4 + (−19) = −13 Žêñ 10011 Z 5−2 -öæ;3.1 1 1 0 1+ 0 0 1 1 01 0 0 0 1 1−3 + 6 = 3 < 16 Žêñ 00011 Z 5−2 -öæ;4.0 0 1 0 1+ 0 0 1 1 10 1 1 0 05 + 7 = 12 < 16, 01100 Z 5−2 -öæ.

605.0 1 1 0 0+ 0 1 0 1 01 0 1 1 012 + 10 = 22 > 16, æéæïŽåãæï, îëé éëåýëãêæèïŽäôãîâIJäâ éëåŽãïáâï ïŽüæîëŽ àŽéëãŽçèëå 2 · 16, 22 − 2 · 16 = −10, Žêñ10110 Z 5−2 -öæ.Žé öâéåýãâãâIJæï àŽŽêŽèæäâIJæå ãýâáŽãå, îëé \àŽáŽãïâIJŽ" (àŽáŽãïâIJæï öâùáëéŽ)àãýãáâIJŽ éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ ýáâIJŽ àŽáŽðŽêŽ åŽêîæàöæ ŽêàŽáŽðŽêŽ êæöêæŽêæ åŽêîæàæáŽê, éŽàîŽé ŽîŽ ëîæãâ âîåŽá.ŽýèŽ áŽãñIJîñêáâå àŽéîŽãèâIJŽï ᎠàŽõëòŽï. àŽêãæýæèëå àŽéîŽãèâIJŽ.àŽãæýïâêëå, îëé Z-öæ (ŽåëIJæå ïæïðâéŽï) îæùýãæï 10-äâ àŽéîŽãèâIJŽ ýáâIJŽõãâèŽ ùæòîæï âîåæ ìëäæùææå éŽîùýêæã \àŽáŽûâãæå" ᎠêñèëãŽê ìëäæùæŽöæ0 ùæòîæï øŽûâîæå. äëàŽáŽá, Z n m-öæ m-äâ àŽéîŽãèâIJŽ öâæúèâIJŽ éëãŽýáæêëåéŽîùýêæã àŽáŽûâãæå.ŽéîæàŽá, Z n 2 -öæ 2-æï áŽáâIJæå ýŽîæïýäâ àŽéîŽãèâIJŽ ýáâIJŽ åæåëâñèæùæòîæï éŽîùýêæã àŽáŽûâãæå ìëäæùæŽåŽ öâïŽIJŽéæï îæùýãäâ.é Ž à Ž è æ å æ 3.3. àŽéëåãèâIJï ãŽðŽîâIJå Z 5−2 -öæ.1. 00011 · 2 = 00110, 00110 · 2 = 01100, Žêñ Z-öæ 3 · 2 2 = 12;2. 11110 · 2 = 11100, 11100 · 2 = 1100 Žêñ Z-öæ −1 · 2 = −4, −4 · 2 = −8;3. 00101 · 2 = 01010, 01010 · 2 = 10100, 10100 · 2 = 01000 Žêñ Z-öæ5 · 2 = 10, 10 · 2 = 20, 20 · 2 = 40, 40 − 2 · 16 = 8.êâIJæïéæâî éåâè îæùýãäâ àŽéîŽãèâIJæïŽï àŽéëãæõâêëå öâçîâIJæï ëìâîŽùææïéæéŽîå àŽéîŽãèâIJæï ëìâîŽùææï áæïðîæIJñùæñèëIJæï åãæïâIJŽ ᎠéŽéîŽãèæûŽîéëãŽáàæêëå, îëàëîù 2-æï ýŽîæïýæï þŽéæ.é Ž à Ž è æ å æ 3.4.1. àŽéëãåãŽèëå 3 · 5, 5 ûŽîéëãŽáàæêëå, îëàëîù 2 2 + 2 0 , éŽöæê 3 · 5 =3·2 2 +3·2 0 , îŽáàŽê 2-æï ýŽîæïýâIJäâ àŽéîŽãèâIJŽ ñçãâ ãæùæå îëàëîù ýáâIJŽ:0 0 0 1 1 { 3 · 2 0+ 0 1 1 0 0 { 3 · 220 1 1 1 1 { 152. àŽéëãåãŽèëå (−5) · 3, Žéæðëé ûŽîéëãŽáàæêëå: 3 = 2 1 + 2 0 ,1 1 0 1 1 { (−5)+ 1 0 1 1 0 { (−5 · 2)1 1 0 0 0 1 Žêñ 10001 { (-15).äñïðŽá Žïâãâ, îëàëîù àŽéîŽãèâIJŽï ãŽýáâêáæå \éŽîþãêæã àŽáŽûâãæå",2-æï áŽáâIJæå ýŽîæïýäâ àŽõëòŽ ýáâIJŽ \éŽîþãêæã àŽáŽûâãæå".é Ž à Ž è æ å æ 3.5. Z 5 2-öæ àŽéëãåãŽèëå

611. 12 4= 3, àãŽóãï:0 1 1 0 0 (12)(0 0 0 1 1 0 0 3 = 12 )4{ ëîæ ìëäæùææå éŽîþãêæã àŽáŽæûæŽ;2. (−6) 1 1 0 1 0 (-6)2 , (0 1 1 0 1 0 13 13 ≠ (−6) ) ;23. 7 0 0 1 1 1 (7)4 , (0 0 0 0 1 1 1 1 ≈ 7 ) .4âîå ìëäæùæŽäâ éŽîþãêæã àŽáŽûâ㎠ŽãðëéŽðñîŽá æûãâãï êâIJæïéæâîæ ñŽîõëòæåæîæùýãæï áŽáâIJæåŽá àŽáŽûâãŽï Z 5−2 -öæ. éŽîþãêæã àŽáŽûâ㎠−16-ïàŽáŽæõãŽêï +8-öæ. âï îëé àŽéëãŽïûëîëå ïŽüæîëŽ, öâáâàï àŽéëãŽçèëå 16,îŽù éëàãùâéï −8-ï (â. æ. − 162). æàæãâ öâæúèâIJŽ éæãæôëå, åñ êæöêæö åŽêîæàŽá0-æï êŽùãèŽá 1-ï áŽãûâîå. ŽéîæàŽá, (−6)2éæãõŽãŽîå 11101 = −3-éáâ.éëýáâIJŽ ñçŽêŽïçêâèæ áŽçŽîàñèæ IJæðæï îæùýãäâ éæéŽðâIJæå. âï öââïŽIJŽéâIJŽáŽéîàãŽèâIJæï øãâñèâIJîæã Žîæåéâðæçñè ûâïï, îŽáàŽê êŽöåæï (Žêñ áŽàŽîàñèæùæòîâIJæï) ìæîãâè IJæðöæ 1 ûŽîéëŽáàâêï 0,5-ï. ŽéàãŽîæ áŽéîàãŽèâIJæåéæãæôâIJå 7 4≈ 2 (îŽù ñòîë Žýèëï Žîæï îæùýãæï äñïð éêæöãêâèëIJŽïåŽê).ï Ž ã Ž î þ æ ö ë 3.1.1. Z n−2 -æï îëéâèæéâ éëìæîáŽìæîâ îæùýãæï (2-éáâ áŽéŽðâIJæï) éëúâIJêæïïûîŽòæ ýâîýæ éáàëéŽîâëIJï öâéáâàöæ: éŽîþãâêŽ IJëèëáŽê áŽûõâIJñèæ õãâèŽêñèæᎠìæîãâèæ öâéýãâáîæ âîåæŽêæ àŽáŽàãŽóãï ñùãèâèŽá, ýëèë õãâèŽáŽêŽîøâêæ IJæðæ æùãèâIJŽ. Žøãâêâå, îëé ñé îŽãèâï öâéåýãâãŽöæ âï ýâîýæ àŽéëïŽõâêâIJâèæŽáŽ àŽêæýæèâå öâéåýãâãŽ, îëùŽ âï Žî éëóéâáâIJï.2. ãåóãŽå, Z 5−2 -öæ ûŽîéëâIJï öâéáâàæ àŽéëåãèâIJæ. æçîæIJâIJŽ ëîæ x áŽy îæùýãæ (éŽåæ þŽéæ Žôãêæöêëå z-æå). åñ z-ï àŽéëãŽçèâIJå y-ï, éæãæôâIJåîŽôŽù c öâáâàï, ýëèë, åñ z-ï àŽéëãŽçèâIJå c-ï, éŽöæê éæãæôâIJå îŽôŽù d îæùýãï.îŽ öâæèâIJŽ æåóãŽï c Ꭰd îæùýãâIJäâ? îëàëîù àŽêïýãŽãáâIJŽ öâáâàâIJæ,åñ àŽéëåãèâIJæ ûŽîéëâIJï Z n−2 -öæ?§ 4. èëàæçñîæ ŽîæåéâðæçŽIJñèæï ŽîæåéâðæçŽ éëóéâáâIJï Z 2 ᎠZ n 2 ïæéîŽãèââIJäâ áŽ, éŽöŽïŽáŽéâ, öâæùŽãïéýëèëá 0 Ꭰ1 îæùýãâIJï. æéæïŽåãæï, îëé ñçâå àŽãâîçãæëå, áŽãæûõëåèëàæçñîæ Žîæåéâðæçæï \öâáŽîâIJæå áæá" Z 5 ïæéîŽãèâäâ àŽêýæèãæå.Žãæôëå Z 5 = {01, 2, 3, 4, 5} ïæéîŽãèâ Ꭰ∨ Ꭰ∧ ëìâîŽùæâIJæ, îëéèâIJæùàŽêïŽäôãîñèæŽ 4.1 ùýîæèöæ.åñ Z 5 -ï áŽãŽèŽàâIJå IJñêâIJîæãæ áŽèŽàâIJæå, áŽãæêŽýŽãå, îëéa ∨ b = max{a, b},a ∧ b = min{a, b}.

63áŽãñIJîñêáâå 4.1 ùýîæèï, îëéâèæù àŽêïŽäôãîŽãï ∨ Ꭰ∧ ëìâîŽùæâIJï.ùýîæèöæ âîåêŽæîæ éêæöãêâèëIJâIJæï éóëêâ âèâéâêðâIJæ êâæðîŽèñîæ âèâéâêðâ-IJæï éæéŽîå àŽêèŽàâIJñèæŽ æïâ, îëàëîù 4.1 êŽýŽää⎠êŽøãâêâIJæ.êŽý. 4.1Žé ïñîŽåäâ çŽîàŽá øŽêï, îëé âï ëìâîŽùæâIJæ âîåéŽêâåæï ïæéâðîæñèæŽ,îŽù ïŽöñŽèâIJŽï àãŽúèâãï îŽæéâ ðëèëIJŽöæ âîåæ ëìâîŽùæŽ öâãùãŽèëå éâëîâåæ.âï Žîæï ëîŽáëIJæï ìîæêùæìæ.àŽêãæýæèëå Z 2 ïæéîŽãèâäâ àŽêïŽäôãîñèæ ∧ Ꭰ∨ ëìâîŽùæâIJæ:∨ 0 10 0 11 1 1,∧ 0 10 0 01 0 1Z 2 -öæ, øãâñèâIJîæã, ∨ ëìâîŽùææï æêðâîìîâðæîâIJŽ ýáâIJŽ, îëàëîù \Žê", ýëèë∧ ëìâîŽùææï, îëàëîù \áŽ". 0 Žîæï êâæðîŽèñîæ âèâéâêðæ \Žê"-æï éæéŽîå,ýëèë 1 { \áŽ"-ï éæéŽîå. âï öâáâàâIJæ öâæúèâIJŽ àŽêãŽãîùëå ñòîëéŽôŽè àŽêäëéæèâIJâIJäâ. éŽàŽèæåŽá, Z n 2 -äâ.é Ž à Ž è æ å æ 4.1.0 1 1 1 0 1 0 0 1∧ 1 1 0 1 0 0 1 0 10 1 0 1 0 0 0 0 10 1 1 1 0 1 0 0 1∨ 1 1 0 1 0 0 1 0 11 1 1 1 0 1 1 0 1ï Ž ã Ž î þ æ ö ë 4.2. Z n ïæéîŽãèâäâ ∧ Ꭰ∨ ëìâîŽùæâIJæ àŽêãïŽäôãîëå,îëàëîù éæêæéñéæ ᎠéŽóïæéñéæ:a ∨ b = max{a, b} Ꭰa ∧ b = min{a, b}.Žøãâêâå, îëé ïîñèáâIJŽ áæïðîæIJñùæñèëIJæï ìæîëIJŽ:a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c).,..

àîŽòåŽ åâëîæŽ (ïŽûõæïæ ùêâIJâIJæ)§ 1. ïŽûõæïæ ùêâIJâIJæãåóãŽå, V ïŽïîñè æïæéîŽãèâŽ. öâéëãæôëå ŽôêæöãêâIJæ:I V = { (v, v) : v ∈ V } , V 2 − = V 2 \ I V = { (v 1 , v ) : v 1 ≠ v 2}.V 2 − ïæéîŽãèâäâ àŽêãïŽäôãîëå éæéŽîåâIJŽ öâéáâàêŽæîŽá:(v 1 , v 2 ) ∼ (w 1 , w 2 ), åñ (v 1 , v 2 ) = (w 1 , w 2 ) Žê (v 1 , v 2 ) ∼ (w 2 , w 1 ).∼ éæéŽîåâIJæï éêæöãêâèëãŽêæ åãæïâIJŽ øŽéëõŽèæIJâIJñèæŽ öâéáâà ûæêŽáŽáâ-IJŽöæ:û æ ê Ž á Ž á â IJ Ž . ∼ éæéŽîåâIJŽ âçãæãŽèâêðëIJæï éæéŽîåâIJŽŽ V 2 −ïæéîŽãèâäâ.á Ž é ð ç æ ù â IJ Ž . ∼ éæéŽîåâIJŽ Žîæï:Ž) îâòèâóïñîæ: êâIJæïéæâîæ (v 1 , v 2 )∈V 2 − ûõãæèæïåãæï (v 1 , v 2 )∼(v 1 , v 2 );IJ) ïæéâðîæñèæ: åñ (v 1 , v 2 ) ∼ (w 1 , w 2 ), éŽöæê (w 1 , w 2 ) ∼ (v 1 , v 2 ).à) ðîŽêäæðñèæ: åñ (v 1 , v 2 ) ∼ (w 1 , w 2 ) Ꭰ(w 1 , w 2 ) ∼ (u 1 , u 2 ), éŽöæê(v 1 , v 2 ) ∼ (u 1 , u 2 ).ŽéàãŽîŽá, àŽêïŽäôãîñèæ âçãæãŽèâêðëIJæï çèŽïâIJæï ïæéîŽãèâ ŽôãêæöêëåV 2 −/ ∼ ïæéIJëèëåæ. âçãæãŽèâêðëIJæï åæåëâñèæçèŽïæ öâæùŽãï äñïðŽá ëîâèâéâêðï, îŽáàŽê, åñ (v 1 , v 2 ) ∈ V 2 −, éŽöæê [(v 1 , v 2 )] Žîæï âçãæãŽèâêðëIJæïçèŽïæ, îëéâèæù öâæùŽãï (v 1 , v 2 ) âèâéâêðï, éŽöæêŽýŽè öâéëãæðŽêëå àîŽòæï ùêâIJŽ.[(v 1 , v 2 )] = [ (v 1 , v 2 ), (v 2 , v 1 ) ] .à Ž ê ï Ž ä ô ã î â IJ Ž . àîŽòæ âûëáâIJŽ G = (V, E) ûõãæèï, ïŽáŽù VûãâîëåŽ ŽîŽùŽîæâèæ ïŽïîñèæ ïæéîŽãèâŽ, ýëèë E Žîæï V 2 −/ ∼ ïæéîŽãèæïóãâïæéîŽãèâŽ.ïý㎠ïæðõãâIJæå öâæúèâIJŽ ãåóãŽå, îëé E ïýãŽáŽïý㎠ûãâîëåŽ ŽîŽáŽèŽàâIJñèûõãæèåŽ ïæéîŽãèâŽ.E ïæéîŽãèâï âûëáâIJŽ àîŽòæï ûæIJëåŽ ïæéîŽãèâ, |V | ŽôêæöêŽãï G àîŽòæïûãâîëåŽ îŽëáâêëIJŽï, ýëèë |E| { ûæIJëåŽ îŽëáâêëIJŽï.öâéáâàæ ûæêŽáŽáâIJŽ ïŽïîñè ïæéîŽãèâäâ àŽêïŽäôãîñè éæéŽîåâIJâIJïŽ áŽàîŽòâIJï öëîæï çŽãöæîï àŽéëýŽðŽãï.65

66û æ ê Ž á Ž á â IJ Ž .Ž) G = (V, E) àîŽòæ V ïæéîŽãèâäâ àŽêïŽäôãîŽãï ŽîŽîâòèâóïñî ïæéâðîæñèéæéŽîåâIJŽï;IJ) V ïŽïîñè ïæéîŽãèâäâ éëùâéñèæ ŽîŽîâòèâóïñîæ ïæéâðîæñèæ éæéŽîåâIJŽàŽêïŽäôãîŽãï àîŽòï.á Ž é ð ç æ ù â IJ Ž . Ž) ãåóãŽå, G = (V, E) àîŽòæŽ. V ïæéîŽãèâäâàŽêãïŽäôãîëå R(E) éæéŽîåâIJŽ öâéáâàêŽæîŽá: v 1 R(E)v 2 éŽöæê ᎠéýëèëáéŽöæê, îëùŽ [v 1 , v 2 ] ∈ E. R(E) éæéŽîåâIJŽ ŽîŽîâòèâóïñîæŽ, îŽáàŽê vR(E)véŽöæê Ꭰéýëèëá éŽöæê, îëùŽ [(v, v)] ∈ E, éŽàîŽé [(v, v)] ∉ E, îŽéáâêŽáŽù(v, v) ∉ V 2 −. R(E) éæéŽîåâIJŽ ïæéâðîæñèæŽ, îŽáàŽê v 1 R(E)v 2 éŽöæê ᎠéýëèëáéŽöæê, îëùŽ [(v 1 , v 2 )] ∈ E, éŽàîŽé[(v 1 , v 2 )] = { (v 1 , v 2 ), (v 2 , v 1 ) } = [(v 2 , v 1 )].éŽöŽïŽáŽéâ, v 1 R(E)v 2 éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ v 2 R(E)v 1 .IJ) ãåóãŽå, R Žîæï V ïæéîŽãèâäâ àŽêïŽäôãîñèæ ŽîŽîâòèâóïñîæ, ïæéâðîæñèæéæéŽîåâIJŽ.R-æï ŽîŽîâòèâóïñîëIJŽ êæöêŽãï, îëé (v, v) ∉ R êâIJæïéæâîæ v ∈ V âèâéâêðæïåãæï.Žéæðëé R ⊂ V 2 −. ïæéâðîæñèëIJŽ êæöêŽãï, îëé v 1 Rv 2 éŽöæê áŽéýëèëá éŽöæê, îëùŽ v 2 Rv 1 .åñ E ïæéîŽãèâï àŽêãïŽäôãîŽãå, îëàëîù R/ ∼ âçãæãŽèâêðëIJæï çèŽïâ-IJæï ïæéîŽãèâï, éŽöæê G(V, E) æóêâIJŽ ïŽúæâIJâèæ àîŽòæ.àîŽòâIJæ öâæúèâIJŽ ûŽîéëãŽáàæêëå IJñèæï âèâéâêðâIJæŽêæ (B = {0, 1})éŽðîæùâIJæï ïŽöñŽèâIJæå. àîŽòâIJæï åãæïâIJâIJæáŽê IJâãîæ öâæúèâIJŽ àŽêãïŽäôãîëåéŽåæ éŽðîæùñèæ ûŽîéëáàâêâIJæå.à Ž ê ï Ž ä ô ã î â IJ Ž . G(V, E) àîŽòæï A ∈ M(n, B) éëéæþêŽãâéŽðîæùæ, ïŽáŽù n = |V |, àŽêæïŽäôãîâIJŽ öâéáâàêŽæîŽá:a ij ={1, åñ [v 1 , v 2 ] ∈ E,0, åñ [v 1 , v 2 ] ∉ E.ŽéIJëIJâê, îëé v i Ꭰv j ûãâîëâIJæ Žîæï éëéæþêŽãâ, åñ a ij = 1. ùýŽáæŽ, îëéa ii = 0 (i = 1, 2, . . . , n) ᎠA = A T , A ïæéâðîæñèæŽ.G(V, E) àîŽòæ öâæúèâIJŽ àŽéëãïŽýëå R 2 ïæIJîðõâäâ öâéáâàêŽæîŽá: åæåëâñèæv ∈ V ûãâîëïåãæï ïæIJîðõâäâ Žôãêæöêëå ûâîðæèæ. ŽéŽïåŽê, åñ[(v, w)] ∈ E, éŽöæê v Ꭰw ûãâîëâIJæ ýŽäæå öâãŽâîåëå.àŽêãæýæèëå éëùâéñèæ àîŽòæï àŽéëïŽýñèâIJŽ ᎠéëéæþêŽãâëIJæï éŽðîæùŽ(êŽý. 1.1)⎡⎡⎤ ⎤0 1 1A = ⎣⎣1 0 1⎦ ⊕⎦ .1 1 0

67êŽý. 1.1é Ž à Ž è æ å æ 1.1.1) ãåóãŽå,V = {v 1 , v 2 , v 3 }, E = { [v 1 , v 2 ], [v 1 , v 3 ], [v 2 , v 3 ] } ,|V | = 3, |E| = 3 [ ⊕ ] .2) V = {v 1 , v 2 , v 3 , v 4 , v 5 },E = { [v 1 , v 2 ], [v 1 , v 5 ], [v 2 , v 3 ], [v 2 , v 4 ], [v 3 , v 5 ], [v 3 , v 4 ], [v 4 , v 5 ] } ,|V | = 5, |E| = 7.àŽêãæýæèëå éëùâéñèæ àîŽòæï éëéæþêŽãâëIJæï éŽðîæùŽ ᎠàŽéëïŽýñèâIJŽ⎡⎤0 1 0 0 11 0 1 1 0A =⎢0 1 0 1 1⎥⎣0 1 1 0 1⎦ .1 0 1 1 0êŽý. 1.2öâãêæöêëå, îëé àîŽòâIJæ ûŽîéëŽáàâêï ñòîë \ðëìëèëàæñî", ãæáîâ\àâëéâðîæñè" ëIJæâóðâIJï, Žêñ àîŽòâIJæ àŽéëýŽðŽãï ñòîë ûãâîëâIJï öëîæï

68éæéŽîåâIJŽï, ãæáîâ ïæãîùâöæ ûãâîëâIJæïŽ áŽ ûæIJëâIJæï àŽêèŽàâIJŽï. ŽéàãŽîŽá,àîŽòæ öâæúèâIJŽ àŽéëãïŽýëå ñïŽïîñèë îŽëáâêëIJæï ïýãŽáŽïýãŽ, éŽàîŽ, \âçãæãŽèâðñîæ"ýâîýæå. ŽéŽïåŽê, êŽýŽääâ ëîæ ûæIJëï àŽáŽçãâåæáŽê Žî àŽéëéáæêŽîâëIJï,îëé àŽáŽçãâåæï ûâîðæèæ Žîæï ûãâîë. ùýŽáæŽ, îëé éëéæþêŽãâëIJæïéŽðîæùæï óãâᎠ(Žê äâáŽ) ïŽéçñåýŽ êŽûæèæ ïŽçéŽîæïæŽ æéæïŽåãæï, îëéàîŽòæ àŽêæïŽäôãîëï⎡⎤0 . . . . . . . . .A = ⎢. . . 0 . . . . . .⎥⎣. . . . . . . . . . . . ⎦ .. . . . . . . . . 0áæŽàëêŽèæï óãâᎠêŽûæèæ àŽêïŽäôãîŽãï G àîŽòï.à Ž ê ï Ž ä ô ã î â IJ Ž . ãæðõãæå, îëé H = (V 1 , E 1 ) àîŽòæ ŽîæïG = (V, E) àîŽòæï óãâàîŽòæ, åñ V 1 ⊆ V ᎠE 1 ⊆ E. åñ V 1 = V , éŽöæêãæðõãæå, îëé H Žîæï G àîŽòæï îñòîâíîé óãâàîŽòæ. åñ V 1 Žîæï G =(V, E) àîŽòæï ûãâîëåŽ ŽîŽùŽîæâèæ óãâïæéîŽãèâ, éŽöæê V 1 ûŽîéëóéêæèæ(V 1 , E 1 ) óãâàîŽòæ àŽêæïŽäôãîâIJŽ öâéáâàêŽæîŽá[v + , w] ∈ E 1 ⇐⇒ v, w ∈ V 1 Ꭰ[v, w] ∈ E.à Ž ê ï Ž ä ô ã î â IJ Ž .Ž) ãåóãŽå, éëùâéñèæŽ G 1 = (V 1 , E 1 ) ᎠG 2 = (V 2 , E 2 ) àîŽòâIJæ. ãæðõãæå,îëé G 1 ᎠG 2 âçãæãŽèâêðñîæŽ, åñ ŽîïâIJëIJï æïâåæ f : V 1 → V 2IJæâóùæŽ, îëévR(E)w =⇒ f(v)R(E 2 )f(w).IJ) ãåóãŽå, G = (V, E) êâIJæïéæâîæ àîŽòæŽ. àŽêãïŽäôãîëåδ : V → N ∪ {0}ŽïŽý㎠öâéáâàêŽæîŽá: δ(v) ïæáæáâ ñáîæï æé ûæIJëåŽ îŽëáâêëIJŽï, îëéâèæùöâæùŽãï v ∈ E ûãâîëï. δ(v)-ï ãñûëáëå v ûãâîëï ýŽîæïýæ.öâéáâàæ ûæêŽáŽáâIJŽ àŽéëýŽðŽãï ëî éŽîðæã, éŽàîŽé éêæöãêâèëãŽê òŽóðïàîŽòâIJæï åãæïâIJâIJæï öâïŽýâIJ.û æ ê Ž á Ž á â IJ Ž .Ž) ∑ δ(v) = 2|E|;v∈VIJ) êâIJæïéæâî àîŽòöæ çâêðæ ýŽîæïýæï ûãâîëåŽ îŽëáâêëIJŽ èñûæŽ.á Ž é ð ç æ ù â IJ Ž . Ž) åæåëâñèæ ûæIJë ëîþâî öâáæï þŽéöæ, ïŽæáŽêŽùàŽéëéáæêŽîâëIJï éðçæùâIJŽ.IJ) ãåóãŽå, V e ⊆ V èñûæ ýŽîæïýæï ûãâîëåŽ ïæéîŽãèâŽ, ýëèë V 0 ⊆ V {çâêðæ ýŽîæïýæï ûãâîëåŽ ïæéîŽãèâ. öâãêæöêëå, îëéV = V e ∪ V 0 ᎠV e ∩ V 0 = ∅.

69éŽöŽïŽáŽéâ,∑δ(v) = ∑ δ(v) + ∑ δ(v).v∈V e v∈V 0v∈Vãæùæå, îëé ∑∑δ(v) = 2|E|, â. æ. èñûæ îæùýãæŽ, δ(v) þŽéöæ åæåëâñèæv∈V∑v∈V eöâïŽçîâIJæ èñûæŽ, Žéæðëé þŽéæù èñûæŽ: δ(v) = 2k (k éåâèæ îæùýãæŽ).v∈V eŽéîæàŽá,∑δ(v) = 2|E| − 2k = 2(|E| − k).v∈V 0éŽàîŽé Žé ñçŽêŽïçêâè þŽéöæ åæåëâñèæ δ(v) öâïŽçîâIJæ çâêðæŽ, Žéæðëé |V 0 |ñêᎠæõëï èñûæ.à Ž ê ï Ž ä ô ã î â IJ Ž .1. ãåóãŽå, S V ᎠS E êæöêâIJæï ïæéîŽãèââIJæŽ. G = (V, E) àîŽòæï éëêæöãꎎêñ êæöêâIJæï àŽêŽûæèâIJŽ âûëáâIJŽ òñêóùæŽåŽ ûõãæèï:f : V → S V ûãâîëâIJæï êæöêâIJæï àŽêŽûæèâIJŽŽ;g : E → S E ûæIJëâIJæï êæöêâIJæï àŽêŽûæèâIJŽŽ.2. ãåóãŽå, G 1 = (V 1 , E 1 ) àîŽòæ éëêæöêñèæŽ f 1 Ꭰg 1 òñêóùæâIJæå, ýëèëG 2 = (V 2 , E 2 ) { f 2 Ꭰg 2 òñêóùæâIJæå.G 1 ᎠG 2 àîŽòâIJï âûëáâIJŽ âçãæãŽèâêðñîŽá êæöŽêáâIJñèæ (éëêæöêñèæ),åñ ŽîïâIJëIJï æïâåæ h : V 1 → V 2 IJæâóùæŽ, îëéŽ) G 1 ᎠG 2 âçãæãŽèâêðñîæŽ, îëàëîù ŽîŽêæöŽêáâIJñèæ (åñ ŽîŽéëêæöêñèæ)àîŽòâIJæ;IJ) öâïŽIJŽéæï ûãâîëâIJï Žóãï âîåæ Ꭰæàæãâ êæöŽêæ:f1(v) = f 2 (h(v)) êâIJæïéæâîæ v ∈ V 1 -åãæï;à) öâïŽIJŽéæï ûæIJëâIJï Žóãï âîåæ Ꭰæàæãâ êæöŽêæ:g 1 ([v, w]) = g 2([h(v), h(w)])õëãâèæ v, w ∈ V1 -åãæï.æé öâéåýãâãŽöæ, îëùŽ éëêæöêñèæŽ éýëèëá ûãâëîâIJæ, àãâóêâIJŽ:{f : V → S V ,g = const .åñ éëêæöêñèæŽ éýëèëá ûãâîëâIJæ, éŽöæê àãâóêâIJŽ{f = const,g : E → S E .àŽêãæýæèëå éŽàŽèæåâIJæ:é Ž à Ž è æ å æ 1.2.1) ãåóãŽå,V = {v 1 , v 2 , v 3 , v 4 }, E = { [v 1 , v 2 ], [v 2 , v 3 ], [v 2 , v 4 ] } ,f : V → ïŽóŽîåãâèëï óŽèŽóâIJæ, g : E → N,

71ï Ž ã Ž î þ æ ö ë 1.1.êŽý. 1.41. àŽéëïŽýâå óãâéëå éëõãŽêæèæ éëéæþêŽãâëIJæï éŽðîæùâIJæå ûŽîéëáàâêæèæàîŽòâIJæ:⎡⎤ ⎡⎤0 1 1 0 00 1 1 1 01 0 0 0 0A 1 =⎢1 0 0 1 1⎥⎣0 0 1 0 0⎦ , A 1 0 0 0 12 =⎢1 0 0 1 0⎥⎣1 0 1 0 1⎦ .0 0 1 0 00 1 0 1 02. öâŽáàæêâå 1.5 êŽýŽääâ ûŽîéëáàâêæèæ àîŽòâIJæï éëéæþêŽãâëIJæï éŽðîæùâIJæ:êŽý. 1.53. áŽýŽäâå 1.5(Ž) êŽýŽääâ éëùâéñèæ àîŽòæï óãâàîŽòæ, ûŽîéëóéêæèæ{v 2 , v 3 , v 4 , v 6 } ûãâîëâIJæå.4. ãåóãŽå, G = (V, E) Žîæï àîŽòæ Ꭰ|V | = n îëàëîæ öâæúèâIJŽ æõëï|E|-ï éŽóïæéŽèñîŽá öâïŽúèë éêæöãêâèëIJŽ.

725. îŽéáâêæ ïýãŽáŽïý㎠àîŽòæ ŽîïâIJëIJï, îëéèâIJïŽù Žóãï n ûãâîë.6. àŽéëïŽýâå K 3 , K 4 , K 5 àîŽòâIJæ.7. ŽŽàâå ëîûæèæŽêæ àîŽòæï éŽàŽèæåæ.8. àŽéëïŽýâå K 3,3 àîŽòæ.9. îŽéáâêæ ûæIJë Žóãï K m,n ëîûæèæŽê ïîñè àîŽòï.§ 2. éŽîöîñðâIJæ, ùæçèâIJæ ᎠIJéñèëIJŽàîŽòâIJæï äëàæâîåæ éêæöãêâèëãŽêæ åãæïâIJŽ àŽéëéáæêŽîâëIJï öâéáâàæ àŽêïŽäôãîâIJæáŽê.à Ž ê ï Ž ä ô ã î â IJ Ž .Ž) ãåóãŽå, G = (V, E) Žîæï àîŽòæ. G àîŽòöæ k ïæàîúæï éŽîöîñðæv ûãâîëáŽê w-öæ âûëáâIJŽ ûãâîëâIJæï (ŽîŽ ŽñùæèâIJèŽá ïýãŽáŽïýãŽ) æïâå〈v 0 , v 1 , . . . , v k 〉 éæéáâãîëIJŽï, ïŽáŽù v 0 = v, v k = w, [v i−1 , v i ] ∈ E õëãâèæi = 1, 2, . . . , k. éŽîöîñðï âûëáâIJŽ øŽçâðæèæ, åñ v 0 = v k . éŽîöîñðïâûëáâIJŽ þŽüãæ, åñ éæïæ õãâèŽ ûãâîë àŽêïýãŽãâIJñèæŽ. øŽçâðæè þŽüãï âûëáâIJŽùæçèæ. ùæçèï âûëáâIJŽ éŽîðæãæ, åñ éýëèëá v 0 = v k , ýëèë õãâèŽáŽêŽîøâêæ ûãâîë àŽêïýãŽãâIJñèæŽ.IJ) åñ ŽîïâIJëIJï éŽîöîñðæ v-áŽê w-öæ, v, w ∈ V , éŽöæê ŽéIJëIJâê, îëé wéæôûâãŽáæŽ v-áŽê.à) àîŽòï, îëéâèæù Žî öâæùŽãï ùæèâIJï, Žùæçèñîæ âûëáâIJŽ.à Ž ê ï Ž ä ô ã î â IJ Ž .Ž) G = (V, E) àîŽòï âûëáâIJŽ IJéñèæ, åñ éæïæ êâIJæïéæâîæ ëîæ àŽêïýãŽãâ-IJñèæ ûãâîëïåãæï ŽîïâIJëIJï Žé ûãâîëâIJæï öâéŽâîåâIJâèæ éŽîöîñðæ.IJ) Žùæçèñî IJéñè àîŽòï ýâ âûëáâIJŽ.à) Îñòîâíûì äåðåâîì âûëáâIJŽ G = (V, E) àîŽòæï éåŽãŽî óãâàîŽòï,îëéâèæù ýâï ûŽîéëŽáàâêï.îëàëîù äâéëå æõë Žôêæöêñèæ, àŽéëåãèâIJæ éëéæþêŽãâëIJæï éŽðîæùâIJäâöâæùŽãï éêæöãêâèëãŽêæ æêòëîéŽùæŽï àîŽòæï IJñêâIJæï öâïŽýâIJ.öâãêæöêëå, îëé óãâéëå éëõãŽêæè åâëîâéŽï Ꭰéæï öâáâàâIJöæ éŽðîæùæïA k ýŽîæïýæ àŽéëæåãèâIJŽ M(n, Z)-öæ ᎠŽîŽ M(n, B)-öæ, Žéæðëé A k éŽðîæùæïâèâéâêðæ öâæúèâIJŽ 1-äâ éâðæ îæùýãæ æõëï.å â ë î â é Ž . ãåóãŽå, A Žîæï G = (V, E) àîŽòæï éëéæþêŽãâëIJæïéŽðîæùŽ Ꭰ|V | = n. A k éŽðîæùæï c ij âèâéâêðæ ñáîæï v i -áŽê v j -éáâ kïæàîúæï éŽîöîñðâIJæï îæùýãï.á Ž é ð ç æ ù â IJ Ž . àŽéëãæõâêëå æêáñóùæŽ k-ï éæéŽîå. îëùŽ k = 1,1 ïæàîúæï éŽîöîñðæ Žîæï äñïðŽá G àîŽòæï ûæIJë. Žé öâéåýãâãŽöæ åâëîâéæïïŽéŽîåèæŽêëIJŽ àŽéëéáæêŽîâëIJï A éŽðîæùæï àŽêïŽäãîâIJæáŽê. ãåóãŽå,A = (a ij ), A k−1 = (b ij ), A k = (c ij ),

73éŽöæêA k = A k−1 · A = (c ij ) =n∑b il a lj .áŽãñöãŽå, îëé åâëîâéŽ ïŽéŽîåèæŽêæŽ (k − 1)-åãæï, Žêñ b il Žîæï v i -áŽê v l -éáâ (k−1)-ïæàîúæï éŽîöîñðâIJæï îæùýãæ, ýëèë a lj , àŽêïŽäôãîâIJæï åŽêŽýéŽá,Žîæï v l -áŽê v j -éáâ 1 ïæàîúæï éŽîöîñðâIJæï îæùýãæ, éŽöŽïŽáŽéâ, b il·a lj êŽéîŽãèææóêâIJŽ v i -áŽê v j -éáâ k ïæàîúæï æé éŽîöîñðâIJæï îŽëáâêëIJŽ, ïŽáŽù IJëèëïûæêŽ ûãâîëŽ v l . ŽóâáŽê àŽéëéáæêŽîâëIJï, îëél=1n ∑l=1b il a lj æóêâIJŽ v i -áŽê v j -éáâk ïæàîúæï õãâèŽ öâïŽúèë éŽîöîñðæï îŽëáâêëIJŽ. Žéæå åâëîâéŽ áŽéðçæùâ-IJñèæŽ.ö â á â à æ .Ž) G = (V, E) àîŽòöæ éŽîöîñðæ v i ûãâîëáŽê v j -éáâ (i ≠ j) ŽîïâIJëIJïéŽöæê Ꭰéýëèëá éŽöæê, îëùŽ n îæàæï çãŽáîŽðñèæA + A 2 + · · · + A n−1éŽðîæùæï (i, j)-ñîæ âèâéâêðæ Žî ñáîæï êñèï.IJ) åñ Žî àŽéëãæõâêâIJå i ≠ j ìæîëIJŽï, éŽöæê éëåýëãêæè éŽðîæùŽï âóêâIJŽA + A 2 + · · · + A n ïŽýâ.á Ž é ð ç æ ù â IJ Ž . ãåóãŽå, 〈v i , v 1 , . . . , v j 〉 Žîæï éŽîöîñðæ v i ûãâîëáŽêv j -éáâ. ãåóãŽå, éëùâéñè éæéáâãîëIJŽöæ ûãâîëâIJæ Žî éâëîáâIJŽ. éŽöæê, îŽáàŽê|V | = n, éŽîöîñðæ öâæùŽãï ŽîŽñéâðâï n − 1 ûæIJëï. åâëîâéæï úŽèæå, A +A 2 + · · · + A n−1 öâïŽçîâIJâIJï öëîæï ŽîïâIJëIJï âîåæ éŽðîæùŽ éŽæêù, îëéèæï(i, j)-ñîæ âèâéâêðæ Žî ñáîæï 0-ï. Žéæå ŽñùæèâIJâèæ ìæîëIJŽ áŽéðçæùâIJñèæŽ.åñ éëùâéñè éŽîöîñðöæ éâëîáâIJŽ ûãâîëâIJæ, éŽöæê éŽï âóêâIJŽ〈 〉vi , . . . , v r , . . . , v} {{ } r , . . . , v j ,øŽçâðæèæ éŽîöîñðæåñ øãâê ŽéëãŽàáâIJå õãâèŽ Žïâå øŽçâðæè éŽîöîñðï, éŽöæê ŽéëùŽêŽ áŽæõãŽêâIJŽûæêŽ öâéåýãâãŽéáâ.ŽýèŽ ìæîæóæå, ãåóãŽå, A + A 2 + · · · + A n−1 éŽðîæùæï (i, j)-ñîæ âèâéâêðæŽî ñáîæï 0-ï. âï éýëèëá éŽöæê Žîæï öâïŽúèâIJâèæ, îëùŽ öâïŽçîâIJæéŽðîæùâIJæï (i, j)-ñîæ âèâéâêðâIJæáŽê âîåæ éŽæêù àŽêïýãŽãáâIJŽ 0-àŽê. âï ñçãâêæöêŽãï, îëé ŽîïâIJëIJï éŽîöîñðæ v i -áŽê v j -éáâ.IJ) åñ áŽïŽöãâIJæŽ i = j, éŽöæê vi-áŽê v j -öæ éŽîöîñðæï ŽîïâIJëIJŽ êæöêŽãï,îëé ŽîïâIJëIJï ûãâîëåŽ 〈v i , v 1 , . . . , v j 〉 éæéáâãîëIJŽ. åñ Žé éæéáâãîëIJŽöæõãâèŽ ûãâîë àŽêïýãŽãâIJñèæŽ (àŽîᎠöâïŽúèë v i = v j öâéåýãâãæïŽ), éŽöæêéŽîöîñðâIJæ öâáàâIJŽ ŽîŽñéâðâï n + 1 ûãâîëïàŽê (ŽîŽñéâðâï n ûæIJëïàŽê).n∑éŽöŽïŽáŽéâ, A k éŽðîæùæï (i, j)-ñîæ âèâéâêðæ Žî ñáîæï 0-ï,k=1A(R + (E)) = A(R(E)) ∨ A(R 2 (E)) ∨ · · · ∨ A(R n (E)) =n∨A(R k (E)),k=1

74A(R ∗ (E)) = I ∨ A(R(E)) ∨ · · · ∨ A(R n−1 (E)) =n−1∨k=0A(R k (E)).àŽãæýïâêëå, îëé êâIJæïéæâîæ R IJæêŽîñèæ éæéŽîåâIJæïåãæï R + ïæáæáâàŽêæïŽäôãîâIJŽ, îëàëîù∞⋃R + = R kk=1áŽ, åñ R ⊂ V × V , |V | = n, éŽöæê ŽóâáŽê àŽéëéáæêŽîâëIJï, îëén∨A(R + ) = A + (R) = A(R k ),ŽêŽèëàæñîŽáA(R ∗ ) = A ∗ (R) =k=1n−1∨k=0A(R k ).ŽôêæöãêâIJæï àŽéŽîðæãâIJæï éæäêæå A(R k (E)) Žôãêæöêëå A(R k )-æå, A(R + (E))Žôãêæöêëå A(R + )-æå, A(R ∗ (E)) çæ { A(R ∗ )-æå.C = A(R ∗ ) éŽðîæùŽï âûëáâIJŽ G = (V, E) àîŽòæï IJéñèëIJæï éŽðîæùŽ.G àîŽòöæ éŽîöîñðæ v i ûãâîëáŽê v j -éáâ (i ≠ j) ŽîïâIJëIJï éŽöæê ᎠéýëèëáéŽöæê, îëùŽ C éŽðîæùæï (i, j)-ñîæ âèâéâêðæ ñáîæï 1-ï. G àîŽòæ IJéñèæŽéŽöæê Ꭰéýëèëá éŽöæê, îëùŽ C éŽðîæùæï õãâèŽ (i, j)-ñîæ âèâéâêðæ ñáîæï1-ï, 1 ≤ i, j ≤ n.û æ ê Ž á Ž á â IJ Ž . R ∗ Žîæï V -äâ âçãæãŽèâêðëIJæï éæéŽîåâIJŽ.á Ž é ð ç æ ù â IJ Ž . îŽáàŽê R ∗ Žîæï R-æï îâòèâóïñîæ øŽçâðãŽ, ŽéæðëéïŽüæîëŽ öâãŽéëûéëå éýëèëá ïæéâðîæñèëIJŽ ᎠðîŽêäæðñèëIJŽ. vR ∗ wéæéŽîåâIJæáŽê àŽéëéáæêŽîâëIJï v-áŽê w-éáâ 〈v, v 1 , . . . , v k , w〉 éŽîöîñðæï ŽîïâIJëIJŽ.â. æ.[v, v 1 ] ∈ E, [v 1 , v 2 ] ∈ E, . . . , [v k , w] ∈ E,Žêñ[w, v k ] ∈ E, [v k , v k−1 ] ∈ E, . . . , [v 2 , v 1 ] ∈ E, [v 1 , v] ∈ E.âï êæöêŽãï, îëé 〈w, v k , v k−1 , . . . , v 1 , v〉 Žîæï w-áŽê v-éáâ éŽîöîñðæ G àîŽòöæ.â. æ. wR ∗ v.ãåóãŽå, vR ∗ w ᎠwR ∗ v, âï ñçŽêŽïçêâèæ êæöêŽãï, îëé ŽîïâIJëIJï〈w, w 1 , . . . , w m , u〉 éŽîöîñðæ w-áŽê u-éáâ G àîŽòöæ, ïŽáŽù[w, w 1 ] ∈ E, [w 1 , w 2 ] ∈ E, . . . , [w m , u] ∈ E.åñ àŽêãæýæèŽãå éæéáâãîëIJŽï 〈v, v 1 , . . . , v k , w, w 1 , . . . , w m , u〉, âï æóêâIJŽéŽîöîñðæ v-áŽê u-éáâ G àîŽòöæ. ŽéîæàŽá, àãâóêâIJŽ, îëé vR ∗ u. R ∗ ŽîæïâçãæãŽèâêðëIJæï éæéŽîåâIJŽ.R ∗ àŽêïŽäôãîŽãï óãâàîŽòâIJæï éêæöãêâèëãŽê çèŽïï.à Ž ê ï Ž ä ô ã î â IJ Ž . ãåóãŽå, {V i : 1 ≤ i ≤ p} Žîæï G àîŽòæïR ∗ éæéŽîåâIJæå àŽêïŽäôãîñèæ áŽõëòŽ, éŽöæê ŽéIJëIJâê, îëé p Žîæï G àîŽòæï

75IJéñèëIJæï îæùýãæ. âçãæãŽèâêðëIJæï çèŽïâIJæå ûŽîéëóéêæè (V i , E i ) óãâàîŽòâIJïâûëáâIJŽ G àîŽòæï IJéñèëIJæï çëéìëêâêðâIJæ.ðõâ âûëáâIJŽ àîŽòï, îëéâèöæù õëãâèæ IJéñèæ çëéìëêâêðæ Žîæï ýâ. G =(V, E) àîŽòæï îñòîâíîé ðõâ Žîæï æïâåæ àŽêùŽèçâãâIJñèæ T i = (V i , E i ) ýââIJæïûãâîëâIJæï âîåëIJèæëIJŽ, îëé V = ⋃ V i ᎠE i ⊂ E õãâèŽ i-åãæï (ûãâîëâIJæïiàŽêùŽèçâãâIJŽ êæöêŽãï, îëé V i ∩ V j = ∅, îëùŽ i ≠ j).1.2 êŽýŽääâ êŽøãâêâIJæŽ àîŽòæ, îëéèæïåãæïŽù p = 2 IJéñèëIJæï îæùýãæŽ.G àîŽòæG àîŽòæï îñòîâíîé ðõâï Ž ã Ž î þ æ ö ë 2.1.1. ãåóãŽå, G = (V, E), ïŽáŽùV = {v 1 , v 2 , v 3 , v 4 },E = { [v 1 , v 2 ], [v 1 , v 3 ], [v 2 , v 4 ], [v 3 , v 4 ] } .G àîŽòæï éëéæþêŽãâëIJæï éŽðîæùæï àŽéëõâêâIJæå àŽêãïŽäôãîëåŽ) 2 ïæàîúæï éŽîöîñðâIJæï îæùýãæ v 3 ûãâîëáŽê v 2 -öæ;IJ) 3 ïæàîúæï éŽîöîñðâIJæï îæùýãæ v 1 -áŽê v 2 -öæ;

76à) Žîæï åñ ŽîŽ G àîŽòæ IJéñèæŽ.2. áŽýŽäâå öâéáâàæ àîŽòâIJæïåãæï îñòîâíûå ýââIJæ.êŽý. 2.1

ìèŽêŽîñèæ àîŽòâIJæ§ 1. ìèŽêŽîñèæ àîŽòâIJæàîŽòåŽ åâëîææï IJâãî êŽöîëéöæ àŽêæýæèŽãâê àîŽòâIJæï ïìâùæŽèñî çèŽïï,îëéèâIJæù öâæúèâIJŽ ûŽîéëãŽáàæêëå êŽýŽäæå R 2 ïæIJîðõâäâ.à Ž ê ï Ž ä ô ã î â IJ Ž .Ž) G àîŽòï ìèŽêŽîñèæ âûëáâIJŽ, åñ öâæúèâIJŽ æï ïæIJîðõâäâ æïâ àŽéëæïŽýëï,îëé éæïæ ûæIJëâIJæ Žî æçãâåâIJëáâï (ëîæ ûæIJëï åŽêŽçãâåŽ öâæúèâ-IJŽ æõëï éýëèëá ûãâîë). Žïâå êŽýŽäï G àîŽòæï îñçŽ âûëáâIJŽ (ïæIJîðõâäâàŽéëïŽýñè Žïâå àîŽòï IJîðõâè àîŽòïŽù ñûëáâIJâê).IJ) G àîŽòæï îñçŽï âûëáâIJŽ IJéñèæ, åñ G IJéñèæŽ.é Ž à Ž è æ å æ 1.1. G = (V, E), V = {v 1 , v 2 , v 3 , v 4 },E = { [v 1 , v 2 ], [v 1 , v 3 ], [v 1 , v 4 ], [v 2 , v 3 ] } .Žé àîŽòæï àŽéëïŽýñèâIJŽ ᎠîñçŽ éëùâéñèæŽ 1.1 êŽýŽääâ.êŽý. 1.1îñçŽ R 2 ïæIJîðõâï ßõëòï ŽîââIJŽá.77

78é Ž à Ž è æ å æ 1.2. àŽêãæýæèëå êŽýŽääâ àŽéëïŽýñèæ îñçŽ. æï R 2ïæIJîðõâï ßõëòï ëåý êŽûæèŽá: r 1 , r 2 , r 3 Ꭰr 4(îñçæï Žîâï ûŽýêŽàâIJïŽù ñûëáâIJâê).êŽý. 1.2ŽîââIJæï ïæéîŽãèâ Žôãêæöêëå R Žïëåæ.â æ è â î æ ï å â ë î â é Ž . êâIJæïéæâîæ IJéñèæ îñçæïåãæï ïîñèáâIJŽ|V | − |E| + R| = 2.á Ž é ð ç æ ù â IJ Ž . ãåóãŽå, |V | = 1, éŽöæê, ùýŽáæŽ, |E| = 1 áŽ|R| = 1. Žé öâéåýãâãŽöæ åâëîâéŽ ïŽéŽîåèæŽêæŽ. ãåóãŽå, éëùâéñèæŽ îñçŽ,îëéèæïåãæïŽù ïîñèáâIJŽ |V 1 | − |E 1 | + |R 1 | = 2, ïŽáŽù |V 1 | éëùâéñèæ îñçæïûãâîëåŽ îæùýãæŽ, |E 1 | ûæIJëåŽ îæùýãæ, |R 1 | { ŽîââIJæï îæùýãæ.àŽêãæýæèëå éëùâéñèæ îñçæï àŽòŽîåëâIJæï ëîæ öâïŽúèë ýâîýæ.Ž) áŽãñéŽðëå âîåæ ûãâîë Ꭰéæãñâîåëå éëùâéñè îñçŽï ûæIJë. Žé öâéåýãâãŽöæ(æý. êŽý. 1.2) 1-æå æäîáâIJŽ |V 1 | ûãâîëåŽ îŽëáâêëIJŽ Ꭰ|E 1 | ûæ-IJëåŽ îŽëáâêëIJŽ, ýëèë |R 1 | îøâIJŽ ñùãèâèæ, |V | = |V 1 | + 1, |E| = |E 1 | + 1,|R| = |R 1 |, îæï öâáâàŽáŽù |V | − |E| + |R| àŽéëïŽýñèâIJæï éêæöãêâèëIJŽ ŽîæùãèâIJŽ,|V | − |E| + |R| == (|V 1 | + 1) − (|E 1 | + 1) + |R 1 | = |V 1 | − |E 1 | + |R 1 | = 2.IJ) öâãŽâîåëå ŽýŽèæ ûæIJëåæ îñçŽäâ ŽîïâIJñèæ ëîæ ûãâîë (æý. êŽý. 1.3).Žé öâéåýãâãŽöæ ûãâîëåŽ îæùýãæ áŽîøâIJŽ ñùãèâèæ, ýëèë ûæIJëåŽ îæùýãæ ᎎîâåŽ îæùýãæ 1-æå àŽæäîáâIJŽ: |V | = |V 1 |, |E| = |E 1 | + 1, |R| = |R 1 | + 1,|V | − |E| + |R| == |V 1 | − (|E 1 | + 1) + (|R 1 | + 1) = |V 1 | − |E 1 | + |R 1 | = 2.öâáâàŽá |V | − |E| + |R| éêæöãêâèëIJŽ æïâã ñùãèâèæ îøâIJŽ.êâIJæïéæâîæ îñçŽ éææôâIJŽ |V | = 1 öâéåýãâãæáŽê Ž) ᎠIJ) ýâîýæï ïŽöñŽèâ-IJæå. åñ ïŽüæîëŽ, ìîëùâáñîŽ éâëîáâIJŽ. ŽéîæàŽá, îŽáàŽê |V |−|E|+|R| = 2,

79êŽý. 1.3êŽý. 1.4îëùŽ |V | = 1, Ꭰ|V | − |E| + |R| àŽéëïŽýñèâIJæï éêæöãêâèëIJŽ æêãŽîæŽêðñèæîøâIJŽ Ž) ᎠIJ) ýâîýæï öâïîñèâIJæå, Žéæðëé êâIJæïéæâîæ IJéñèæ îñçæïåãæïàãŽóãï|V | − |E| + |R| = 2.ùýŽáæŽ, îëé îñçæï åæåëâñèæ Žîâ öâéëïŽäôãîñèæŽ öâçîñèæ éŽîöîñðæå.à Ž ê ï Ž ä ô ã î â IJ Ž . ãåóãŽå, G = (V, E) ìèŽêŽîñèæ àîŽòæŽ. GàîŽòæï îñçæïåãæï r Žîæï ∆ r ýŽîæïýæ ãñûëáëå Žé Žîæï öâéëéïŽäôãîâèæöâçîñèæ éŽîöîñðæï ïæàîúâï.

80é Ž à Ž è æ å æ 1.3. êŽýŽääâ àŽéëïŽýñèæ îñçæïåãæï r 1 Žîæï öâéëéïŽäôãîâèæöâçîñèæ éŽîöîñðæŽ 〈v 1 , v 2 , v 3 , v 5 , v 3 , v 4 , v 1 〉, ýëèë r 2 -æï {〈v 1 , v 2 , v 3 , v 4 , v 1 〉, Žéæðëé ∆ r1 = 6, ∆ r2 = 4.êŽý. 1.5û æ ê Ž á Ž á â IJ Ž . ãåóãŽå, G = (V, E) Žîæï ìèŽêŽîñèæ àîŽòæçæ,éŽöæê∑Ž) ∆ r = 2E êâIJæïéæâîæ G îñçæïåãæï;r∈RIJ) åñ |V | ≥ 3, éŽöæê |E| ≤ 3|V | − 6.àŽêãæýæèëå K 5 ïîñèæ àîŽòæ ᎠK 3,3 ëîûæèæŽêæ ïîñèæ àîŽòæêŽý. 1.6

81û æ ê Ž á Ž á â IJ Ž . K 5 ᎠK 3,3 Žî Žîæï ìèŽêŽîñèæ àîŽòâIJæ.á Ž é ð ç æ ù â IJ Ž . åñ K 5 ìèŽêŽîñèæŽ, éŽöæê ûæêŽ öâáâàæáŽê àãâóêâIJŽ,îëé |E| ≤ 3|V | − 6, éŽàîŽé, îŽáàŽê K 5 -åãæï |V | = 5, |E| = 10, éæãæôâIJå10 ≤ 3 · 5 − 6, îŽù ŽîŽïûëîæŽ. ŽéîæàŽá, æéæï áŽöãâIJŽ, îëé K 5 ìèŽêŽîñèæŽ,Žî Žîæï ïûëîæ.ŽýèŽ áŽãñöãŽå, îëé K 3,3 ìèŽêŽîñèæŽ. àãâóêâIJŽ ∑ ∆ r = 2|E|. K 3,3 -öæŽîù âîåæ ïŽéæ ûâãîæ Žî Žîæï âîåéŽêâååŽê öââîåâIJñèæ, éŽöŽïŽáŽéâ, ∆ r ≥ 4,êâIJæïéæâîæ r ∈ R-åãæï.∑âæèâîæï òëîéñèæáŽê |R| = 2+|E|−|V | = 2+9−3 =5. éŽöŽïŽáŽéâ, ∆ r ≥ 5 · 4 = 20, ŽóâáŽê, 2|E| ≥ 20, |E| ≥ 10, éæãæôâår∈RûæꎎôéáâàëIJŽ. ŽéîæàŽá, áŽãïâIJŽ æéæïŽ, îëé K 3,3 ìèŽêŽîñèæŽ ŽîŽïûëîæŽ.K 5 ᎠK 3,3 àîŽòâIJæ æéæå Žîæï ïŽæêðâîâïë, îëé æïæêæ ŽîïâIJæåŽá \âîåŽáâîåæ"ŽîŽìèŽêŽîñèæ àîŽòâIJæŽ. õãâèŽ ïý㎠ŽîŽìèŽêŽîñè àîŽòâIJï Žóãå K 5Žê K 3,3 -æï \éïàŽãïæ" óãâàîŽòâIJæ.à Ž ê ï Ž ä ô ã î â IJ Ž .Ž) G = (V, E) àîŽòæï âèâéâêðŽîñèæ éëüæé㎠ýáâIJŽ éæïæ îëéâèæéâ[v i , v j ] ûæIJëï ŽéëàáâIJæå Ꭰv i Ꭰv j ûãâîëâIJæï àŽæàæãâIJæå.IJ) G àîŽòï âûëáâIJŽ éæüæéãŽáæ G ′ àîŽòäâ, åñ G ′ éææôâIJŽ G-àŽê âèâéâêðŽîñèæéëüæéãâIJæï éæéáâãîëIJæå öâïîñèâIJæå.é Ž à Ž è æ å æ 1.4. 1.7 êŽýŽääâ àŽéëïŽýñèæŽ G ᎠG ′ àîŽòâIJæ. ŽéŽïåŽê,G àîŽòæ éëüæéãŽáæŽ G ′ -äâ.r∈RêŽý. 1.7G àîŽòæáŽê ŽéëàáâIJñèæŽ [v 1 , v 2 ] ûæIJë (v 1 Ꭰv 2 ûãâîë öâùãèæèæŽ âîåæw 1 ûãâîëåæ). [v 4 , v 5 ] ûæIJë (v 4 Ꭰv 5 ûãâîë öâùãèæèæŽ âîåæ w 4 ûãâîëåæ),[v 6 , v 7 ] ûæIJë (v 6 Ꭰv 7 öâùãèæèæŽ w 6 ûãâîëåæ).

82å â ë î â é Ž (ç ñ î Ž ð ë ã ï ç æ ) . àîŽòæ ìèŽêŽîñèæŽéŽöæê Ꭰéýëèëá éŽöæê, îëùŽ æï Žî öâæùŽãï K 5 ᎠK 3,3 àîŽòâIJäâ éëüæéãŽáóãâàîŽòâIJï.à Ž ê ï Ž ä ô ã î â IJ Ž .Ž) G = (V, E) àîŽòæï àŽòâîŽáâIJŽ âûëáâIJŽ G-ï ûãâîëâIJæïåãæï òâîæïéæùâéŽï æïâ, îëé åñ [v, w] ∈ E, éŽöæê v Ꭰw ûãâîëâIJï ïýãŽáŽïý㎠òâîæ ñêáŽßóëêáâï.IJ) G àîŽòæï χ(G) óîëéŽðñèæ îæùýãæ âûëáâIJŽ æé òâîâIJæï éæêæéŽèñîîŽëáâêëIJŽï, îëéâèæù ïŽüæîëŽ G àîŽòæï àŽïŽòâîŽáâIJèŽá.à Ž ê ï Ž ä ô ã î â IJ Ž . ãåóãŽå, M îñ玎, àŽêãïŽäôãîëå M ′ -æïéæéŽîå ëîŽáñèæ M ′ îñçŽ öâéáâàêŽæîŽá: M-æï åæåëâñè Žîâöæ ŽãæîøæëåöæàŽ ûâîðæèæ; åñ ëî Žîâï Žóãï ïŽâîåë ûæIJë, éŽöæê àŽãŽðŽîëå ŽéŽîââIJöæ Žîøâñèæ öæàŽ ûâîðæèâIJæïåãæï áŽéŽçŽãöæîâIJâèæ îçŽèæ. âï ìîëùâïæàŽêïŽäôãîŽãï M ′ îñçŽï.é Ž à Ž è æ å æ 1.5. 1.8 êŽýŽääâ àŽéëïŽýñèæŽ M îñçŽ áŽ éæïæ ëîŽáñèæM ′ îñçŽêŽý. 1.8M ′ -æï àŽòâîŽáâIJŽ öââïŽIJŽéâIJŽ M-æï ŽîââIJæï æïâå àŽòâîŽáâIJŽï, îëùŽïŽâîåë ûæIJëï éóëêâ ŽîââIJï Žóãï ïýãŽáŽïý㎠òâîæ. éŽöŽïŽáŽéâ, åâëîâéŽ àŽòâîŽáâIJæïöâïŽýâIJ öâàãæúèæŽ øŽéëãŽõŽèæIJëå öâéáâàêŽæîŽá.å â ë î â é Ž . M îñçæï ŽîââIJæï æïâåêŽæîæ àŽòâîŽáâIJæïåãæï, îëùŽéëéæþêŽãâ ŽîââIJï ïýãŽáŽïý㎠òâîæ Žóãï, ïŽüæîëŽ ŽîŽñéâðâï ëåýæ òâîæïŽ.ï Ž ã Ž î þ æ ö ë 1.3.1. ãåóãŽå, T = (V, E) Žîæï ýâ, |V | = n. ᎎéðçæùâå, îëé |E| = n − 1.

832. öâŽéëûéâå âæèâîæï òëîéñèæï éŽîåâIJñèëIJŽ öâéáâàæ àîŽòæïåãæï3. ãåóãŽå, G = (V, E) ìèŽêŽîñèæ àîŽòæŽ, ∆ r çæ { r Žîæï ýŽîæïýæ.ᎎéðçæùâå, îëé∑∆ r = 2|E|.r∈R4. ãåóãŽå, G = (V, E) IJéñèæ ìèŽêŽîñèæ àîŽòæŽ áŽ |V | ≥ 3. ᎎéðçæùâå,îëé |E| ≤ 3|V | − 6.5. éëæõãŽêâå æéæï éŽàŽèæåæ, îëé Žé ïŽãŽîþæöëï ûæêŽ ŽéëùŽêŽöæ éëùâéñèæñðëèëIJŽ ŽîŽïûëîæŽ, îëùŽ |V | ≤ 3.6. àŽêïŽäôãîâå àîŽòæ, îëéèæïåãæïŽù χ(G) = 4.7. ãåóãŽå, T Žîæï ýâ. îŽ æóêâIJŽ χ(T )-ï éêæöãêâèëIJŽ.§ 2. éëêŽùâéåŽ ïðîñóðñîñèæ àîŽòæï ûŽîéëïŽáàâêŽáL v -æå Žôãêæöêëå v ∈ V ûãâîëï éëïŽäôãîâ ûãâîëåŽ êñïýŽ.é Ž à Ž è æ å æ 2.1. 2.1 êŽýŽääâ éëùâéñèæŽ àîŽòæ ᎠêñïýâIJæ, îëéèâIJæùöâæúèâIJŽ àŽéëãæõâêëå àîŽòæï ûŽîéëïŽáàâêŽá.êŽý. 2.1à Ž ê ï Ž ä ô ã î â IJ Ž . V = {v 1 , v 2 , . . . , v n } ûãâîëåŽ ïæéîŽãèâïûãâîëâIJæï áŽèŽàâIJñè ûõãæèåŽ {L v1 , L v2 , . . . , L vn } áŽèŽàâIJñè \êñïýŽåŽ"ïæéîŽãèâïåŽê âîåŽá áŽèŽàâIJñèæ àîŽòæ âûëáâIJŽ.

84ïŽüæîëŽ L v êñïýâIJï éëãåýëãëå àŽîçãâñèæ ìæîëIJâIJæ æéæïŽåãæï, îëéàŽêýæèñèæ ïðîñóðñîâIJæ æõëï àîŽòâIJæ:Ž) (v, v) ∉ L v õëãâèæ v ∈ V -åãæï;IJ) (w, u) ∈ L w =⇒ (u, w) ∈ L u .2.2 êŽýŽääâ àŽéëïŽýñèæ àîŽòæ öâæúèâIJŽ áŽèŽàâIJñèæ àîŽòæï ðâîéæêâIJöæöâéáâàêŽæîŽá øŽæûâîëï:({v 1 , v 2 , v 3 , v 4 },{ ((v1, v 2 ), (v 1 , v 3 ), (v 1 , v 4 ) ) , ( (v 2 , v 1 ), (v 2 , v 3 ), (v 2 , v 4 ) ) ,((v3 , v 1 ), (v 3 , v 2 ), (v 3 , v 4 ) ) , ( (v 4 , v 1 ), (v 4 , v 2 ), (v 4 , v 3 ) )}) .êŽý. 2.2áŽèŽàâIJñèæ àîŽòæ àŽêïŽäôãîŽãï âîåŽáâîå ŽîŽáŽèŽàâIJñè àîŽòï. öâ-IJîñêâIJñèæ éðçæùâIJŽ éùáŽîæŽ, îŽáàŽê, ïŽäëàŽáëá, öâïŽúèâIJâèæŽ àîŽòæïáŽèŽàâIJŽ ïýãŽáŽïý㎠ýâîýæå.à Ž ê ï Ž ä ô ã î â IJ Ž . G 1 ᎠG 2 áŽèŽàâIJñè àîŽòï âçãæãŽèâêðñîæâûëáâIJŽ, åñ ŽîïâIJëIJï f : V 1 → V 2 IJæâóùæŽ ûãâîëåŽ ïæéîŽãèââIJï öëîæï,îëéâèæù æêŽîøñêâIJï \êñïýŽåŽ" ïðîñóðñîŽï. ïý㎠ïæðõãâIJæå, åñ()L v = (v, w 1 ), (v, w 2 ), . . . , (v, w k )Žîæï G 1 -ï ûæIJëåŽ \êñïýŽ", éŽöæê( (f(v),L f(v) = f(w1 ) ) , ( f(v), f(w 2 ) ) , . . . , ( f(v), f(w k ) ))Žîæï G 2 -ï ûæIJëåŽ \êñïýŽ".é Ž à Ž è æ å æ 2.2. 2.3 êŽýŽääâ àŽéëïŽýñèæ àîŽòâIJæ âçãæãŽèâêðñîæŽ,éŽàîŽé æïæêæ, îëàëîù áŽèŽàâIJñèæ àîŽòâIJæ, Žî Žîæï âçãæãŽèâêðñîæ§ 3. àîŽòæï öâéëãèŽ3.1. öâïŽãŽèæ. àîŽòâIJåŽê áŽçŽãöæîâIJñè éîŽãŽè ŽéëùŽêŽöæ éëæåýëãâIJŽàîŽòâIJæï öâéëãèŽ, îŽù êæöêŽãï, îëé àîŽòæï åæåëâñè ûãâîëïåŽê \éæïãèŽ"ñêᎠéëýáâï éýëèëá âîåýâè. ŽéàãŽîŽá, àîŽòæï öâéëãèŽ öâàãæúèæŽûŽîéëãŽáàæêëå ûãâîëâIJæï àŽîçãâñèæ éæéáâãîëIJæï ïŽýæå.

85êŽý. 2.3åñ G = ( {v 1 , v 2 , . . . , v n }, E ) àîŽòæï Ꭰσ : N n → N n àŽáŽêŽùãèâIJŽ,éŽöæêt = v σ(1) , v σ(2) , . . . , v σ(n)éæéáâãîëIJŽ àŽêïŽäôãîŽãï G àîŽòæï öâéëãèŽï. îŽáàŽê ŽîïâIJëIJï N n -æï n!ïýãŽáŽïý㎠àŽáŽêŽùãèâIJŽ, öâïŽIJŽéæïŽá, ñêᎠŽîïâIJëIJáâï n ûãâîëæŽêæ àîŽòæïöâéëãèæï n! àŽêïýãŽãâIJñèæ ýâîýæ, Žêñ ŽîïâIJëIJï àîŽòæï ûãâîëåŽ áŽèŽàâ-IJæï n! ýâîýæ. v σ(1) ûãâîëï âûëáâIJŽ σ-åæ àŽêïŽäôãîñèæ öâéëãèæï ïŽûõæïæûãâîë.Žó éëãæõãŽêå àîŽòâIJæï öâéëãèæï éýëèëá ëî éâåëáï.3.2. àîŽòæï öâéëãèŽ ïæôîéæå. ãåóãŽå,()G = {v 1 , v 2 , . . . , v n }, {L v1 , L v2 , . . . , L vn }áŽèŽàâIJñèæ àîŽòæŽ. Žãæîøæëå îëéâèæôŽù v s (1 ≤ s ≤ n) ïŽûõæïæ ûãâîëᎠáŽãñöãŽå σ(1) = s. öâéáâà t éæéáâãîëIJæï ûãâîëâIJæ àŽêæïŽäôãîâIJŽ öâéáâàêŽæîŽá:v σ(2) Žîæï ìæîãâèæ ûãâîë L vδ(1) éëéæþêŽãâëIJæï êñïýæáŽê, (â. æ.v s -æï éëïŽäôãîâ), v δ(3) Žîæï ìæîãâèæ ûãâîë L vσ(2) -áŽê, îëéâèæù þâî ŽîŽîæï t éæéáâãîëIJŽöæ ᎠŽ. ö. v δ(k) Žîæï ìæîãâèæ ûãâîë L vσ(k−1) -áŽê, îëéâèæùþâî Žî Žîæï t-öæ. ŽéŽïåŽê, åñ öâàãýãáâIJŽ æïâåæ u ûãâîë, îëé õãâèŽûãâîë L u -áŽê ñçãâ öâïñèæŽ t-öæ, éŽöæ ìîëùâïæ àŽêéâëîáâIJŽ w ∈ t ûãâîëáŽê,ïŽáŽù w Žîæï æïâåæ ñçŽêŽïçêâèæ ûãâîë t éæéáâãîëIJŽöæ, îëé L w öâæùŽãïûãâîëâIJï, îëéèâIJæù Žî Žîæï t-öæ. öâéëãèŽ éåŽãîáâIJŽ éŽöæê, îëùŽ Žîùâîåæûãâîë V \ t-áŽê Žî öâæúèâIJŽ éæôûâñèæ æóêâï t éæéáâãîëIJæï ûãâîëâIJæáŽê.åñ àîŽòæ IJéñèæŽ, éŽöæê äâéëå Žôûâîæèæ ìîëùâïæ àŽêïŽäôãîŽãï G-ïöâéëãèŽï, ûæꎎôéáâà öâéåýãâãŽöæ G àîŽòæï éýëèëá âîå çëéìëêâêðï (îëéâèæùöâæùŽãï v σ(1) -ï). åñ àîŽòæ Žî Žîæï IJéñèæ, éŽöæê G-ï ïîñèæ öâéëãèæïéæïŽôâIJŽá ìîëùâïæ ñêᎠáŽãæûõëå G àîŽòæï åæåëâñè IJéñè çëéìëêâêðöæ.Žé éâåëáæï ïŽöñŽèâIJæå öâàãæúèæŽ àŽêãïŽäôãîëå àîŽòæï IJéñèæ çëéìëêâêðâIJæïîŽëáâêëIJŽ. IJéñè àîŽòöæ ïŽûõæïæ ûãâîëï åæåëâñèæ öâîøâãæï áîëïéæãæôâIJå âîåŽáâîå öâéëãèŽï. ŽéîæàŽá, öâïŽúèâIJâèæŽ áŽèŽàâIJñèæ IJéñèæàîŽòæï n öâéëãèŽ (ïæôîéæå). åñ G-ï Žóãï V i (1 ≤ i ≤ p) IJéñèæ çëéìëêâêðâIJæ,ïŽáŽù |V i | = n i , éŽöæê àãâóêâIJŽ n 1 · n 2 · n p ïæôîéæå öâéëãèŽ.

86é Ž à Ž è æ å æ 3.1. àŽêãæýæèëå 3.1 êŽýŽääâ àŽéëïŽýñèæ áŽèŽàâIJñèæàîŽòæ. v 1 ïŽûõæïæ ûãâîëáŽê ïæôîéæå öâéëãèŽ àŽêæïŽäôãîâIJŽ öâéáâàêŽæîŽá:v 1 , v 2 , v 4 , v 8 , v 5 , v 6 , v 3 , v 7 .êŽý. 3.13.3. àŽêæãæ öâéëãèŽ. ãåóãŽå,()G = {v 1 , v 2 , . . . , v n }, {L v1 , L v2 , . . . , L vn }Žîæï áŽèŽàâIJñèæ àîŽòæ. Žãæîøæëå ïŽûõæïæ v s ûãâîë ᎠáŽãñöãŽå, îëéL vs = ( (v s , w 1 ), (v s , w 2 ), . . . , (v s , w k ) ) .t éæéáâãîëIJæï ìæîãâèæ k+1 ûâãîæ àŽêæïŽäôãîâIJŽ öâéáâàêŽæîŽá: v σ(1) =v s , v σ(2) = w 1 , . . . , v σ(k+1) = w k , ýëèë v σ(k+1+i) Žîæï L w1 -öæ i-ñîæ ûãâîæ,îëéâèæù Žî Žîæï t-öæ. Žéæå ŽéëæûñîâIJŽ L w1 , ìîëùâïæ áŽæûõâIJŽ L w2 -áŽêᎠŽ. ö. öâéëãèŽ öâûõáâIJŽ, îëù㎠õãâèŽ ûãâîë, îëéèâIJæù v σ(1) -áŽê éæôûâãŽáæŽ,öâ㎠t-öæ. åñ àîŽòæ IJéñèæŽ, éŽöæê Žé ìîëùâïæå v σ(1) -áŽê éææôûâãŽõãâèŽ ûãâîë. â. æ. ýáâIJŽ àîŽòæï ïîñèæ öâéëãèŽ. åñ Žî Žîæï IJéñèæ, éŽöæêåæåëâñèæ çëéìëêâêðæïåãæï ñêᎠöâæîøâï ïŽûõæïæ ûâîðæèæ. IJéñèæ àîŽòæïåãæïïŽûõæïæ v σ(1) ûâîðæèæï öâîøâ㎠àŽêïŽäôãîŽãï âîåŽáâîå öâéëãèŽï.ŽéîæàŽá, åñ àîŽòæ n ûãâîëï öâæùŽãï, ŽîïâIJëIJï G àŽêæãæ öâéëãèŽ (îëàëîùïæôîéæï öâéëãèæï áîëï). åñ G Žî Žîæï IJéñèæï Ꭰéæïæ çëéìëêâêðâIJæŽ V i(1 ≤ i ≤ p), |V i | = n i , éŽöæê ŽîïâIJëIJï n 1 · n 2 · · · n p àŽêæãæ öâéëãèŽ.é Ž à Ž è æ å æ 3.2. 3.1 êŽýŽääâ àîŽòæï àŽêæãæ öâéëãèŽ ïŽûõæïæ v 1ûâîðæèæáŽê öâïîñèáâIJŽ öâéáâàêŽæîŽá:t = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 , v 7 , v 8 ).3.4. ïæôîéæå ᎠàŽêæãæ öâéëãèâIJæï îñòîâíûå ðõââIJæ. ãåóãŽå,G = ( {v 1 , v 2 , . . . , v n }, E ) Žîæï àîŽòæ, ýëèë t = v σ(1) , v σ(2) , . . . , v σ(n) ŽîæïG-ï öâéëãèŽ. éŽöæê t öâéëãèŽ àŽêïŽäôãîŽãï E ûæIJëåŽ ïæéîŽãèâï E t óãâïæéîŽãèâïöâéáâàêŽæîŽá: [v, w] ∈ E t éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ [v, w] ûæIJë

87àŽéëæõâêâIJŽ öâéëãèæï ŽïŽàâIJŽá. ŽêŽèëàæñîŽá, àŽêæïŽäôãîâIJŽ áŽèŽàâIJñèæàŽîòâIJæï L v ïæéîŽãèââIJæïåãæï L t v óãâïæéîŽãèââIJæ.û æ ê Ž á Ž á â IJ Ž . ãåóãŽå,G = ( {v 1 , v 2 , . . . , v n }, [L v1 , L v2 , . . . , L vn ] )áŽèŽàâIJñèæ IJéñèæ àîŽòæ, ýëèë t öâéëã莎, éŽöæêG t = ( {v 1 , v 2 , . . . , v n }, {L t v 1, L t v 2, . . . , L t v n} )æóêâIJŽ G-ï áŽèŽàâIJñèæ îñòîâíîå ýâ.á Ž é ð ç æ ù â IJ Ž . îŽáàŽê G Žîæï IJéñèæ àîŽòæ, Žéæðëé G t -ù ŽàîâåãâIJéñèæŽ áŽ ûŽîéëŽáàâêï G-ï øëêøýï ( îñòîâ) (îŽáàŽê G t öâæùŽãï G-ï õãâèŽûãâîëï). åñ G t öâæùŽãï øŽçâðæè éŽîöîñðï, éŽöæê äëàæâîåæ ûãâîë t-öæàŽéëøêáâIJŽ âîåäâ éâðþâî. éŽàîŽé, îŽáàŽê t Žîæï öâéëãèŽ âï öâñúèâIJâèæŽ,Žéæðëé G t Žîæï Žùæçèñîæ. éŽöŽïŽáŽéâ, G t Žîæï ýâ.ö â á â à æ . õãâèŽ IJéñè àîŽòï Žóãï îñòîâíîå ýâ.êŽý. 3.2é Ž à Ž è æ å æ 3.3. 3.1 êŽýŽääâ àŽéëïŽýñèæ àîŽòæïåãæï v 1 ïŽûõæïæûãâîëáŽê ïæôîéæå ᎠàŽêæãæ öâéëãèâIJæå àŽêïŽäôãîñèæ îñòîâíûå ýââIJææóêâIJŽ, öâïŽIJŽéæïŽá, 3.2(Ž) Ꭰ3.2(IJ) êŽýŽäâIJäâ àŽéëïŽýñèæ ýââIJæ.ŽîŽIJéñèæ àîŽòâIJæïåãæï ïîñèæ öâéëãèŽ àŽêïŽäôãîŽãï îñòîâíîé ðõâï.ï Ž ã Ž î þ æ ö ë 3.1.1. ãåóãŽå, G = ( {v 1 , v 2 , v 3 , v 4 , v 5 , v 6 }, {L v1 , L v2 , L v3 , L v4 , L v5 , L v6 } ) áŽèŽàâIJñèæàîŽòæŽ, ïŽáŽùL v1 = ( (v 1 , v 2 ), (v 1 , v 3 ), (v 1 , v 4 ) ) ,L v2 = ( (v 2 , v 1 ), (v 2 , v 5 ) ) ,L v3 = ( (v 3 , v 1 ), (v 3 , v 6 ) ) ,L v4 = ( (v 4 , v 1 ), (v 4 , v 6 ), (v 4 , v 5 ) ) ,L v5 = ( (v 5 , v 2 ), (v 5 , v 6 ), (v 5 , v 4 ) ) ,L v6 = ( (v 6 , v 3 ), (v 6 , v 4 ), (v 6 , v 5 ) ) .

88àŽêïŽäôãîâåŽ) ïæôîéæå öâéëãèŽ v 2 ïŽûõæïæ ûâîðæèæáŽê;IJ) àŽêæãæ öâéëãèŽ v 6 ïŽûõæïæ ûâîðæèæáŽê.2. áŽýŽäâå ûæêŽ ŽéëùŽêŽöæ éëõãŽêæèæ G àîŽòæï öâéëãèâIJæå àŽêïŽäôãîñèæîñòîâíîå ýââIJæ.3. ãåóãŽå, G àîŽòæï éëéæþêŽãâëIJæï éŽðîæùŽï Žóãï IJèëçñîæ ïðîñóðñîŽ,⎡⎤A 1 0 . . . 0A = ⎢ 0 A 2 . . . 0⎥⎣. . . . . . . . . . . . . . . . . ⎦ ,0 0 . . . A kïŽáŽù åæåëâñèæ A i Žîæï IJñèæï âèâéâêðâIJæŽêæ çãŽáîŽðñèæ éŽðîæùŽ, ýëèëõãâèŽ áŽêŽîøâêæ âèâéâêðæ 0-æï ðëèæŽ. îŽ öâæúèâIJŽ æåóãŽï G àîŽòæïåãæïâIJâIJäâ? éëæõãŽêâå âîåæ éŽàŽèæåæ ᎠŽŽàâå öâïŽIJŽéæïæ êŽýŽäæ.

ëîæâêðæîâIJñèæ àîŽòâIJæ§ 1. ëîæâêðæîâIJñèæ àîŽòâIJæ1.1. öâïŽãŽèæ. àîŽòåŽ åâëîææï àŽéëõâêâIJæïŽï ýöæîŽá éëæåýëãâIJŽ,îëé àîŽòæï ûæIJëâIJï ßóëêáâï éæéŽîåñèâIJŽ.à Ž ê ï Ž ä ô ã î â IJ Ž . ëîæâêðæîâIJñèæ àîŽòæ Žîæï G = (V, E)ûõãæèæ, ïŽáŽù V Žîæï ûãâîëâIJæï ïŽïîñèæ ïæéîŽãèâ, ýëèë E çæ { V × VáâçŽîðñèæ êŽéîŽãèæï êâIJæïéæâîæ óãâïæéîŽãèâ.û æ ê Ž á Ž á â IJ Ž . Ž) ëîæâêðæîâIJñèæ G = (V, E) àîŽòæ àŽêïŽäôãîŽãïéæéŽîåâIJŽï V ïæéîŽãèâäâ;IJ) ãåóãŽå, V ïŽïîñèæ ïæéŽîãèâŽ. éŽöæê V ïæéîŽãèâäâ IJæêŽîñèæ éæéŽîåâIJŽàŽêïŽäôãîŽãï ëîæâêðæîâIJñè àîŽòï, îëéèæï ûãâîëåŽ ïæéîŽãè⎠V .á Ž é ð ç æ ù â IJ Ž . Ž) R(E) éæéŽîåâIJŽ àŽêãïŽäôãîëå öâéáâàêŽæîŽá:vR(E)w éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ (v, w) ∈ E. ùýŽáæŽ, R(E) æóêâIJŽIJæêŽîñèæ éæéŽîåâIJŽ.IJ) ãåóãŽå, R Žîæï IJæêŽîñèæ éæéŽîåâIJŽ R ïæéîŽãèâäâ, éŽöæê ûæIJëåŽïæéîŽãèâ ŽãŽàëå öâéáâàêŽæîŽá: (v, w) ∈ E éŽöæê Ꭰéýëèëá éŽöæê, îëùŽvRw. ŽéàãŽîŽá ŽàâIJñèæ G = (V, E) æóêâIJŽ ëîæâêðæîâIJñèæ àîŽòæ.ûæIJëï éæéŽîåñèâIJŽ àŽêæïŽäôãîâIJŽ ûõãæèæï áŽèŽàâIJæå. éŽàŽèæåŽá, åñ(v, w) ∈ E, ãæðõãæå, îëé ûæIJë àŽéëáæï v-áŽê Ꭰöâáæï w-öæ. áæŽàîŽéŽäâéæéŽîåñèâIJæï éæïŽåæåâIJèŽá æïîâIJï æõâêâIJâê.é Ž à Ž è æ å æ 1.1.Ž) ãåóãŽå,V = {v 1 , v 2 , v 3 , v 4 },{}E 1 = (v 1 , v 2 ), (v 1 , v 4 ), (v 2 , v 3 ), (v 3 , v 1 ), (v 3 , v 4 ) .êŽýŽääâ éëùâéñèæŽ Žé àîŽòæï éëéæþêŽãâëIJæï éŽðîæùŽ ᎠàŽéëïŽýñèâIJŽIJ) ãåóãŽå,{}E 2 = (v 1 , v 2 ), (v 1 , v 3 ), (v 1 , v 4 ), (v 2 , v 3 ), (v 3 , v 1 ), (v 3 , v 4 ), (v 4 , v 4 ) ,G = (V, E 2 ) ëîæâêðæîâIJñèæ àîŽòæï éëéæþêŽãâëIJæï A 2 éŽðîæùŽ ᎠàŽéëïŽýñèâIJŽéëùâéñèæŽ 1.2 êŽýŽääâ:89

90êŽý. 1.1êŽý. 1.2öâãêæöêëå, îëé ëîæâêðæîâIJñèæ àîŽòæå àŽêïŽäôãîñèæ IJæêŽîñèæ éæéŽîåâIJŽ,ïŽäëàŽáëá, Žî Žîæï ïæéâðîæñèæ, ŽîŽîâòèâóïñîæ. öâïŽIJŽéæïŽá,éëéæþêŽãâëIJæï éŽðîæùæïåãæï öâæúèâIJŽ A ≠ A ′ Ꭰa ii ≠ 0. (v, v) ïŽýæï ûæ-IJëâIJï éŽîõñíï ñûëáâIJâê. v ∈ V ûãâîëï δ(v) ýŽîæïýæ öâæúèâIJŽ øŽæûâîëïδ(v) = δ − (v) + δ + (v) þŽéæï ïŽýæå, ïŽáŽù δ − (v) Žîæï v-öæ öâéŽãŽèæ ûæ-IJëâIJæï îŽëáâêëIJŽ, ýëèë δ + (v) Žîæï v-áŽê àŽéëéŽãŽè ûæIJëåŽ îŽëáâêëIJŽ.{w : (w, v) ∈ E} Ꭰ{w : (v, w) ∈ E} ïæéîŽãèââIJï ñûëáâIJâê v ûãâîëïöâéŽãŽè ᎠàŽéëéŽãŽè çãŽêúâIJï, öâïŽIJŽéæïŽá.âçãæãŽèâêðëIJæïŽ áŽ éëêæöãêæï ùêâIJâIJæ ëîæâêðæîâIJñè àîŽòâIJäâ IJñêâIJîæãŽáàŽêäëàŽáâIJŽ.1.2. éŽîöîñðâIJæ ᎠIJéñèëIJŽ ëîæâêðæîâIJñè àîŽòâIJöæ.à Ž ê ï Ž ä ô ã î â IJ Ž . k ïæàîúæï éŽîöîñðæ v ûãâîëáŽê w-öæ ëîæâêðæîâIJñèG = (V, E) àîŽòöæ âûëáâIJŽ ûæIJëâIJæ öâéáâàæ ïŽýæï éæéáâãîëIJŽï:(v, w 1 ), (w 1 , w 2 ), . . . , (w k−2 , w k−1 ), (w k−1 , w).â. æ. åæåëâñèæ ûæIJëï éâëîâ ûãâîë âéåýãâ㎠öâéáâàæ ûæIJëï ìæîãâè ûãâîëï.ýöæîŽá éëýâîýâIJñèæŽ éŽîöîñðæ ûŽîéëãŽáàæêëå ûãâîëâIJæï æév, w, w 2 , . . . , w k−2 , w k−1 , w

91éæéáâãîëIJæå, îëéèâIJæù éŽï àŽêïŽäôãîŽãï. åñ v = w, éŽîöîñðï âûëáâIJŽøŽçâðæèæ. éŽîöîñðæ Žêñ ùæçèæ. ëîæâêðæîâIJñè àîŽòï, îëéâèæù ùæçèï ŽîöâæùŽãï, Žùæçèñîæ âûëáâIJŽ.ãåóãŽå, D ŽôêæöêŽãï õãâèŽ ëîæâêðæîâIJñèæ àîŽòæï ïæéîŽãèâï, ýëèëG { õãâèŽ àîŽòæï ïæéîŽãèâï. öâàãæúèæŽ àŽêãïŽäôãîëå F : D → G ŽïŽýãŽöâéáâàêŽæîŽá.à Ž ê ï Ž ä ô ã î â IJ Ž . ãåóãŽå, G = (V, E) ∈ D ëîæâêðæîâIJñèæàîŽòæŽ. éŽöæê F (G) ∈ G àîŽòæï ûãâîëåŽ ïæéîŽãèâ âéåýãâ㎠V -ï, ýëèëF (G)-æï ûæIJëåŽ ïæéîŽãèâ àŽêæïŽäôãîâIJŽï E-äâ øŽðŽîâIJñèæ öâéáâàæ ëìâîŽùæâIJæå:Ž) E-áŽê éëãŽöëîëå õãâèŽ éŽîõñíæ;IJ) (v, w) öâãùãŽèëå [v, w]-æå õãâèŽ (v, w) ∈ E-åãæï. éŽöæê F (G)àîŽòæ æóêâIJŽ G ëîæâêðæîâIJñè àîŽòåŽê áŽçŽãöæîâIJñèæ àîŽòæ.ëîæâêðæîâIJñèæ àîŽòâIJæïåãæï IJéæï ùêâIJŽ ñòîë éâðŽá öæꎎîïëIJîæãæŽ,ãæáîâ àîŽòâIJæïåãæï. àŽêãïŽäôãîëå ëîæâêðæîâIJñèæ àîŽòâIJæï IJéæï ïŽéæ ðæìæ:à Ž ê ï Ž ä ô ã î â IJ Ž . ãåóãŽå, G = (V, E) Žîæï ëîæâêðæîâIJñèæàîŽòæ. ãæðõãæå, îëéŽ) G ïñïðŽá IJéñèæŽ, åñ F (G) àîŽòæ Žîæï IJéñèæ.IJ) G ùŽèéýîæãŽá IJéñèæŽ, åñ v, w ∈ V ïýãŽáŽïý㎠ûãâîëï õëãâèæûõãæèæïåãæï ŽîïâIJëIJï éŽîöîñðæ v-áŽê w-öæ Žê w-áŽê v-öæ.à) G úèæâîŽá IJéñèæŽ, åñ v, w ∈ V ïýãŽáŽïý㎠ûãâîëï õëãâèæûõãæèæïåãæï ŽîïâIJëIJï éŽîöîñðæ v-áŽê w-öæ Ꭰw-áŽê v-öæ.ùýŽáæŽ, îëéG úèæâîŽá éIJñèæŽ =⇒ G ùŽèéýîæãŽá IJéñèæŽ =⇒ G IJéñèæŽ.é Ž à Ž è æ å æ 1.2. 1.3 êŽýŽääâ àŽéëïŽýñèæŽ ëîæâêðæîâIJñèæ àîŽòæŽ) éýëèëá ïñïðŽá IJéñèæ (æý. êŽý. 1.3(Ž));IJ) ùŽèéýîæãŽá IJéñèæŽ (æý. êŽý. 1.3(IJ));à) úèæâîŽá IJéñèæŽ (æý. êŽý. 1.3(à));C = A(R ∗ ) IJéñèëIJæï éŽðîæùæï ðâîéæêâIJöæ ëîæâêðæîâIJñèæ G àîŽòæúèæâîŽá IJéñèæŽ éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ c ij = 1, õëãâèæ 1 ≤ i, j ≤ n-åãæï; G ùŽèéýîæãŽá IJéñèæŽ éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ c ij ∨ c ji = 1,õëãâèæ 1 ≤ i, j ≤ n-åãæï.é Ž à Ž è æ å æ 1.3. àŽêãæýæèëå 1.4 êŽýŽääâ áæŽàîŽéæå ûŽîéëáàâêæèæëîæâêðæîâIJñèæ àîŽòæ

92êŽý. 1.3êŽý. 1.4Žé àîŽòæïåãæï àãŽóãï:⎡⎤⎡⎤0 1 0 1 00 1 1 0 10 1 1 0 0A(R) =⎢1 1 0 0 1⎥⎣0 0 1 0 1⎦ , 1 1 1 0 1A(R2 ) =⎢1 1 1 1 0⎥⎣1 1 0 0 1⎦ ,1 0 0 0 00 1 0 1 0

Žéæðëé⎡⎤⎡⎤1 1 1 0 11 1 1 1 11 1 1 1 1A(R 3 ) =⎢1 1 1 1 1⎥⎣1 1 1 1 0⎦ , 1 1 1 1 1A(R4 ) =⎢1 1 1 1 1⎥⎣1 1 1 1 1⎦ ,0 1 1 0 11 1 1 0 1C =4∨A(R 4 ) = I ∨ A(R) ∨ A(R 2 ) ∨ A(R 3 ) ∨ A(R 4 )k=0éŽðîæùæïåãæï c ij = 1 õãâèŽ 1 ≤ i, j ≤ 5-åãæï, ŽéîæàŽá, éëùâéñèæ àîŽòæŽîæï úèæâîŽá IJéñèæ.åñ G = (V, E) Žîæï ëîæâêðæîâIJñèæ àîŽòæŽ, éŽöæê âçãæãŽèâêðëIJæï ρéæéŽîåâIJæå V ûãâîëåŽ ïæéîŽãèâ öâæúèâIJŽ áŽãõëå çèŽïâIJŽá öâéáâàêŽæîŽá:vρw, åñ v = w Žê ŽîïâIJëIJï éŽîöîñðæ v-áŽê w-öæ Ꭰìæîæóæå. åñ {V i L 1 ≤i ≤ p} Žîæï V -ï áŽõëòŽ Ꭰ{E i : 1 ≤ i ≤ p}, ïŽáŽù E i = (V i × V i ) ∩ E ŽîæïûæIJëåŽ öâïŽIJŽéæïæ ïæéîŽãèâ, éŽöæê G i = (V i , E i ) (1 ≤ i ≤ p) óãâàîŽòâIJïâûëáâIJŽ G àîŽòæï úèæâîŽá IJéñèæ çëâòæùæâêðâIJæ.ùýŽáæŽ, îëé ρ ⊆ R ∗ ᎠA(ρ) öâæúèâIJŽ àŽêæïŽäôãîëï A(R ∗ )-áŽê, îëàëîùA(ρ) ij = A(R ∗ ) ij ∧ A(R ∗ ) ji .G àîŽòæ úèæâîŽá IJéñèæŽ éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ G-ï Žóãï éýëèëáâîåæ úèæâîŽá IJéñèæ çëéìëêâêðæ, â. æ. åñ p = 1.é Ž à Ž è æ å æ 1.4. àŽêãæýæèëå 1.5 êŽýŽääâ àŽéëïŽýñèæ ëîæâêðæîâ-IJñèæ àîŽòæ. àãâóêâIJŽ⎡ ⎤ ⎡ ⎤1 1 1 11 0 0 0A(R ∗ ) = ⎢0 1 1 0⎥⎣0 0 1 0⎦ , A(p) = ⎢0 1 0 0⎥⎣0 0 1 0⎦ .0 0 1 10 0 0 193êŽý. 1.5ŽéîæàŽá, G i = ({v i }, ∅) (1 ≤ i ≤ 4) Žîæï G àîŽòæï úèæâîŽá IJéñèæçëéìëêâêðâIJæ.

94ãåóãŽå, G = (V, E) Žùæçèñîæ àîŽòæŽ. v ∈ V ûãâîëï âûëáâIJŽ òëåëèæ,åñ δ + (v) = 0. åñ (v, w) ∈ E, éŽöæê v Žîæï w-ï ñöñŽèë ûæêŽìŽîæ,ýëèë w çæ { v-ï ñöñŽèë öåŽéëãŽèæ. åñ ŽîïâIJëIJï éŽîöîñðæ v-áŽê w-öæ,éŽöæê ŽéIJëIJâê, îëé v Žîæï w-ï ûæêŽìŽîæ, ýëèë w çæ { v-ï öåŽéëéŽãŽèæ.Žé ùêâIJâIJï Žäîæ Žî Žóãï ùæçèæï öâéùãâèæ àîŽòâIJæïåãæï.é Ž à Ž è æ å æ 1.5. 1.6 êŽýŽääâ àŽéëïŽýñèæŽ ëîæâêðæîâIJñèæ ŽùæçèñîæàîŽòæ.êŽý. 1.6v 3 Žîæï v 5 -æï ûæêŽìŽîæ.v 4 Žîæï v 5 -æï ñöñŽèë ûæêŽìŽîæ.v 5 Žîæï v 3 -æï ñöñŽèë öåŽéëéŽãŽèæ.éüæáîë çŽãöæîæ ŽîïâIJëIJï Žùæçèñîæ ëîæâêðæîâIJñè àîŽòâIJïŽ áŽ êŽûæèëIJîæãáŽèŽàâIJñè éæéŽîåâIJâIJï öëîæï.û æ ê Ž á Ž á â IJ Ž . Ž) ãåóãŽå, \

95Ž) G Žîæï ýâ.IJ) F (G) àîŽòæ Žîæï IJéñèæ ᎠŽîïâIJëIJï v r ûãâîë, îëéâèïŽù Žî ŽóãïûæêŽìŽîæ, ýëèë õãâèŽ áŽêŽîøâê ûãâîëï Žóãï éýëèëá âîåæ ñöñŽèëûæêŽìŽîæ.à) G-ï Žóãï v r ûãâîë, îëéâèæù êâIJæïéæâî ïý㎠ûãâîëï ñâîåáâIJŽ âîåŽáâîåæéŽîöîñðæå.á) G-ï Žóãï v r ûãâîë, îëéâèïŽù Žî Žóãï ûæêŽìŽîæ; õã âèŽ ïý㎠ûãâîëïŽóãï éýëèëá âîåæ ñöñŽèë ûæêŽìŽîæ; ŽîïâIJëIJï éŽîöîñðæ v r -áŽêåæåëâñè ûãâîëéáâ.1.3. áŽèŽàâIJñèæ ëîæâêðæîâIJñèæ àîŽòâIJæ ᎠöâéëãèâIJæ. éëéæþêŽãâëIJæïêñïýâIJæ éëéæþêŽãâëIJæï éŽðîæùâIJæï éæéŽîå ëîæâêðæîâIJñèæ àîŽòâ-IJæï ûŽîéëáàâêæï ŽèðâîêŽðæñèæ òëî鎎.à Ž ê ï Ž ä ô ã î â IJ Ž . áŽèŽàâIJñèæ ëîæâêðæîâIJñèæ àîŽòæ âûëáâIJŽG = (V, E) ûõãæèï, ïŽáŽù V ûãâîëâIJæï ïŽïîñèæ ïæéîŽãèâŽ, ýëèë E ëîæâêðæîâIJñèæûæIJëâIJæ áŽèáŽàâIJñè êñïýŽåŽ ïæéîŽãèâŽ. E-ï âèâéâêðâIJï Žóãïöâéáâàæ ïŽýâ:()L v = (v, w 1 ), (v, w 2 ), . . . , (v, w k ) ,ïŽáŽù v, w k ∈ V .é Ž à Ž è æ å æ 1.6. àŽêãæýæèëå áŽèŽàâIJñèæ ëîæâêðæîâIJñèæ àîŽòæG =((v 1 , v 2 , v 3 , v 4 , v 5 ),{ ((v1, v 2 ), (v 1 , v 3 ) ) , ( (v 2 , v 3 ) ) , ( (v 4 , v 3 ), (v 4 , v 5 ), (v 4 , v 4 ) )}) .G àîŽòæ öâæúèâIJŽ ûŽîéëãŽáàæêëå éëéæþêŽãâëIJæï êñïýâIJæå:êŽý. 1.7ᎠöâæúèâIJŽ ûŽîéëãŽáàæêëå áæŽàîŽéæå (êŽý. 1.9)áŽèŽàâIJñèæ G ëîæâêðæîâIJñèæ àîŽòæ àŽêïŽäôãîŽãï âîåŽáâîå ŽîŽáŽèŽàâIJñèëîæâêðæîâIJñèæ àîŽòï: åæåëâñèæ áŽèŽàâIJñèæ((v, w1 ), (v, w 2 ), . . . , (v, w k ) )

96êŽý. 1.8êñïýŽ ñêᎠöâãùãŽèëå { (v, w 1 ), (v, w 2 ), . . . , (v, w k ) } ïæéîŽãèæå. ŽïâåêŽæîŽáàŽêïŽäôãîñè ëîæâêðæîâIJñè àîŽòï âûëáâIJŽ G-ï áŽóãâéáâIJŽîâIJñèæëîæâêðæîâIJñèæ àîŽòæ.é Ž à Ž è æ å æ 1.7. T Žîæï áŽèŽàâIJñèæ ëîæâêðæîâIJñèæ ýâT =( {v1, v 2 , v 3 , v 4 , v 5 , v 6 , v 7 , v 8},{ ((v1, v 2 ), (v 2 , v 3 ) ) , ( (v 2 , v 4 ), (v 2 , v 5 ) ) , ( (v 3 , v 6 ), (v 3 , v 7 ), (v 3 , v 8 ) )}) .êŽý. 1.9v 1 Žîæï T ýæï òâïãæ (æý. êŽý. 1.10).à Ž ê ï Ž ä ô ã î â IJ Ž . Ž) ãåóãŽå, S V ᎠS E ïæéîŽãèââIJæŽ. G = (V, E)áŽèŽàâIJñèæ ëîæâêðæîâIJñèæ àîŽòæï ïîìåòêà (êæöŽêæ, éëêæöãêŽ) âûëáâIJŽ(f, g), ïŽáŽùf : V −→ S V∞⋃g : E −→k=1Žîæï ûãâîæåŽ êæöŽêæ,S k EŽîæï ûæIJëåŽ êæöŽêæ.

97g ŽïŽýãŽï Žóãï öâéáâàæ ïŽýâ:()g (v, w 1 ), (v, w 2 ), . . . , (v, w k ) = (α 1 , α 2 , . . . , α k ) ∈ SE.k2) ŽéIJëIJâê, îëé ëîæ G 1 = (V 1 , E 1 ) ᎠG 2 = (V 2 , E 2 ) êæöêæŽêæ àîŽòæ(f 1 , g 1 ) Ꭰ(f 2 , g 2 ) éëêæöãêâIJæï òñêóùæâIJæå, âçãæãŽèâêðñîæŽ, åñ ŽîïâIJëIJïæïâåæ h : V 1 → V 2 IJæâóùæŽ, îëéŽ) ( (v, w 1 ), (v, w 2 ), . . . , (v, w k ) ) ∈ E 1 éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ((h(v), h(w1 ) ) , . . . , ( h(v), h(w k ) )) ∈ E 2 (âçãæãŽèâêðñîæŽ îëàëîùáŽèŽàâIJñèæ àîŽòâIJæ);IJ) f 1 (v) = f 2 (h(v)) õãâèŽ v ∈ V ûãâîëïåãæï (ûãâîëåŽ êæöêâIJæ âéåýãâãŽ);à) õãâèŽ ( (v, w 1 ), (v, w 2 ), . . . , (v, w k ) ) ∈ E 1 ûæIJëïåãæï àãŽóãï) ( (h(v),g 1((v, w 1 ), (v, w 2 ), . . . , (v, w k ) = g 2 h(w1 ) ) , . . . , ( h(v), h(w k ) ))(â. æ. ûæIJëåŽ êæöêâIJæ âéåýãâãŽ).àŽêãïŽäôãîëå ëîæâêðæîâIJñèæ àîŽòâIJæï öâéëãèŽ. áŽèŽàâIJñèæ ëîæâêðæîâIJñèæàîŽòâIJæï öâéëãèŽ äñïðŽá æïâãâ àŽêæïŽäôãîâIJŽ, îëàëîù ŽîŽëîæâêðæîâIJñèæàîŽòâIJæïåãæï. áŽèŽàâIJñèæ ëîæâêðæîâIJñèæ ýââIJæïåãæï ýöæîŽáéëïŽýâîýâIJâèæŽ ïý㎠öâéëãèâIJæ.à Ž ê ï Ž ä ô ã î â IJ Ž . ãåóãŽå, T = ( {v 1 , v 2 , . . . , v n }, E ) ŽîæïáŽèŽàâIJñèæ ëîæêâðæîâIJñèæ ýâ áŽL v = ( (v, w 1 ), (v, w 2 ), . . . , (v, w k ) ) ∈ E.{w 1 , w 2 , . . . , w k } ïæéîŽãèâäâ àŽêãïŽäôãîëå < éæéŽîåâIJŽ öâéáâàêŽæîŽá: w i

98åñ x ≠ y, éŽöæê àãâóêâIJŽ δ − (w) = 2 ((x, w) Ꭰ(y, w)), îŽù öâñúèâIJâèæŽ,îŽáàŽê T Žîæï ýâ. ŽéîæàŽá, x = y áŽ()L x = (x, w 1 ), . . . , (x, v), . . . , (x, w), . . . , (x, u), . . . , (x, w k ) .â. æ. v < u.ŽéîæàŽá, < éæéŽîåâIJŽ Žîæï êŽûæèëIJîæãæ áŽèŽàâIJæï éæéŽîåâIJŽ V -äâ.< éæéŽîåâIJŽ ŽáŽîâIJï éýëèëá âîåæ ûãâîëáŽê àŽéëïñè ûãâîëâIJï.é Ž à Ž è æ å æ 1.8. áŽèŽàâIJñèæ ëîæâêðæîâIJñèæ T ýæïåãæï (æý. êŽý.1.11) àãŽóãï:v 2 < v 3 , v 4 < v 5 , v 6 < v 7 , v 7 < v 8 , v 6 < v 8 .êŽý. 1.10ö â ê æ ö ê ã Ž . v ∈ V , Γ + (v)-æå Žôãêæöêëå õãâèŽ æé ûãâîëåŽ ïæéîŽãèâ,îëéèâIJæïåãæïŽù v Žîæï \ûæêŽìŽîæ". ŽêŽèëàæñîŽá, Γ − (v)-æå Žôãêæöêëå æéûãâîëåŽ ïæéîŽãèâ, îëéèâIJæïåãæïŽù v Žîæï \öåŽéëéŽãŽèæ".à Ž ê ï Ž ä ô ã î â IJ Ž . < éæéŽîåâIJŽï ñûëáâIJâê áŽèŽàâIJñèæ ëîæâêðæîâIJñèæT = (V, E) ýæï ûãâîëâIJæï ðîŽêïãâîïŽèñî áŽèŽàâIJŽï.ãåóãŽå, < éæéŽîåâIJŽ àŽêïŽäôãîŽãï < l éæéŽîåâIJŽï V -äâ öâéáâàêŽæîŽá:åñ w i < w j , éŽöæê w ′ i < l w ′ j õãâèŽ w′ i ∈ Γ+ (w i ) ∪ {w i } ᎠõãâèŽ w ′ j ∈Γ + (w j ) ∪ {w j }-åãæï.û æ ê Ž á Ž á â IJ Ž .1)

99âï êæöêŽãï, îëé δ − (v) = 2 Žê δ − (w) = 2 îëéâèæôŽù w ∈ Γ − (v)-åãæï. âïöâñúèâIJâèæŽ, îŽáàŽê T Žîæï ýâ. ŽéîæàŽá, v ≮ l v l õãâèŽ v ∈ V -åãæï.IJ) ãåóãŽå, àãŽóãï v < l w Ꭰw < l u. v < l w êæöêŽãï, îëé v < w ŽêŽîïâIJëIJï æïâåæ x, y ∈ V , îëé x < y Ꭰv ∈ Γ + (x) ∪ {x}, w ∈ Γ + (y) ∪ {y}.w < l u êæöêŽãï, îëé w < u Žê ŽîïâIJëIJï æïâåæ r, s ∈ V ûãâîëâIJæ, îëé r < sᎠw ∈ Γ + (r) ∪ {r}, u ∈ Γ + (s) ∪ {s}. w ∈ Γ + (r) ∪ {r} Ꭰw ∈ Γ + (y) ∪ {y}àãŽúèâãï, îëé Žê r ∈ Γ + (y), Žê y ∈ Γ + (r). éŽöŽïŽáŽéâ, ýâï Žóãï 1.12 êŽýŽääâàŽéëïŽýñèæ âîå-âîåæ òëîéŽ.êŽý. 1.11åñ r ∈ Γ + (y), éŽöæê s ∈ Γ + (y) Ꭰx < y àãŽúèâãï x < l s. éŽàîŽéu ∈ Γ + (s), öâïŽIJŽéæïŽá, x < l u áŽ, îŽáàŽê v ∈ Γ + (z) àãŽóãï v < l u. åñ y ∈Γ + (r), éŽöæê x ∈ Γ + (r) Ꭰr < s àãŽúèâãï x < e l s. éŽàîŽé, u ∈ Γ + (s) ∪ {s},éŽöŽïŽáŽéâ, x < l u, ýëèë v ∈ Γ + (x), éæãæôâIJå v < l u.ŽéîæàŽá, < l Žîæï êŽûæèëIJîæãæ áŽèŽàâIJæï éæéŽîåâIJŽ V -äâ.äëàþâî v < l w-ï çæåýñèëIJâê, îëàëîù \v æéõëòâIJŽ w-äâ éŽîùýêæã".û æ ê Ž á Ž á â IJ Ž . ãåóãŽå, T = (V, E) Žîæï áŽèŽàâIJñèæ ëîæâêðæîâIJñèæýâ. éŽöæê v i , v j ∈ V (i ≠ j) Žê v i < l v j , Žê v j < l v i , Žê v i Ꭰv j âîåéŽîöîñðäâŽ.á Ž é ð ç æ ù â IJ Ž . ãåóãŽå, v i , v j ∈ V Ꭰv i ≠ v j . éŽöæê ŽîïâIJëIJïæïâåæ v 0 ûãâîë, îëé v i ∈ Γ + (u α ) ∪ {u α } Ꭰv j ∈ Γ + (u α ) ∪ {u α }. åñv i = a α Žê v j = v α , éŽöæê v i Ꭰv j âîå éŽîöîñðäâŽ. ûæꎎôéáâà öâéåýãâãŽöæàŽêãæýæèëå v a l-áŽê ìæîáŽìæî àŽéëéŽãŽèæ w 1 , w 2 , . . . , w k ûãâîëâIJæ. éŽöæê ŽêŽêv i ∈ Γ + (w l ) Ꭰv j ∈ Γ + (w m ), w l ≠ w m ,v i , v j ∈ Γ + (w k ).

100ìæîãâè öâéåýãâãŽöæ àãŽóãï: v i < l v j Žê v j < l v i æéæï éæýâáãæå w l

101à Ž ê ï Ž ä ô ã î â IJ Ž . ãåóãŽå, T = (V, E) Žîæï ïîñèæ IJæêŽîñèæýâ. V -äâ àŽêãïŽäôãîëå < s ïæéâðîæñèæ áŽèŽàâIJŽ öâéáâàêŽæîŽá: åæåëâñèæûãâîëïåãæï, îëéâèæù òëåëèï Žî ûŽîéëŽáàâêï ᎠŽóãï w 1 Ꭰw 2 (w 1 < w 2 )ìæîáŽìæî àŽéëéŽãŽèæ ûãâîëâIJæ. áŽãñöãŽå:w ′ 1 < 3 v, õãâèŽ w ′ 1 ∈ Γ + (w 1 ) ∪ {w 1 }-åãæï,v < w ′ 2, õãâèŽ w ′ 2 ∈ Γ + (w 2 ) ∪ {w 1 }-åãæï.< s áŽèŽàâIJŽ æóêâIJŽ < l áŽèŽàâIJæï àŽòŽîåëâIJŽ.

algebruli struqturebiCven gamoviyenebT Semdeg aRniSvnebs: _ naturaluri ricxvebisaTvis; _ mTeli ricxvebisaTvis; _ racionaluri ricxvebisaTvis; _ namdvili ricxvebisaTvis; _ kompleqsuri ricxvebisaTvis;_ dadebiTi racionaluri ricxvebisaTvis;_ dadebiTi namdvili ricxvebisaTvis.binaruli operaciebimaTematikaSi xSirad vxvdebiT situaciebs, rodesac mocemuliaraime simravle da am simravlis nebismieri ori elementisagan amavesimravlis axali elementis miRebis wesi. aseTi situaciebis magali-Tebia:m a g a l i T i 1 . Tu mocemulia namdvil ricxvTa simravle da misi raime ori elementi (e. i. ori namdvili ricxvi), Cven SegviZliaes ricxvebi SevkriboT an gavamravloT erTmaneTze, raTa miviRoTaxali elementi (e. i. axali namdvili ricxvi).m a g a l i T i 2 . Tu mocemulia nnzomis matricebis simravle,maSin ori aseTi matricis SekrebiT an maTi erTmaneTze gamravlebiTmiiReba axali nnzomis matrici.m a g a l i T i 3 . ganvixiloT raime X simravle da misi TavisTavSi yvela asaxvis simravle E X . nebismieri ori aseTi asaxviskompozicia aris isev X simravlis Tavis TavSi asaxva, anu E X simravlis elementi.103

m a g a l i T i 4 . ganvixiloT raime X simravle da E X simravlisis qvesimravle, romelic Sesdgeba mxolod bieqciuri asaxvebisgan.es simravle aRvniSnoT X simboloTi. radgan nebismieriori bieqciis kompozicia isev bieqciaa, amitom X simravlisnebismieri ori elementis (anu X simravlis nebismieri ori TavisTavze bieqciuri asaxvis) kompozicia aris isev am simravlis elementi.Tanamedrove algebra Seiswavlis iseT simravleebs, romelbicaRWurvilia misi nebismieri ori elementisagan axali elementis mi-Rebis erTi an ramdenime wesiT, romelTac b i n a r u l i o p e r a c i -e b i ewodeba.g a n m a r t e b a 1 . mocemul X simravleze gansazRvruli b i -n a r u l i o p e r a c i a ewodeba asaxvas X X dekartuli namravlidanX simravleSi.magaliTad, namdvili ricxvebis Sekrebis da gamravlebis Cveulebrivioperaciebi, romlebic ganxiluli iyo magaliTi 1-Si, arisasaxvebi simravlidan simravleSi, romlebic namdvilricxvTa x,ywyvils uTanadebs x yda x yricxvebs, Sesabamisad.rodesac mocemulia raime zogadi operacia f : X X X , namdvilricxvebze gansazRvruli Sekrebisa da gamravlebis cnobilibinaruli operaciebis analogiurad, f asaxvis mniSvnelobas x,ywyvilze (anu X simravlis f x,yelements) xy simboloTiaRniSnaven. maSasadame, Tu x,y X X , maSin f x,y x y .m a g a l i T i 5 . ganvsazRvroT mTel ricxvTa simravleze binarulioperacia Semdegnairad:nm nm 2 .maSin, magaliTad,2 3 2 3 2 6 2 84 5 4 5 2 18 . da m a g a l i T i 6 . vTqvaT, X raime simravlea, ℘() ki misi yvelaqvesimravlis simravle. Tu , ∈ ℘() (e. i. Tu A da B104

aris X simravlis qvesimravleebi), maSin A B da A B arisagreTve ℘() simravlis elementi (anu simravlis qvesimravle).es niSnavs, rom simravlis ori qvesiravlis gaerTianeba da TanakveTa℘() simravleze gansazRvravs binarul operaciebs. ori qvesimravlisA \ B sxvaoba ki gansazRvravs mesame binarul operacias℘()-ze.m a g a l i T i 7 . Cven SegviZlia ganvsazRvroT binaruli opraciasibrtyis yvela wertilTa simravleze Semdegnairad: Tu P da Qaris sibtryis nebismieri ori wertili, maSin P Qiyos PQ monakveTisSuawertili.m a g a l i T i 8 . ganvixiloT gamoklebis operacia naturalurricxvTa simravleze. igi araa binaruli operacia, radgan rodesacerT naturalur ricxvs akldeba sxva naturaluri ricxvi, sxvaobaSeiZleba aRar iyos isev naturaluri ricxvi. magaliTad,9 10 1da 1 .rodesac X simravle sasrulia, maSin masze mocemuli binarulioperacia SeiZleba warmodgindes am binaruli operaciis SesabamisicxriliT. qvemoT moyvanili cxrili warmoadgens X x, y,zsimravleze gansazRvruli binaruli operaciis gamravlebis cxrils,romlis drosac x x y , x y z , x z y , y x y da a. S. x y zx y z yy y x zz x z zkomutaciuri da asociuri binaruli operaciebiCven axla ganvixilavT im zogierT kerZo saxis binarul operaciebs,romlebic saWiroa monoidisa da jgufis cnebebis gansasaz-Rvravad.g a n m a r t e b a 2 . raime X simravleze gansazRvrul binaruloperacis ewodeba k o m u t a c i u r i , Tux y y x yvela x,y X elementisTvis105

da ewodeba a s o c i u r i , Tu x y z x y zyvela ,x y XelementisTvis.m a g a l i T i 9 . namdvil ricxvebze gansazRvruli Sekrebisa dagamravlebis cnobili operaciebi aris komutaciuric da asociuric.m a g a l i T i 1 0 . mocemuli simravlis qvesimravleebis Tanakve-Tisa da gaerTianebis binaruli operaciebi (ix. magaliTi 6) asevearis komutaciuric da asociuric.m a g a l i T i 1 1 . nebismieri naturaluri n ricxvisTvis, gamravlebisada Sekrebis operaciebisimravleze aris agreTve komutaciuricda asociuric. nm a g a l i T i 1 2 . Tu X raime simravlea, maSin kompoziciiTgansazRvruli operacia E X -ze (ix. magaliTi 4) aris asociuri,magram araa komutaciuri (ratom?); e. i. arsebobs iseTi binarulioperacia, romelic asociuria, magram araa komutaciuri.m a g a l i T i 1 3 . magaliTi 7-Si ganxiluli sibrtyis wertilTasimravleze gansazRvruli binaruli operacia aris komutaciuri,magram araa asociuri (ratom?). e. i. arsebobs iseTi binarulioperacia, romelic komutaciuria, magram araa asociuri.m a g a l i T i 1 4 . gamoklebis operacia namdvil ricxvebze arckomutaciuria da arc asociuri. marTlac, komutaciuroba niSnavs,rom x y y x nebismieri x , y namdvili ricxvebisTvis, xoloasociuroba ki, rom x y z x y zyvela x , y , z namdviliricxvebisTvis. cxadia, rom am ori tolobidan arc erTi araaigiveoba. magaliTad, 5 3 3 52 5 3 4 5 3 4 6 . da vTqvaT, X raime simravlea, xolo ki masze gansazRvruliraime binaruli operacia. Tu x, y, w,z X , maSin gamosaxulebebi , da x y w z x y w z x y wzsazogadod, erTmaneTisgan gansxvavebulia, magram Tu aris asociuribinaruli operacia, maSin asociurobis ZaliT gvaqvs: x y w z x y w z x y w z .106

amis gamo, Cven SegviZlia es sami gamosaxuleba ubralod ase CavweroT:x y w z .zogadad, rodesac binaruli operacia asociuri, maSin nebismierirTuli gamosaxuleba, romelic Sedgenilia X simravlis elementebis, operaciisa da frCxilebis gamoyenebiT,. SegviZlia gava-TavisufloT frCxilebisagan im pirobiT, rom elementebis Tanamimdevrobaucvleli rCeba. amitom nebismieri xX elementisa danebismieri naturaluri n ricxvisaTvis azri aqvs gamosaxulebas n -jer x x xromelsac mokled ase CavwerT: xn (aq cxadia igulisxmeba, rom aris asociuri binaruli operacia). maSin advili sanaxavia, rom nebismierin,m ricxvebisTvis, n x m x n xm .SevniSnoT, rom Tu asociuri araa, maSin xn gamosaxulebasazri ara aqvs. marTlac, ganvixiloT magaliTi 8-Si gansazRvrulibinaruli operacia. x3ori SesaZlebloba:magram, radgandaamitom x x xda xarisxis gansasazRvravad gvaqvs mxolod da x x x x x xx x x x y zx x x y x y ,x x x gamosaxulebebi erTmaneTis toligamosaxu-araa da amitom gaugebaria, Tu ra unda vigulisxmoT x3lebis qveS.neitraluri elementivTqvaT, raime simravle, xolo ki masze gansazRvruli binarulioperacia.g a n m a r t e b a 3 . eX elements ewodeba neitrauli ( operaciismimarT), Tu nebismieri ∈ elementisaTvis gvaqvs tolobebi:x e e x x .107

m a g a l i T i 1 5 . ganvixiloT namdvil ricxvTa simravleze gamravlebisoperaciiT gansazRvruli binaruli operacia. radgan nebismierix namdvili ricxvisaTvis, 1 x x1 x , amitom 1 aris neitraluri elementi Cveulebrivi gamravlebiT -ze gansazRvrulibinaruli operaciis mimarT.m a g a l i T i 1 6 . Tu wina magaliTSi, gamravlebis operaciasSevcvliT Sekrebis operaciiT, miRebul binarul operaciasac eqnebaneitraluri elementi, kerZod, 0 (ratom?).m a g a l i T i 1 7 . analogiurad, Tu n raime naturaluriricxvia, maSin gamravlebis (Sekrebis) operacias n-ze aqvs neitralurielementi 1 (0).m a g a l i T i 1 8 . ganvixiloT gamoklebiT gansazRvruli binarulioperacia simravleze. mas ara aqvs neitraluri elementi.marTlac, Tu e aris neitraluri elementi gamoklebis mimarT,maSin nebismieri x namdvili ricxvisaTvis gveqneboda tolobebie x x e x ,rac cxadia, SeuZlebelia (Tu x 0 , miviRebT, rom e e 0 damaSin 5 0 5 da 0 5 5toli ricxvebi unda yofiliyo).rogorc vnaxeT, binarul operacias SeiZleba neitraluri elementiar gaaCndes.w i n a d a d e b a 1 . Tu raime simravleze gansazRvrul binaruloperacias gaaCnia neitraluri elementi, maSin is erTaderTia.d a m t k i c e b a . vTqvaT, aris X simravleze gansazRvrulibinaruli operacia. Tu e da e orive aris am operaciis mimarTneitraluri elementi, maSin x e e x x tolobaSi x e mniSvnelobisCasmiT miviRebT, rom ee e. magram, radgan e ariaagreTve neitraluri elementi, amitom e x x nebismieri xXelementisaTvis. kerZod, ee e da amitom e ee e.naxevarjgufebig a n m a r t e b a 4 . n a x e v a r j g u f i aris X , wyvili, sadacX raime simravlea, xolo ki masze gansazRvruli asociuribinaruli operacia.108

m a g a l i T i 1 9 . ,da ,xolo ,ki ara.orive aris naxevarjgufi,m a g a l i T i 2 0 . nebismieri n ricxvisTvis, , , norive naxevarjgufia.m a g a l i T i 2 1 . Tu X raime simravlea, maSin (℘(), ∪) da(℘(), ∩) orive naxevarjgufia.rogorc zemoT moyvanili magaliTebidan Cans, erTi da igive simravlezeSeiZleba gansazRvruli iyos ramodenime gansxvavebulinaxevarjgufis struqtura.m a g a l i T i 2 2 . vTqvaT, X aris erTelementiani simravle,e. i. X x . ganvsazRvroT masze binaruli operacia formuliT:x x x , maSin X , aris naxevarjgufi.m a g a l i T i 2 3 . vTqvaT, X aracarieli simravlea. ganvsaz-RvroT masze binaruli operacia Semdegnairad:x y y nebismieri x,y X elementebisTvis.maSin X , aris naxevarjgufi. marTlac, Tu x, y,z X , maSin ; x y z y z z x y z x z z .amitom aris asociuri binaruli operacia da, maSasadame, X , ki naxevarjgufi.monoidebig a n m a r t e b a 5 . monoidi aris naxevarjgufi, romlis binaruloperacias gaaCnia neitraluri elementi.m a g a l i T i 2 4 . , ,0aris monoidi, radgan ,ndanaxevarjgufia,xolo 0 ki neitraluri elementi Sekrebis mimarT (ix.magaliTi 19). analogiurad, , ,1monoidia.m a g a l i T i 2 5 . nebismieri n ricxvisTvis, , ,0 , ,1norive monoidia.nda109

m a g a l i T i 2 6 . Tu X raime simravlea, maSin (℘(), ∪) da(℘(), ∩) agreTve monoidebia.m a g a l i T i 2 7 . mTel ricxvTa simravleze ganvsazRvroTbinaruli operacia Semdegnairad:a b a b ab,maSin ,monoidia. marTlac, Tu a, b,c , maSin gvaqvs:da a b c a b ab c a b ab c a b ab c a b ab c ac bc abc a b c ab ac bc abc a bc a b c bc a b c a b c bc a b c bc ab ac abc a b c ab ac bc abc.maSasadame, a bc a b c, rac niSnavs, rom aris asociuri.Semdeg, radgan nebismieri a mTel ricxvisTvis,0a 0 a 0a 0 a adaa 0 a 0 a 0 a ,amitom 0 aris neitraluri elementi operaciis mimarT. sabolood, aris asociuri binaruli operacia simravleze, romelsacgaaCnia neitraluri elementi, saxeldobr, 0. amitom , ,0arismonoidi.jgufebig a n m a r t e b a 6 . jgufi ewodeba iseT X , ,emonoids, romlisnebismieri x X elementisTvis arsebobs iseTi x X elementi,rom x x x x e .x elements ewodeba x elementis S e b r u n e b u l i , amitomjgufi SeiZleba, ganisazRvros rogorc monoidi, romlis yvelaelements gaaCnia Sebrunebuli.jgufis ganmartebaSi ar moiTxoveba, rom misi ganmsazRvreli binarulioperacia iyos komutaciuri. magram, rodesac es pirobasruldeba, maSin jgufs ewodeba k o m u t a c i u r i an a b e l u r i .110

m a g a l i T i 2 8 . , ,0, ,0viciT, rom aris abeluri jgufi. marTlac, Cvenaris monoidi. amitom sakmarisia vaCvenoT, romyovel elements gaaCnia Sebrunebuli. magram, Tu x -is rolSi avi-RebT x-s, advili sanaxavia, rom xaris x -is SebrunebuliSekrebiT gansazRvruli operaciis mimarT.m a g a l i T i 2 9 . , ,1 monoidia, magram is araa jgufi.marTlac, ar arsebobs 0-is Sebrunebuli gamravlebiT gansazRvrulibinaruli operaciis mimarT, radgan Tu aseTi 0 elementi iarsebebda,maSin gveqneboda toloba0 0 0 0 1.magram 0 0 0 da 0 0 0 , saidanac miviRebdiT, rom 0 1, racSeuZlebelia.m a g a l i T i 3 0 . wina magaliTSi vnaxeT, rom , ,1araajgufi. magram (R\∅, ∙, 1) ki aris abeluri jgufi: nebismieri xelementis Sebrunebuli xarisx 1,xolo abeluroba gamomdinareobsnamdvili ricxvebis gamravlebis komutaciurobidan.m a g a l i T i 3 1 . nebismieri n ricxvisTvis, , ,0abelurijgufia, xolo , ,1m a g a l i T i 3 2 . Tu X , ,enki ara.njgufia, romelic mxolod erTelements Seicavs, maSin is elementi unda iyos e . amitomXe .maSin X jgufis binaruli operacia srulad gansazRvruliaee e tolobiT. piriqiT, Tu X xerTelementiani simravleada Tu masze ganvsazRvravT binarul operacias x x x tolobiT,maSin X , ,xaris jgufi (Seadare magaliTi 22-s).jgufs ewodeba s a s r u l i , Tu misi elementebis raodenoba sasrulia,da ewodeba u s a s r u l o , Tu misi elementebis raodenobausasruloa.111

Tu X , ,ejgufi sasrulia, misi elementebis raodenoba XsimboloTi aRiniSneba. X -s ewodeba X simravlis rigi.m a g a l i T i 3 3 . Tu X , ,emonoidia, maSin × aRvniSnoT Xsimravlis im elementebis simravle, romelTac gaaCnia Sebrunebuli.maSin X , ,earis jgufi.m a g a l i T i 3 4 . , ,0aris sasruli (abeluri) jgufi nebismierin ricxvisTvis, xolo , ,1nki usasrulo.m a g a l i T i 3 5 . nebismieri n ricxvisTvis, , ,1 n jgufi.arismocemuli X simravlis gadaadgileba ewodeba am simravlis bieqciasTavis Tavze. maSasadame, X -is gadaadgileba arisf : X X asaxva, romelic erTdroulad ineqciacaa da sureqciac.magaliTad, X simravlis igivuri asaxva 1X: X XTavis Tavzearis gadaadgileba. Tu f da g X simravlis gadaadgilebebia, ma-Sin maTi gf kompozicia isev X -is gadaadgilebaa, radganac oribieqciuri asaxvis kompozicia isev bieqciuri asaxvaa. Xyvela gadaadgilebebs simravle S X simboloTi aRiniSneba.simravlism a g a l i T i 3 6 . ganvixiloT da davafiqsiroT misi raimea elementi. maSin asaxva fa: , romelic gansazRvruliaformuliTaris Tvis, f S aa ,f x a x xsimravlis gadaadgileba; e. i. nebismieri a elementis- .m a g a l i T i 3 7 . vTqvaT, X raime simravlea da ganvixiloTmisi gadaadgilebebis simravle S X . maSin S X aris araabelurijgufi masze asaxvebis kompoziciiT gansazRvruli binarulioperaciis mimarT, romlis neitraluri elementia 1 X, xolof : X X gadaadgilebis Sebrunebuli f gadaadgileba ki f112

asaxvis Seqceuli funqcia1f . radgan kompoziciis operacia arakomutaciuria,amitom S X araa abeluri jgufi.jgufTa TeoriaSi miRebulia, rom binaruli operaciebi Caiweroskargad cnobili Sekrebis ` ~ da gamravlebis ` ~ niSnebisgamoyenebiT. maSasadame, xy gamosaxulebis nacvlad xSirad werenx yan x ygamosaxulebebs. magaliTad, Sekrebisa da gamravlebisniSnebis gamoyenebiT komutaciuroba, Sesabamisad, Caiwereba ase:x y y x da x y y x .SevniSnoT, rom Sekrebis niSani ZiriTadad abeluri jgufebis binarulioperaciebis Casawerad gamoiyeneba. am dros miRebulia, rom + gamosaxulebas ewodeba x da y elementis jami, 0-iT aRiniSnebaneitraluri elementi, xolo x elementis Sebrunebulielementi ki xsimboloTi aRiniSneba. magaliTad, x y Caiwereba-jerx y saxiT, xolo n x x xxy aRniSnavs xy -s;1 _ neitralur elements;1x _ x elementis x Sebrunebuls;nx-jerki n x x xgamosaxulebas.ki nx saxiT. analogiurad:nx da xn xarisxebi SeiZleba ganmartebul iqnas agreTve yvelamTeli ricxvisaTvis Semdegnairad: 0 x x da n x nx ;nn 10 n 1x da x x x 1 .SevniSnoT, rom bolo tolobis misaRebad Cven gamoviyeneT winadadeba6.jgufebis elementaruli TvisebebivTqvaT, G jgufia (e. i. G aris raime simravle, romelzedacgansazRvrulia gamravlebis operacia, romelsac gaaCnia neitralurielementi 1). rogorc viciT, neitraluri elementi aris erTaderTi.axla vaCvenoT, rom nebismieri x elementiserTaderTia. amisaTvis jer davamtkicoT Semdegi1x Sebrunebulic113

w i n a d a d e b a 2 . G jgufis nebismieri ori x,y GelementebisaTvisarsebobs erTaderTi iseTi elementi a G , rom ax yda es elementiaelementi, rom xb y1yx . analogiurad, arsebobs erTaderTi iseTi bda es elementia1x y .d a m t k i c e b a . Tu ax y , maSin am tolobis orive mxarisax x yx1 1-ze gamravlebiT miiReba, rom x 1. magram asociurobisda Sebrunebuli elementis ganmartebis ZaliT1 1 ax x a xx a 1 a . amitom, Tu ax y , maSinpiriqiT, Tua yx 11 1 , maSin ax yx x y xx y 1 y .winadadebis meore nawili mtkicdeba analogiurad.Tu am winadadebaSi y 1, maSin gveqneba:a yx 1 .w i n a d a d e b a 3 . Tu xGda arsebobs iseTi a G , romax 1, maSinbx 1, maSinradganiT, xolo x -s kix a 1 . analogiurad, Tu arsebobs iseTi bG , romb x 1 .1xx 1, amitom Tu winadadeba 3-Si a -s SevcvliT x -1x -iT, miviRebT:w i n a d a d e b a 4 . Tu x G1 , maSin 1x x . w i n a d a d e b a 5 . Tu x,y G , maSin 1 1 1xy y x .d a m t k i c e b a . radgan 1 1 1 1 xy y x x y y x x yy 1 x 1 x 1 x 1 xx1 1,xy y xamitom 1 1 1toloba gamomdinareobs winadadeba 3-dan.am winadadebis samarTlianoba advilad zogaddeba n cali elementisSemTxvevaSi da gvaqvs: 1 1 1 1x x x x x x .1 2 n n n1 1axla, Tu x 1 x 2 x n, miviRebT:114

w i n a d a d e b a 6 . Tu xG , maSin nebismieri naturaluri n1nricxvisTvis, x x .1 nS e n i S v n a 1 . adiciuri Caweris gamoyenebisas, winadadeba 2 aseCamoyalibdeboda: Tu x,y G , maSin arsebobs erTaderTi iseTia Gelementi, rom a x y da es elementia analogiurad, arsebobs erTaderTi iseTi bGx b y da es elementia x y y x . winadadeba 4 da winadadeba5-dan miiReba, romy x y x .elementi, romx x da x y x y x y .m a g a l i T i 3 8 . neitraluri elementis Sebrunebuli isevneitraluri elementia (ratom?).vTqvaT, G raime jgufia.qvejgufebig a n m a r t e b a 7 . G simravlis raime aracarieli H qvesimravlesewodeba G jgufis qvejgufi, Tu sruldeba Semdegi oripiroba:1. Tu x,y H , maSin xy H .2. Tu x H , maSin ∈ .maSasadame, H Garis G jgufis qvejgufi, Tu is Caketiliagamravlebis da Sebrunebulis povnis operaciebis mimarT.adiciuri Cawerisas, qvejgufeobis piroba ase gamoiyureba:1. Tu x,y H , maSin x y H .2. Tu x H , maSin x H .w i n a d a d e b a 7 . Tu H aris G jgufis qvejgufi, maSin Haris jgufi.d a m t k i c e b a . 1. pirobis gamo, sakmarisia vaCvenoT, rom1 H . amisaTvis ganvixiloT H simravlis nebismieri xH elementi(aseTi elementi yovelTvis arsebobs, radgan ganmartebiT H115

aris Gsimravlis aracarieli qvesimravle). maSin 2. pirobis ZaliT,xxH 1 . magramw i n a d a d e b a 8 . H G1xx 1. amitom 1 H .aracarieli qvesimravle aris Gjgufis qvejgufi maSin da mxolod maSin, rodesac sruldebaSemdegi piroba:Tu x,y H , maSinxyH 1 .d a m t k i c e b a . davuSvaT, rom H aris G -s qvejgufi da romx,y H . radgan y H , amitom 2. pirobis ZaliTSemdeg 1. pirobis ZaliT,xyH 1 .piriqiT, davuSvaT, rom Tu x,y H , maSinradgan xH , amitom11 xx HxyyH 1 daH 1 . maSin, . garda amisa, radgan 1, x H ,1 1amitom x 1 x H . maSasadame, imisaTvis, rom vaCvenoT H -isqvejgufoba, saWiroa vaCvenoT, rom Tu x,y H , maSin xy H .magram, radgan y H , amitomyH 1 da axla, radganx,y1amitom x y 11 H . magram winadadeba 4-is ZaliT, y 1amitom xy H .H 1 , y .S e n i S v n a 2 . adiciuri Cawerisas, wina piroba ase Caiwerebo-Tu x,y H , maSin x y H .da:xHm a g a l i T i 3 9 . Tu H aris G jgufis qvejgufi da Tu, maSin xn H, sadac n nebismieri mTeli ricxvia.m a g a l i T i 4 0 . nebismier G jgufs gaaCnia sul mcire oriqvejgufi: TviTon G da 1 . t r i v i a l u r i qvejgufi ewodeba.m a g a l i T i 4 1 . ,aris ,,ki ,jgufis qvejgufi.1 qvejgufs G jgufisjgufis qvejgufi, xolom a g a l i T i 4 2 . vTqvaT, n raime naturaluri ricxvia. maSinsimravle, 2 n, n,0, n,2 n,116

aris ,jgufis qvejgufi.m a g a l i T i 4 3 . simravle 0,2,4aris 6,0,3simravle.qvejgufi. misi qvejgufia agreTve jgufisw i n a d a d e b a 9 . Tu H da H aris G jgufis oriqvejgufi, maSin H H aris agreTve G -s qvejgufi.d a m t k i c e b a . vTqvaT, x,y H H , maSin x,y Hamitom winadadeba 8-is ZaliT,amitomH H 1xy H HxyH 1 . analogiurad,xy1da H, da maSin isev winadadeba 8-is ZaliT,aris G jgufis qvejgufi.vTqvaT, G raime jgufi da gki misi raime elementi. ganvixiloTsimravleH x G : x g n , n .w i n a d a d e b a 1 0 . H aris G jgufis qvejgufi.d a m t k i c e b a .n m nmxy g g g HTunx g H da da 1winadadeba 6). amitom H1 n nx g g Hmx g H , maSin (ixilearis G jgufis qvejgufi.g a n m a r t e b a 8 . vTqvaT, G jgufia da g ki misi raimeelementi. maSin H g n , n qvejgufs ewodeba g elementiTwarmoqmnili G jgufis qvejgufi da gsimboloTi aRiniSneba.G jgufis H qvejgufs ewodeba c i k l u r i , Tu arsebobsiseTi g Gelementi, rom H g . TviTon g elements ki Hqvejgufis w a r m o m q m n e l i ewodeba. kerZod, Tu raime g GelementisTvis, G gg -s misi warmomqmneli., maSin G -s ewodeba c i k l u r i jgufi, da117

Tu G g, maSin G , g , g 1 , g 0 1, g 1 , g2 , da radgann m nm mn g g g gm n g g , amitom G aris abeluri jgufi.maSasadame, gvaqvs:w i n a d a d e b a1 1 . nebismieri cikluri jgufi abeluria.S e n i S v n a 3 . Tu gamoviyenebdiT adiciur Caweras, cikluriG jgufis elementebi ase warmodgindeboda, 2 a, a,0 a 0, a, 2 a,m a g a l i T i 4 4 . ,aris cikluri jgufi. marTlac,radgan nebismieri mTeli ricxvi n warmoidgineba n1amitom 1 .saxiT,m a g a l i T i 4 5 . Tu n raime naturaluri ricxvia, maSin n -isjeradi mTeli ricxvebis n simravle aris ,jgufis cikluriqvejgufi da es qvejgufi warmoqmnilia n ricxviT.SevniSnoT, rom zogadad ciklur jgufs SeiZleba erTze metiwarmomqmneli hqondes. magaliTad, , 1 da , 1.m a g a l i T i 4 6 . mTeli luwi ricxvebis E simravle aris,jgufis cikluri qvejgufi da E 2 .imisda mixedviT, cikluri jgufis warmomqmnelis yvelaxarisxebi gansxvavdeba Tu ara erTmaneTisgan, arsebobs ori saxiscikluri jgufi: sasruli da usasrulo. Tu G g2 1 0 1 2, g , g , g e, g , g , simravlis yvela elementi erTmane-Tisgan gansxvavdeba, e. i.gn m g , Tu n mda, maSin amboben, romg Gelementis rigi aris usasrulo, an rom g elements aqvsusasrulo rigi. am faqts simbolurad ase Caweren: g . Tu arsebobsiseTi mTeli n m ricxvebi, rom gn gm , maSin1 m 1 1nm n m n n m n ng g g g g g g g g eda amitom arsebobs iseTi dadebiTi mTeli ricxvi kk n m ), romkg(magaliTad, e . aseTi Tvisebis umcires ricxvs g118

elementis r i g i ewodeba. im faqts rom g elementis rigi arisk , ase Caweren: g k .w i n a d a d e b a 1 2 . vTqvaT, g Gelementis rigiaÀ k . Tum,n mTeli ricxvebia, maSin gm gn , maSin da mxolod maSin,rodesac m niyofa k -ze.d a m t k i c e b a . Tu m n kl , l , maSin m n kl daamitom lm n kl n kl n k n k n ng g g g g g g e g e g (aq gamoviyeneTis faqti, rom gmaSinpiriqiT, vTqvaT,g k ).m n g da m n lk r l , sadac 0 r k .1 1n n m n m n mn lk re g g g g g g g g magram, radgan giyofa k -ze.lk r k r l r r g g g g 1 g g . k da radgan r k , amitom r 0 , anu m nTu g k , maSin nebismieri n SeiZleba ase Caiweros:n mk s , sadac 0 s k . amitom gvqeneba:maSasadame, gm 1g g g g g g g gn mk s mk s k s m s swina winadadebaSi n 0ciklur jgufs aqvs zustad k elementi:g g g g g 0 1 2 k 1, , , ,1 .mniSvnelobis CasmiT miviRebTmw i n a d a d e b a 1 3 . Tu g k , maSin g 1 maSin da mxolodmaSin, rodesac m iyofa k -ze.m a g a l i T irigia 1.m a g a l i T i 4 8 . ,4 7 . nebismier jgufSi misi neitraluri elementisjgufis nebismieri elementis rigiusasruloa, radgan Tu ng 0 , maSin aucilebald g 0 .119

m a g a l i T i 4 9 . , ,1jgufs aqvs 1-gan gansxvavebulimxolod erTi sasruli rigis mqone elementi. es elementia 12misi rigia 2, radgan , ,1jgufis Sesaxeb.1 1 1 1 . igive SeiZleba iTqvaswinadadeba 14. cikluri jgufis nebismieri qvejgufi isevcikluria.d a m t k i c e b a . vTqvaT, G aris cikluri jgufi, xolo H kimisi raime qvejgufi. Tu G g , g G , maSin n -iT aRvniSnoT isumciuresi naturaluri ricxvi, romlisTvisacmgngda H . Tu H , maSin n m da amitom m nl s , sadac 0 s n . maSinm nl s nl s n l snl n ls nl da radgan 1 mg g gg g g g g g da radgang g H , amitom qvejgufis ganmartebisgamo,gsg , amitom s 0 , rac niS-m nnavs, rom l H . radgan 0 s nng H , amitom g . e. i. H qvejgufis nebismieri elementi arisnng elementis raime xarisxi, amitom H g .lagranJis TeoremavTqvaT, G jgufia, xGmisi raime elementi, xolo H kiG simravlis raime qvesimravle. ganvixiloT Semdegi simravleHx hx,h H .Hx kvlav G simravlis qvesimravlea da Tu Hsimravle x1, , xn, maSin Hx x1 x, x2x, , xnx.aris sasrulianalogiurad ganimarteba xH simravle. SevniSnoT, rom arakomutaciurobisgamo, zogadad, Hx xH .Tu x,y H , maSin asociurobis gamoHx y H xy,x yH xy H xH y xHycxadia, Tu G komutaciuria, maSin Hxda . xH .120

g a n m a r t e b a 8 . vTqvaT, H aris G jgufis qvesimravle. Hxsaxis simravles, sadac xG,ewodeba H qvesimravlis m a r j v e -n a m o s a z R v r e k l a s i da G H simboloTi aRiniSneba. analogiurad,xH saxis simravles ewodeba H qvesimravlis m a r c x e n am o s a z R v r e k l a s i da G \ H simboloTi aRiniSneba.w i n a d a d e b a 1 5 . vTqvaT, G aris raime jgufi, x,y Gmisiori nebismieri elementi da H ki G simravlis qvejgufi, maSin anHx Hy an Hx Hy . garda amisa, Hx Hy maSin da mxolodmaSin, rodesacyxH 1 .d a m t k i c e b a . Tu Hx Hy , maSin arsebobs iseTi g Gelementi, rom g Hx da g Hy . es niSnavs, rom arsebobs iseTih1 , h2 H ori elementi, rom h 1x h 2y . aqedan x h 1h y1 2. axlaTu z Hx , maSin arsebobs iseTi h H elementi, rom z hx , daamitom 1 1 1 2 1 2 z h h h y hh h y . magram, radgan H aris Gjgufis qvejgufi, amitom11 2hh h11 2 H da, maSasadame,z hh h y Hy . miviReT, rom Hx Hy . simetriulad,Hy Hx da, maSasadame, Hx Hy .Tuyxamitom yH 1 , maSinyx 1 h raime hHelementisaTvis, da hx , rac niSnavs, rom y Hx . radgan y 1radgan 1 H , amitom y Hy da miviReT, rom y Hx Hy , raczemoT damtkicebulis Tanaxmad niSnavs Hx HypiriqiT, Tu HxTvis, radgan y Hy1yx h H . amiT Teorema damtkicebulia.aqedan miiReba Semdegitolobas. Hy , maSin y hx raime hH. amitomyda,elementisa-w i n a d a d e b a 1 6 . Tu H aris G jgufis qvejgufi, maSinG H Hx,x Garis G simravlis dayofa (e.i. G simravlewarmoidgineba rogorc wyvil-wyvilad TanaukveTi G HraodenobisHx saxis simravleebs gaerTianebad).121

analogiurad miiReba, rom G \ H aris agreTve G simravlisdayofa.SevniSnoT, rom, radgan eH He H , amitom H aris rogorcmarcxena, ise marjvena mosazRvreobis klasi.H m a g a l i T i 5 0 . ganvixiloT 6,da misi qvejgufi0 , 2 , 4. maSin0 H H ,2 H H3 H 1 , 3 , 5 ,4 H H ,5 H 2 , 3 , 5 .SevniSnoT, rom radgan 6,adiciur Caweras.abeluri jgufia, Cven viyenebTl a g r a n J i s T e o r e m a . Tu G sasruli jgufia, xolo Hki misi qvejgufi, maSin H -is rigi aris G -s rigis gamyofi daG H G H .d a m t k i c e b a . jer vaCvenoT, rom nebismieri xGelementisaTvis, H da Hx Sedgeba erTi da igive raodenobiselementisagan. amisaTvis ganvixiloT asaxva f : H Hx , romelicgansazRvruli formuliTf hbieqciuri. Tu f h f htolobas orive mxarise. i. f aris ineqciuri.1 2imisaTvis, rom vaCvenoT, rom fganvixiloT nebismieri y Hxes niSnavs, rom fbieqciuri. hx , da vaCvenoT, rom f aris , h 1, h 2 H , maSin h 1x h 2xda am1x -ze gamravlebiT miviRebT, rom 1 2elementi. maSin1 1 1 h h .aris agreTve sureqciuli,yxf yx yx x y x x y 1 y .H 1 da gvaqvsyofila sureqciulic da, maSasadame, is aris122

axla, radgan Gsimravle warmoidgineba rogorc wyvil-wyviladTanaukveTi G Hraodenobis Hx saxis simravleebs gaerTianebadda, radgan TiToeuli aseTi simravlis elementebis raodenobaaH , amitomsimetriulad miiReba, romamitom G \ HG G H H .G G \ H H . G H , rac niSnavs, rom H -s marcxena da marjvenamosazRvre klasebis raodenobebi tolia.xS e d e g i 1 . Tu G raime sasruli jgufia da xG , maSinaris G -s gamyofi.S e d e g i 2 . yvela sasruli jgufi, romlis rigic martivia,aris cikluri.d a m t k i c e b a . vTqvaT, G aris iseTi sasruli jgufi, rom Garis martivi da ganvixiloT misi neitraluri elementisagan gansxvavebulinebismieri xGelementi da misgan warmoqmnili ciklurix qvejgufi. maSin lagranJis Teoremis Tanaxmad, x arisG -s gamyofi da radgan x e , amitom x 1, saidanac davaskvniT,rom x G . es ki niSnavs, rom G x .paf e r m i s T e o r e m a . Tu a da p martivi, ricxvia, maSin a iyofa p -ze.normaluri jgufebig a n m a r t e b a 9 . vTqvaT, G jgufia da H misi raime qvejgufi.amboben, rom H aris G jgufis n o r m a l u r i q v e j g u f ian, rom H aris normaluri G jgufSi, Tu nebismieri xGelementisaTvis,Hx xH .123

maSasadame, H aris normaluri G -Si, Tu H -is marcxena damarjvena mosazRvre klasebi erTmaneTs emTxveva. SevniSnoT, rom TuG abeluria, maSin misi nebismieri qvejgufi aris normaluri.1Tu Hx xH tolobis orive mxares gavamravlebT x -ze marcxnidan,miviRebT, rom H aris normaluri G -Si maSin da mxolod1maSin, rodesac x Hx H nebismieri xGelementisaTvis.nebismieri jgufisaTvis, TviT es jgufi da trivialuri qvejgufiaris normaluri. Tu erTze meti rigis mqone jgufsmxolod es ori normaluri qvejgufi aqvs, mas m a r t i v i ewodeba.Sedegi 1-is Tanaxmad, Tu sasruli jgufis rigi martiviricxvia, maSin es jgufi martivia.w i n a d a d e b a 1 7 . H aris normaluri G -Si maSin da mxolodmaSin, rodesac nebismieri xGd a m t k i c e b a .davuSvaT, rom1x HxelementisaTvis, maSinda amitom,aqedan miiReba, rommiviRebT,romelementisaTvis,1x Hx H .erTi mxare zeviT SevamowmeT ukve, amitom H CarTva sruldeba nebismieri xGx x Hx x xHx 1 1 1 H H xx H xx xHx1x Hx1 1 11 1 .H aris normaluri G -Si.1x Hxda radgan pirobiT1x Hx H H nebismieri x elementisaTvis, anu rom Hw i n a d a d e b a 1 8 . Tu H da K aris G jgufis ori normaluriqvejgufi, maSin H K aris agreTve G -s normaluri qvejgufi.d a m t k i c e b a . winadadeba 9-is ZaliT, rom H K aris Gjgufis qvejgufi. ganvixiloT axla G jgufis nebismieri xelementi. maSin H K H CarTvis ZaliT,x H K x K . maSa- 1 x H K x x 11 Hx H . analogiurad, 1sadame, x H K x H Kwina winadadebidan. da sasurveli Sedegi axla miiReba124

faqtor-jgufivTqvaT, G jgufia da H ki misi raime normaluri qvejgufi.G \ H simravleze ganvsazRvroT binaruli operacia Semdegnairad:Tu Hx da Hy aris H qvejgufis ori nebismieri marjvena mosaz-Rvre klasi, maSin Hx Hy H xy. SevamowmoT, rom es ganmartebakoreqtuli, e. i. rom Tu Hx Hx da Hy Hy , maSin ∗ = ∗ , anu rom Hxy Hxy . radgan H normaluria G -Si, amitom nebismieri z GelementisaTvis, zH Hz . amitomgvaqvs: radgan xHy radganxH Hx xHy radganHy HyH xy H x y Hx Hxw i n a d a d e b a 1 9 . G H, ,H Hxy radganxH Hx.aris jgufi.d a m t k i c e b a . Tu x, y,z G , maSin gvaqvs Hx Hy Hz H xy Hz . H xy z H x yz Hx H yz Hx Hy HzmaSasadame, aris asociuri binaruli operacia. radgan nebismierixGelementisaTvisdaamitom Hamisa, Tu xG , maSindamaSasadame, G H, ,HHx H Hx H1 H x 1 HxH Hx H1 Hx H 1 x Hx ,aris neitraluri elementi operaciis mimarT. garda1Hx da Hx klasis Sebrunebuli, radganHx Hx 1 H xx 1 H 1 H1 1Hx Hx H x x H 1 H .aris jgufi.125

m a g a l i T i 5 1 . m,aris ,,aris abeluri jgufi, amitom m,qvejgufi. maSin m simravleze Sekrebis operacia ase ganimartebajgufis qvejgufi. radganaris misi normalurim x m y m x yadvili saCvenebelia, rom m , , . .jgufTa homomorfizmiori jgufis Sesadareblad gamoiyeneba iseTi asaxvebi erTi jgufidanmeoreSi, romlebic inaxavs jgufis operaciebs. aseT asaxvebsh o m o m o r f i z m e b i ewodeba.g a n m a r t e b a 1 0 . homorfizmi G,jgufidan G,Si aris iseTi asaxva f : G G , rom nebismieri x,y GelementebisaTvisf x y f xf y.mjguf-sxva sityvebiT rom vTqvaT, homomorfizmi ori elementisnamravls maTi anasaxebis namravlSi asaxavs.m a g a l i T i 5 2 . nebismieri jgufidan nebismier sxva jgufSiyovelTvis arsebobs erTi mainc homomorfizmi. es aris e. w. t r i -v i a l u r i homomorfizmi, romelsac yvela elements uTanadebsneitralur elements. radgan 11 1 nebismier jgufSi, amitomadvili sanaxavia, rom aseTi asaxva namdvilad homomorfizmia.f : G Ghomomorfizms ewodeba:a) m o n o m o r f i z m i , Tu f asaxva ineqciuria;b) e p i m o r f i z m i , Tu f asaxva sureqciulia;g) i z o m o r f i z m i , Tu f asaxva aris bieqcia, e. i.erTdroulad sureqciac da ineqciac.Tu G G , maSin f homomorfizms ewodeba e n d o m o r f i z m ida ewodeba a v t o m o r f i z m i , Tu faris izomorfizmi.m a g a l i T i 5 3 . vTqvaT, H aris G jgufis normaluri qvejgufida ganvixiloT asaxva p : G G H , romelic xGelementsuTanadebs misi mosazRvreobis Hxklass. maSin es asaxva126

aris homomorfizmi, radgan Tu x,y G , maSinp xy H xy Hx Hy p x p y. radgan cxadia, rom p arissureqciuli asaxva, amitom p aris epimorfizmi.m a g a l i T i 5 4 . Tu G nebismieri jgufia, maSin G -s TavisTavze igivuri asaxva 1G: G Garis bieqcia, amitom 1 Garis homomorfizmi. radgan 1 Gfaqtiurad aris avtomorfizmi.m a g a l i T i 5 5 . vTqvaT, n raime naturaluri ricxvia da ganvixiloTasaxva fn:nf m nm . radganamitomis aris , , romelic gansazRvrulia Semdegnairad:fn m m nm m nm nm fn n fnmf aris homomorfizmi ,n ,jgufi endomorfizmi.jgufidan Tavis TavSi, anum a g a l i T i 5 6 . vTqvaT, G jgufia, x ki misi raime elementi.ganvixiloT asaxva f : G , romelic gansazRvrulia Semdeg-nnairad: f n x . maSin f aris homomorfizmi ,nnn nG jgufSi. marTlac, f n n x x x f n f n .jgufidanSevniSnoT, rom wina magaliTi aris am magaliTis kerZo SemTxveva.m a g a l i T i 5 7 . vTqvaT, n 1. ganvixiloT asaxva p : n,romelic gansazRvrulia Semdegnairad: pm m, sadac maris m -is ekvivalentobis klasi moduliT n , maSin paris epimorfizmi.m a g a l i T i 5 8 . vTqvaT, G jgufia da x ki misi raime elementi.ganvixiloT asaxva fx: G G1 gansazRvrul Semdegnairad:fxy x yx . fxaris homomorfizmi. marTlac, Tu y,y G ,maSin1 1 1 f yy x yy x x yx x y x f y f y .x x x127

odesac G da G jgufebis binaruli operaciebi mutiplikaciuradaaCawerili, maSin f : G G asaxvis homomorfulobis pirobaSemdegnairad Caiwereba:maSinf xy f x f yyvela ,x y GelementebisaTvis.w i n a d a d e b a 2 0 . Tu f : G G homomorfizmi da x,y G ,d a m t k i c e b a . radgan f1 1f xy f x f y .homomorfizmia, amitom gvaqvs1 1 f xy f y f xy y f x .f xy f x f y .1aqedan miviRebT, rom 1Tu am winadadebaSi vigulisxmebT, rom x y 1 , xolo Semdegki, rom x 1, miviRebT:maSinTvis.z zw i n a d a d e b a 2 1 . Tu f : G G jgufebis homomorfizmia,111f da 1m a g a l i T i 5 9 . asaxva f :cxadia, rom ff y f y nebismieri y G TanadobiT, aris homomorfizmi, radgan 1 2 1 2 1 2 1 2elementisa- , romelic gansazRvruliaf z z z z z z f z f z .aris sureqcia: Tu x , maSin f araa ineqcia. marTlac f z f z, radgan z z .f x x . magramw i n a d a d e b a 2 2 . Tu f : G G da f : G Ghomomorfizmia, maSin maTi kompozicia gf : G Ghomomorfizmi.d a m t k i c e b a . Tu x,y G , maSin gvaqvs: gf xy g f xy orivearis agreTve g f x f y g f x g f y ( gf ) x ( gf ) y .128

w i n a d a d e b a 2 3 . Tu f : G Gf 1 : G G aris agreTve izomorfizmi.izomorfizmia, maSinrodesac G da G jgufebis binaruli operaciebi adiciurisaxiTaa warmodgenili, maSin winadadeba 20 da winadadeba 21-damiviRebT, rom f x y f x f yda izomorfuli jgufebif 0 0.g a n m a r t e b a 1 1 . or G da G jgufs ewodeba izomorfuli(da amis simbolurad ase Caweren G G ), Tu arsebobsizomorfizmi f : G G .m a g a l i T i 6 0 . ganvixiloT asaxva f :xganmartebuli: f x e . radgan xy x yf x y e e e f x f yamitom f aris homomorfizmi , ,0xSi. rogorc cnobilia, f x e , , romelic asea jgufidan , ,1jguf-funqcia aris bieqciuri. maSasadame,f aris jgufebis izomorfizmi da amitom , ,0 , ,1jgufebi izomorfulia.m a g a l i T i 6 1 . radgan g x ln xxedafunqciis Seqceuli funqciaafunqcia, amitom winadadeba 23-is ZaliT asaxvaaris jgufebis izomorfizmi.m a g a l i T i 6 2 . vTqvaT, C 3 : , lnf x x xx C 3iseTi elementia, rom C 3vTqvaT, f : C3 C3romRvruliaTxveva:112f . radgan 2f mesame rigis cikluri jgufia. Tu x , maSin C 23 1, x,x . axlaaris nebismieri endomorfizmi, maSin viciT,f x f x , amitom f srulad gansaz-x -is mniSvnelobis codniT. amitom gvaqvs sami Sem-129

f x . maSin f x 2 f x f x1. 1aris trivialuri homomorfizmi.2. f xx . maSin 11 1 da amitom f2 2f x f x f x x da am SemTxvevaSif aris C 3-is igivuri homomorfizmi.23. f x x . maSin pirvel SemTxvevaSi cxadia, rom fmesame SemTxvevaSi ki es asea. maSasadame, C 32 4 1 3f x f x f x x x x x 1 x .araa izomorfizmi. meore daaris Tavisi Tavis izomorfulisul mcire ori gansxvavebuli izomorfizmis saSualebiT.jgufebis raime Tvisebas ewodeba s a k u T r i v i , Tu nebismieriori izomorfuli G da G jgufebisaTvis, G jgufs aqvs esTviseba maSin da mxolod maSin, rodesac G jgufs aqvs igiveTviseba. magaliTad, sasruli jgufis rigi aris sakuTrivi Tviseba.marTlac, Tu sasruli G da G jgufebi izomorfulia, maSinarsebobs raime izomorfizmi f : G G . radgan yoveli izomorfizmibieqciaa, amitom f asaxva aris bieqciuri. e. i. G da Gsimravleebs aqvT erTi da igive raodenobis elementebi.m a g a l i T i 6 3 . jgufis Tviseba iyos abeluri, arissakuTrivi. marTlac, Tu f : G G izomorfizmia da Tu Gabeluria, maSin G jgufis nebismieri ori x,yGelementebisaTvis, gvaqvs:1 1 1 xy f f xy f f x f y1 1 1 1 f f y f x ff y ff x yx,rac niSnavs, rom G jgufic abeluria. aq gamoviyeneT is faqti,rom izomorfizmis Sebrunebuli asaxva isev izomorfizmia (ix. winadadeba23).m a g a l i T i 6 4 . samelementiani simravlis gadaadgilebebs S 3jgufs da 6 ,jgufs aqvT erTi da igive raodenobis elementebianu maTi rigi erTi da igivea da, kerZod, aris 6, magram isiniizomorfuli jgufebi ar aris.130

m a g a l i T i 6 5 . Tu f : G G izomorfizmia da f x x ,nmaSin 1x maSin da mxolod maSin, rodesac 1nTu x 1, maSinxpiriqiT, Tu 1radgan1f n nnn x f x f x f 1 1. , maSinn n n 1 1 x f x f x f1 1 1,xn . marTlac,aris agreTve homomorfizmi. maSasadame, izomorfulijgufebis Sesabamis elementebs aqvT erTi da igive rigi.m a g a l i T i 6 6 . ganvixiloT jgufi G , romelic SedgebaoTxi e , t , u da v elementebisagan da romlis gamravlebisoperaciis cxrili gamoiyureba Semdegnairade t u ve e t u vt t e v uu u u e tv v u t eam jgufis neitraluri elementia e da misi rigi cxadia, aris 4.miuxedavad imisa, rom 4 , jgufis rigic 4-ia da rom esorive jgufi abeluria, isini araizomorfuli jgufebia. marTlac, 4 , jgufSi arsebobs elementi, romlis rigia 4 (kerZod,1 ), maSin, rodesac G jgufSi neitraluri elementis gardayvela sxva elementis rigi aris 2.homomorfizmis birTvi da anasaxinebismieri homomorfizmi or jgufs Soris gamoxatavs raime kav-Sirs am jgufebis struqturebs Soris. magaliTad, jgufebis izomorfizmisSemTxvevaSi, erTi jgufis struqtura SeiZleba sruladaRdges meore jgufis struqturis saSualebiT. rodesac homomorfizmiizomorfizmi araa, amdenis gakeTebis saSualeba aRara gvaqvs,magram nebismieri homomorfizmis dros yovelTvis gvaqvs izomor-131

fizmi am homomorfizmiT dakavSirebuli jgufebis faqtor-jgufsada qvejgufs Soris.g a n m a r t e b a 1 2 . vTqvaT, f : G Gjgufebis homomorfizmia.simravlesewodeba fki misi a n a s a x i . : Ker f x G f x ehomomorfizmis b i r T v i , xolo simravles Im f f x : x Gw i n a d a d e b a 2 4 . Tu f : G GmaSin misi birTvi aris GjgufTa homomorfizmia,jgufis normaluri qvejgufi, xolomisi anasaxi ki G jgufis (ara aucileblad normaluri) qvejgufi.d a m t k i c e b a . Tu x,y Ker f , maSin f xy ef x f y e . ami-1tom winadadeba 20-is ZaliT, rac niSnavs, rom Ker f 1 , e. i. xy Ker f ,aris G jgufis qvejgufi. rom vaCvenoTmisi normaluroba G -Si, ganvixiloT nebismieri x Ker f nebismieri y G . maSin, radgan1 1 1 1f yxy f y f x f y f y e f y f y f y e,1amitom yxy Ker f , rac niSnavs, rom Ker f G -Si.winadadeba 24-is ZaliT, Im f daaris normaluriaris G jgufis qvejgufi, Tunebismieri x, yIm f elementebsaTvis, x y 1Im f aris agreTve-is elementi. magram anasaxis ganmartebis ZaliT, arsebobsiseTi x,y G , rom f xx da f y y . maSin1 1 1 1f xy f x f y f x f y x y ,rac niSnavs, rom x y 1 Im f , e. i. Im f qvejgufi.aris G jgufis132

l e m a . vTqvaT, G da G jgufebi, f : G Gki maTihomomorfizmia. Tu x,y Garis G jgufis nebismieri orielementi, maSin f x f yKx Ky , sadac K Ker f .maSin da mxolod maSin, rodesacd a m t k i c e b a . Tu f x f y, maSin gvaqvs:1 1 1 1f xy f x f y f x f y f x f x e1amitom xy K Ker f da, maSasadame, Kx Ky winadadeba 15-is ZaliT. piriqiT, Tu Kxy kxda amitom Ky , maSin arsebobs iseTi k K , rom f y f kx f k f x e f x f x .h o m o m o r f i z m i s T e o r e m a . vTqvaT, f : G Ghomomorfizmia da K ker( f ) . maSin asaxvaaris izomorfizmi.d a m t k i c e b a . Tu kx kyTanaxmad da amitom f f : G K Im f ,f kx f x , maSin f x f ygansazRvrulia koreqtulad.axla, Tu f x f y, maSin Kx KyTanaxmad, amitom f jgufTawina lemisisev wina lemisaris ineqciuri asaxva. misi gansazRvrebidangamomdinare, f aris agreTve sureqciuli. e. i. f asaxva aris bieqciuri.amitom sakmarisia, vaCvenoT, rom f aris homomorfizmi.amisaTvis ganvixiloT G jgufis ori nebismieri x da yelementi. gveqneba:f *( KxKy) f *( K ( xy)) f ( xy) F( x) f ( y) f *( Kx) f *( Ky),rac niSnavs, rom f aris homomorfizmi.advili sanaxavia, rom kanonikuri CadgmiT inducirebuli asaxvai : Im f G aris monomorfizmi.133

T e o r e m a . vTqvaT, f : G GTu K ker( f ) , maSin diagramaaris jgufebis homomorfizmi.sadac p : G G K aris kanonikuri proqcia (ix. magaliTi 53),aris komutaciuri. sxva sityvebiT rom vTqvaT, fd a m t k i c e b a . radgan nebismieri xGp( x) Kx,f *( Kx) f ( x) da i f x f x if p ., amitomelementisaTvis( if * p)( x) ( if *)( p( x)) ( if *)( Kx) i( f *( Kx)) i( f ( x)) f ( x).maSasadame, jgufebis nebismieri homomorfizmi SeiZlebawarmodgenili iqnas rogorc sami homomorfizmis kompozicia,romelTagan pirveli epimorfizmia, meore _ izomorfizmi, xolomesame ki monomorfizmi.m a g a l i T i 6 7 . radgan p : G G / K epimorfizmia (ix.magaliTi53), xolo f : G K Im f ki izomorfizmi (da,kerZod, epimorfizmi), da radgan ori epimorfizmis kompoziciakvlav epimorfizmia, homomorfizmebis Teoremidan gamomdinareobs,rom f aris epimorfizmi maSin da mxolod maSin, rodesaci : Im f G aris izomorfizmi, anu rodesac Im f G .m a g a l i T i 6 8 . ganvixiloT jgufebis homomorfizmi(rad-f : G G . Tu f aris monomorfizmi, maSin Ker f egan viciT, rom f e e ). piriqiT, Tu Ker f elema 1-is ZaliT, Tu f x f y, maSin x ex ey y , anu faris monomorfizmi., maSinTu homomorfizmebis TeoremaSi f aris G jgufis igivuriendomorfizmi, maSin Ker f eda Im f G e G . G , amitom134

m a g a l i T i 6 9 . vTqvaT, G cikluri jgufia, xolo x kimisi iseTi elementi, rom G x. ganvixiloT homomorfizmi f : G,f n x(ix. magaliTi 56). cxadia, rom Im f x n : n n : n . magram jgufix n aris zustad x , amitom im( f ) x H .ganvixiloT axla f -is birTvis. cxadia, romn : Ker f n x e .amitom, Tu H usasruloa, maSin Ker f 0(ix. magaliTi 67).amitom homomorfizmis Teoremis Tanaxmad, 0 G . e. i. Hda aris izomorfuli jgufebi.Tu H sasrulia da misi rigia m , maSin Ker f aris m -isyvela jeradebis qvejgufi m , amitom m G . miviReT, romnebismieri cikluri jgufi izomorfulia Semdegi cikluri jgufiizomorfulia Semdegi cikluri jgufebidan erT-erTis:, , 2 , 3 , , m , (*)am jgufebidan arcerTi ori izomorfuli araa, radgan maTi rigebigansxvavdeba. amitom (*) aris cikluri jgufebis sruli sia.rgolebiCven aqamde ganvixilavdiT simravleebs maTze gansazRvruli erTibinaruli operaciiT. axla ganvixiloT simravleebi, romlebzedacerTdroulad gansazRvrulia ori binaruli operacia, romlebicerTmaneTTanaa SeTanxmebuli garkveuli azriT.g a n m a r t e b a 1 3 . r g o l i aris simravle R da maszeganmartebuli Sekrebis ` ~ ga gamravlebis ` ~ ori binarulioperacia ise, rom R,aris abeluri jgufi, R,aris monoidida sruldeba Semdegi d i s t r i b u c i u l o b i s kanonebi:1) r s t rs rt ,2) r s t st st ,r, s,t R .135

m a g a l i T i 7 0 . , , , simravleebi CveulebriviSekrebisa da gamravlebis binaruli operaciebis mimarT aris rgoli.m a g a l i T i 7 1 . nnzomis matricaTa simravle matricebisSekrebisa da gamravlebis cnobili operaciebis mimarT aris rgoli.m a g a l i T i 7 2 . orelementiani simravle a,bmaszeganmartebuli Sekrebisa da gamravlebis Semdegi operaciebis mimarTaris rgoli:R, a b a ba a b b a ab b a b a bg a n m a r t e b a 1 4 . R rgolis ewodeba k o m u t a c i u r i , Tuaris komutaciuri monoidi.m a g a l i T i 7 3 . , , , komutaciuri rgolebia.m a g a l i T i 7 4 . magaliTi 70-Si moyvanili yvela rgoli ariskomutaciuri, magram magaliTi 71-Si moyvanili ki ara, radganviciT, rom matricebis gamravleba araa komutaciuri.m a g a l i T i 7 5 . ganvixiloT raime R,abeluri jgufi damasze ganvsazRvroT gamravlebis operacia Semdegnairad: r s 0yvela r,s R elementebisaTvis. advili Sesamowmebelia, rom esgansazRvravs komutaciuri rgolis struqturas R -ze.R rgols ewodeba rgoli gayofiT, Tu R \ 0 , aris jgufi,anu Tu R rgolis nulisagan gansxvavebul yovel elements gaaCniaSebrunebuli gamravlebis mimarT. Tu rgoli gayofiT komutaciuria,maSin mas v e l i ewodeba.m a g a l i T i 7 6 . araa rgoli gayofiT, radgan magaliTad 2-s ar gaaCnia Sebrunebuli.m a g a l i T i 7 7 . , , samive velia.m a g a l i T i 7 8 . qvaternionebis algebra aris rgoligayofiT, magram araa veli.136

w i n a d a d e b a2 5 . vTqvaT, R rgolia. maSin1) nebismieri r R elementisaTvis sruldeba tolobar 0 0r 0 ;2) 1 1r r r .r 0 r 0 0 r 0 r 0 .d a m t k i c e b a . 1) gvaqvs: Tu am tolobas orive mxares gamovaklebT r 0-s, miviRebT, romr 0 0 . analogiurad, 0r 0 .3) radgan r 0 0 , amitom gvaqvs 0 r 0 r 1 1 r 1 r 1 r r 1 .am tolobis orive mxridan r -is gamoklebiT miviRebT, romr ( 1) r .S e d e g i 3 . Tu R rgolia. maSinr s rsda , ,r s rs r s R .SevniSnoT, rom rgolis ganmartebidan ar gamomdinareobs, rom 0da 1 gansxvavebuli elementebia. Tu 0 1, maSin nebismieri r RelementisaTvis, gveqneba:r r 1 r 0 0 ,amitom mocemul R rgols gaCnia erTaderTi elementi,saxeldobr 0. aseT rgols t r i v i a l u r i an n u l o v a n i rgoliewodeba.w i n a d a d e b a 2 6 . vTqvaT, R rgolia. Semdegi ori winadadebaeqvivalenturia:1) nebismieri r, s,t R elementebisavis, Tu r 0 da rs rt ,maSin s t .2) nebismieri r,s R elementebisavis, Tu rs 0 , maSin r 0an s 0 .d a m t k i c e b a . 1)-Si t 0 mniSvnelobis CasmiT miiReba 2).piriqiT, Tu rs rt , maSin r s t 0 da amitom 2)-is ZaliT,s t 0 , e. i. s t .137

g a n m a r t e b a 1 5 . R rgolis nulisagan gansxvavebul relements ewodeba n u l i s g a m y o f i , Tu arsebobs iseTi sRelementi, rom rs 0 .g a n m a r t e b a 1 6 . rgols, romelic winadadeba 26-is erT-erT(da, maSasadame, orive) pirobas akmayofilebs, a r e ewodeba.komutaciur ares m T e l o b i s a r e ewodeba.m a g a l i T i 7 9 . aris mTelobis are (da saxelwodebac ammagaliTidan modis).m a g a l i T i 8 0 . nebismieri rgoli gayofiT aris are danebismieri veli aris mTelobis are.w i n a d a d e b a 2 7 . Tu raime R rgoli sasrulia (e. i. Tu masaqvs sasruli raodenobis elementebi) da aris mTelobis are, maSinis aris veli.d a m t k i c e b a . vTqvaT, R aris sasruli mTelobis are dar R aris misi aranulovani raime elementi. ganvixiloT simravlers,s R . Tu 2s1 2s2raime s 1, s 2Ss s1 2elementebisaTvis, maSinwinadadeba 26-is ZaliT. amitom yvela rs aris gansxvavebuli.es ki niSnavs, rom R -is nebismier elements aqvs aseTi saxe.kerZod, 1 rs', s' R.es ki niSnavs, rom r aris Sebrunebuli da,maSasadame, R aris veli.g a n m a r t e b a 1 7 . vTqvaT, R rgolia. R simravlis aracarielS R simravles ewodeba R rgolis q v e r g o l i , Tu Saris rgoli R -is binaruli operaciebis mimarT.SevniSnoT, rom Tu S R aris R rgolis qvergoli, maSinS,aris R,abeluri jgufis qvejgufi, xolo S,R,monoidis qvemonoidi.m a g a l i T i 8 1 . rgolis nebismieri qvergoli emTxveva -s. marTlac, Tu S aris raime qvergoli, maSin 1S da radgann-jerS TviTon aris rgoli, amitom 1 11 n1SmTeli n elementisaTvis. amitom S .kinebismieri138

m a g a l i T i 8 2 . aris , da rgolebis qvergoli, aris da rgolebis qvergoli, xolo aris rgolisqvergoli.w i n a d a d e b a 2 8 . vTqvaT, R rgolia, xolo S ki misisakuTrivi qvesimravle. S aris R rgolis qvergoli maSin damxolod maSin, rodesac1 . S Caketilia R rgolis Sekrebisa da gamravlebisoperaciebis mimarT, e. i. Tu r,s S , maSin r s,rs S .2 . nebismieri sSelementisaTvis, s S .g a n m a r t e b a 1 8 . vTqvaT, R rgolia. R rgolis Sebrunebadielementebis jgufi aris R,monoidis Sebrunebadi elementebisjgufi. is R simboloTi aRiniSneba.m a g a l i T i 8 3 . 1da n m n , , 1n n u.s.g. .m a g a l i T i 8 4 . Tu R rgolia gayofiT, maSin R R \ 0.idealebijgufebis SemTxvevis msgavsad, faqtor-rgoli rom ganvixiloT,gvWirdeba Semdegig a n m a r t e b a 1 9 . vTqvaT, R rgolia. I Rqvesimravlesewodeba R rgolis m a r c x e n a i d e a l i , Tu I aris R,jgufis qvejgufi da nebismieri r R da xIelementebisaTvis,. analogiurad, R,jgufis raime qvejgufs ewodeba R rgolismarjvena ideali, Tu nebismieri r R da xI elementebisaTvis,xr I . I R qvesimravles ewodeba R rgolis ( o r m x r i v i )i d e a l i , Tu is erTdroulad marcxena idealicaa da marjvenac.SevniSnoT, rom Tu R rgoli komutaciuria, maSin misimarcxena da marjvena da ormxrivi idealebi erTmaneTs emTxveva daamitom Cven maT movixseniebT ubralod rogorc idealebs.Rm a g a l i T i 8 5 . Tu R nebismieri rgolia, maSin 0 R R orive aris R rgolis ormxirivi ideali. maT R rgolsda139

arasakuTrivi idealebi ewodeba, yvela sxva ideals ki sakuTrivi.rgols, romelsac mxolod arasakuTrivi idealebi aqvs, m a r t i v iewodeba.m a g a l i T i 8 6 . luwi ricxvebis simravle aris rgolis(ormxrivi) ideali.m a g a l i T i8 7 . radgan ori kenti ricxvis jami luwiricxvia, amitom kenti ricxvebis simravle araa rgolis ideali.g a n m a r t e b a 2 0 . komutaciuri R rgolis I idealsewodeba martivi, Tu nebismieri r,s R elementebisaTvis, Turs I , maSin an r I , an s I .m a g a l i T i 8 8 . Tu R rgolia da r R, maSin simravleRr sr,s R aris R rgolis marcxena ideali. faqtiurad, Rraris r elementis Semcveli minimaluri ideali. am ideals re l e m e n t i T w a r m o q m n i l i m T a v a r i m a r c x e n a i d e a l iewodeba. analogiurad, rR rs,s R simravle aris R rgolisr elementiT warmoqmnili marjvena mTavari ideali.SevniSnoT, rom rodesac R komutaciuria, maSin Rr rR . amSemTxvevaSi, es ideali rsimboloTi aRiniSneba.m a g a l i T i 8 9 . rgolis nebismieri ideli aris mTavari,anu aqvs nsaxe, sadac n raime mTeli ricxvi (radgan komutaciuria, amitom marcxena da marjvena idealebi erTmaneTsemTxveva).w i n a d a d e b a 2 9 . nebismier rgols gayofiT aqvs mxolodarasakuTrivi idealebi.d a m t k i c e b a . vTqvaT, R rgolia gayofiT da I Rraime (vTqvaT) marcxena ideali. Tuki misiI 0, maSin arseobos iseTielementi s I , rom s 0 . radgan s aranulovania da radgandaSvebiT R aris rgoli gayofiT, amitom arsebobs s elementisSebrunebulissR 1 elementi. maSin, radgan I aris marcxena11ideali, amitom s s I . magram s s 1 , e. i. 1 I . maSin isev I -is marcxena idealobidan miiReba, rom nebismieri r R140

elementisaTvis r r 1 I , rac niSnavs, rom I R . damtkicebaanalogiuria marjvena I idealis SemTxvevaSic.m a g a l i T i9 0 . rgolSi1) ideali 7 7 n : n 7 aris martivi imitom, rom Tumn , maSin mn 7k, k , da radgan 7 martiviricxvia, amitom an m iyofa 7-ze (e. i. aniyofa 7-ze (e. i.n 7).2) rgolis ideali 14 14 n : n radgan, magaliTad, 28 4 7 (14)arc 7 14.m 7), an n araa martivi, , magram arc 414am magaliTis 1) nawilSi moyvanili msjelobis analogiurimsjelobiT mtkicdeba Semdegiw i n a d a d e b a 3 0 . rgolis nmxolod maSin, rodesac n martivi ricxvia.daideali martivia maSin dag a n m a r t e b a 2 1 . R rgolis I ideals ewodeba maqsimaluri,Tu ar arsebobs R rgolis iseTi sakuTrivi ideali, romelicSeicavs I -s da ar emTxveva mas.m a g a l i T i 9 1 . Cven viciT, romaris nsaxis. radgan n mm yofs n -s, amitom n rgolis nebismieri idealimaSin da mxolod maSin, rodesacaris maqsimaluri ideali maSin damxolod maSin, rodesac n 1 da n -s ara aqvs sakuTrivigamyofebi, e. i. rodesac n aris martivi.maxasiaTebelig a n m a r t e b a 2 2 . R rgolis maxasiaTebeli char Rumciresi naturaluri ricxvi, romlisTvisac sruldeba toloban-jern1 111 0 .aris is141

Tu aseTi naturaluri ricxvi ar arsebobs, maSin vambobT, romchar R 0 .m a g a l i T i 9 2 . char char char char 0m a g a l i T i 9 3 . char . n .nw i n a d a d e b a 3 1 . Tu R mTelobis area, maSin char R 0an char Rmartivia.d a m t k i c e b a . davuSvaT, rom char R n 0 . Tu nSedgenilia, maSin n kl , sadac 1 k n da 1 l n . maSinmaxasiaTeblis gansazRvris Tanaxmad, k 1 0 da l 1 0 . magramk 1 l 1 k l 1 n1 0 , rac niSnavs, rom R rgolsmaSin gaaCnia nulis gamyofebi, rac pirobas ewinaaRmdegeba. maSasadame, nunda iyos martivi.rgolTa homomorfizmig a n m a r t e b a 2 3 . vTqvaT, R da R ori rgolia. asaxvasf : R R ewodeba homomorfizmi R rgolidan R rgolSi, Tuf erTdroulad aris R,da R,jgufebisa da R,R,rommonoidebis homomorfizmi. ufro konkretulad es niSnavs, f rs f r f sf r s f r f snebismieri r,s RgansazRvrebidan gamomdinareobs, romgarda amisa, f r rnebismieri r Rf : R R homomorfizms ewodeba: endomorfizmi, Tu R R. epimorfizmi, Tu f asaxva sureqciulia. monomorfizmi, Tu f asaxva ineqciuria.elementebisTvis.f 11da elementisaTvis. avtomorfizmi, Tu R R da f aris izmorfizmi.daf 0 0.142

or R da R rgols ewodeba izomorfuli, Tu arsebobs raimeizomorfizmi f : R R .g a n m a r t e b a 2 4 . vTqvaT, f : R R rgolTa raimehomomorfizmi. f -is birTvi, ker f , ewodeba f : R, R,jgufTa homomorfizmis birTvs, e. i. f r R f rker : 0 .w i n a d a d e b a 3 2 . Tu f : R RmaSin ker f aris R rgolis ormxrivi ideali.rgolTa homomorfizmia,d a m t k i c e b a . radgan ker f aris f : R, R,jgufTa homomorfizmis birTvic, amitom ker f aris R rgolisadiciuri jgufis qvejgufi. Tu axla r ker( f ) , maSin nebismierisRdaelementisaTvis gveqneba:rac niSnavs, rom , ker( )ormxrivi ideali.f sr f s f r f s0 0 sr rs f , e. i. f rs f r f s 0 f s 0 ,w i n a d a d e b a 3 3 . Tu f : R RmaSin f asaxvis anasaxi Im f f sker f aris R rgolisrgolTa homomorfizmia,aris R rgolis qvergoli.d a m t k i c e b a . Tu r, s Im f , maSin f r s raime , rdar s R elementebisaTvis. maSin da Im da rsIm f f r s f r f s r srs fniSnavs, rom Im f f rs f r f s rs, amtiom, rac winadadeba 28-is ZaliTaris R rgolis qvergoli.m a g a l i T i 9 3 . nebismieri R rgolisTvis, igivuri asaxva1: R R aris rgolTa homomorfizmi.143

m a g a l i T i 9 4 . vTqvaT, n da ganvixiloT asaxvafn: , m nm . Cven viciT (ix. winadadeba 55), rom es asaxvaaris ,jgufis endomorfizmi. magram, sazogadod, is araaf m m nm m , xolorgolTa homomorfizmi. marTlac,n 1 2 1 22fnm1 nm1da fnm2 nm2, amitomn n maSasadame, f m m f m f mn 1 2 n 1 n 2f m f m n m m .1 2 1 2 yvela m1m 2elementisaTvis maSin da mxolod maSin, rodesacrodesac n 0 an n 1.2n n , e. i.w i n a d a d e b a 3 4 . Tu R rgolia gayofiT, maSin nebismierif : R R homomorfizmisaTvis, an f asaxvaa ineqciuri, anR 0.d a m t k i c e b a . radgan ker f aris R -is ormxrivi idealiwinadadeba 32-is ZaliT, amitom is aris an 0an R (ix.winadadeba 29). pirvel SemTxvevaSi fmagaliTi 68), xolo meore SemTxvevaSi, radganR 0.vTqvaT, R rgolia, xolo I RmaSin, kerZod, I aris R,(ratom?) da amitom SegviZlia ganvixiloT R Iaris ineqciuri asaxva (ix. 0 f 1 1, amitommisi raime ormxrivi ideali.jgufis normaluri qvejgufifaqtor-jgufi.viciT, rom am faqtor-jgufSi Sekreba ganimarteba Semdegnairad:r I s I r s I .ganvsazRvroT axla R I simravleze gamravlebis operacia ase:r I s I rs I .w i n a d a d e b a 3 5 . Sekrebisa da gamravlebis zemoT moyvanilioperaciebi R Isimravleze gansazRvravs rgolis struqturas.garda amisa, asaxva p : R R I , r r I aris rgolTahomomorfizmi, romlis birTvia I .144

d a m t k i c e b a . sakmarisia vaCvenoT, rom gamravlebis operaciagansazRvrulia koreqtulad, e. i. rom gamravlebis operacia araadamoukidebeli eqvivalentobis klsasebidan warmomadgenlebisarCevaze. amitom davuSvaT, romr I r I da s I s I .maSin r r x da s s y , sadac x,y I . amitom gveqneba:rs r xs y rs z ,sadac z xs ry xy . radgan I ormxrivi idealia, amitom z Ida, maSasadame,rs I rs I .SevniSnoT, rom Tu R komutaciuria, maSin R I agreTve komutaciuria.R I rgols ewodeba R rgolis f a q t o r - r g o l i I idealiT.w i n a d a d e b a 3 6 . vTqvaT, f : R RrgolTa homomorfizmia.1) Tu I R aris R rgolis marcxena (marjvena, ormxrivi)ideali da f aris sureqciuli asaxva, maSin f I Raris R rgolis marcxena (marjvena, ormxrivi) ideali.2) Tu I Raris R rgolis marcxena (marjvena, ormxrivi)1ideali, maSin f Iormxrivi) ideali.3) Tu S R aris R rgolis marcxena (marjvena, aris R rgolis marcxena qvergoli, maSin f Saris R rgolis qvergoli. kerZod, fR rgolis qvergoli.4) Tu SRasaxvis anasaxi aris1 aris R rgolis qvergoli, maSin f R arisR rgolis qvergoli.5) Tu g : R ' R"rgolTa homomorfizmia, maSin gf : R R"aseve rgolTa homomorfizmia.w i n a d a d e b a 3 7 . vTqvaT, f : R R . maSin asaxva f : R I Rda I ker f rgolTa homomorfizmia , r I f r arisrgolTa homomorfizmi, romelic aris izomorfizmi maSin damxolod maSin, rodesac f aris sureqciuli asaxva.145

d a m t k i c e b a . sakmarisia aRvniSnoT, rom f homomorfizmisbirTvia 0 da amitom f aris ineqciuri asaxva.m a g a l i T i 9 5 . asaxva n, m mmodnrgolTa sureqciuli asaxva, romlis birTvia n . n .nrgolTa lokalizaciaarisamitomvTqvaT, F velia da R ki misi qvergoli. maSin R arismTelobis are da advili Sesamowmebelia, rom F -is is umciresiqveveli, romelic moicavs R -s, aris simravlecxadia, Tu 01 , , da 0Q R ab F a b R b1c , maSin ab acbc 1 . amitom, Tuab 1namravls aRvniSnavT a bsimboloTi, maSinaQ R , a, b R da b 0.bgarda amisa, a ac , Tu c 0 . rogorc vxedavT, R rgolidanb bcQ R veli aigeba igivenairad, rogorc mTeli ricxvebisrgolebidan igeba racionalur ricxvTa veli. ganvixiloT axlaSebrunebuli amocana: vTqvaT, mocemulia mTelobis are R .SegviZlia vipovoT iseTi veli, romlis qvergolic iqneboda R ? amkiTxvaze pasuxi dadebiTia.ganvixiloT raime mTelobis are R . vTqvaT, a,bda c,d aris R elementebis iseTi wyvilebi, rom b 0 da d 0 .vuwodoT am wyvilebs e q v i v a l e n t u r i (da es ase CavweroTa, b ~ c,d ), Tu ad bc . R mTelobis aris elementebisganSedgenil wyvilebs Soris ganmartebuli es mimarTeba ariseqvivalentobis mimarTeba.marTlac, refleqsuroba da simetriuloba advili saCvenebelia,amitom davuSvaT, rom , ~ , a b c d da c, d ~ , e f . gansazRvre-146

iT maSin ad bc da cf de . Tu pirvel tolobas gavamravlebTf -ze, meores ki b -ze, miviRebT:saidanac adfadf bcf da bcf bde bde . maSin adf bde 0 da amitom d afbe 0 .radgan R mTelobis area, mas nulebis gamyofebi ara aqvs daradgan d 0 Cveni daSvebiT, amitom af be 0 , e. i. af be . racniSnavs, rom a, b ~ e,f . maSasadame, ` ~ ~ aris eqvivalentobismimarTeba.a,b wyvilis eqvivalentobis klasi aRvniSnoT a b simboloTi.axla ganvsazRvroT, Tu rogor SevkriboT da rogor gavamravloTes ekvivalentobis klasebi: Tu a b da c dekvivalentobis nebismieriori klasia, maTi jami ganvsazRvroT ase:a c ad bc ,b d bdxolo namravli ki ase:a c ac .b d bdvaCvenoT, rom Sekrebis da gamravlebis es operaciebi koreqtuladaagansazRvruli, e. i. rom am operaciebis gansazRvra araa damokidebulia b da c dekvivalentobis klasebidan maTi warmomadgenlebisamorCevaze. vTqvaT, a a da c c . unda vaCvenoT, rom maSinb b d d ad bc ad bc ,bd bdmagram es toloba sworia maSin da mxolod maSin, rodesac bd ad bc bd ad bcanubdad bdbc bdad bdbc.es toloba ki samarTliania, radganab ab da cd dc. (1)147

analogiurad, a c a c maSin da mxolod maSin, rodesacb d bdac ac . es toloba ki sruldeba maSin da mxolod maSin,bd b d rodesac acbd bdacanu, rodesac abcd abdc, es kisruldeba (1)-is ZaliT.axla advili Sesamowmebelia, rom Sekrebisa da gamravlebis zemoTgansazRvruli operaciebi a beqvivalentobis klasebis simravleze,romelsac Cven Q( R)simboloTi aRvniSnavT, gansazRvravsrgolis sturquras. am rgolis erTeulia 1 1klasi, xolo 0 ki _01 . sinamdvileSi Q( R)aris veli. amis dasamtkiceblad, sakmarisiavaCvenoT, rom nebismieri aranulovan elements gaaCnia Sebrunebulielementi. radgan nulovani elementia 0 1 , amitom Tu a 0b 1, maSina a 1 b0 0 . amitom a bklasi aranulovania maSin da mxolodmaSin, rodesac a 0 (da aseve b 0 ). maSin misi Sebrunebulielementi iqneba b a , radgan a b ab ab 1 b a ba ab 1(radgan R rgolikomutaciuria, amitom ab ba ).SevniSnoT, rom asaxvaq : R Q R,aa 1aris rgolTa ineqciuri homomorfizmi.Q R vels ewodeba R mTelobis aris w i l a d T a v e l i .rodesac R , maSin QR .w i n a d a d e b a 3 8 . vTqvaT, R mTelobis area, K raime veli,xolo f : R K ki ineqciuri homomorfizmi. maSin arsebobserTaderTi rgolTa homomorfizmi : Semdegi diagrama aris komutaciuri (anu ff Q R K , romlisTvisac f p ):148

fd a m t k i c e b a . pirdapiri Semowmeba aCvenebs, rom a f a b f b winadadebis pirobas.formuliT gansazRvruli asaxva akmayofilebssasruli rgolebi da velebirgols ewodeba sasruli, Tu misi elementebis raodenobasasrulia. sasruli rgolebis SemTxvevaSi, maTi gamravlebisa daSekrebis operaciebi cxrilis saSualebiT moicema. magaliTad, 0,1simravleze Semdegi operaciebi 0 1 0 10 0 1 0 0 01 1 0 1 0 1gansazRvravs rgolis struqturas. SevniSnoT, rom rgolis esstruqtura aris ubralod 2 rgoli.rogorc viciT, n aris rgoli nebismieri nricxvisaTvis, romelsac gaaCnia sasruli raodenobis elementebi.magram yvela sasruli rgoli aseTi saxis araa.m a g a l i T i 9 6 . ganvixiloT simravle 0,1, a,bganvsazRvroT maTze Sekrebisa da gamravlebis operaciebiSemdegnairad: 0 1 a b 0 1 a b0 0 1 a b 0 0 0 0 01 1 0 b a 1 0 1 a ba a b 0 1 a 0 a a 0b b a 1 0 b 0 b 0 bda149

advili Sesamowmebelia, rom 0,1, a,bsimravle maszegansazRvruli aseTi operaciebiT aris rgoli da rom is araa 4 rgolis izomorfuli (ratom?).vTqvaT, R aris rgoli. ganvixiloT asaxvan-jer111, Tu n 0f : R, f n n1 0, Tu n 0 . n-jer 1 1 1 , Tu n 0 rogorc viciT, f aris ,jgufSi. vaCvenoT, rom faqtobrivad f jgufis homomorfizmi R,aris rgolTa homomorfizmi.marTlac, nebismieri n,m dadebiTi ricxvebisaTvisf nm nm1 nm1 n1 m1 f n f m.radgan f k f k , wina toloba samarTliania imSemTxvevaSic, rodesac an n , an maris uaryofiTi. amitom faris rgolTa homomorfizmi.ganvixiloT axla nebismieri rgolTa homomorfizmi f : R .rgolTa homomorfizmis ganmartebis ZaliT,nebismieri n 0mTeli ricxvisTvis gvaqvs:f 11da amitom n-jer n-jer f n f 111 f 1 f 1 n f 1 n 1. amitom, roca n 0 , maSin f m f m f m m 1 m 1 m 1 .amrigad, arsebobs erTaderTi rgolTa homomorfizmi f :da is moicema formuliT: f n n 1davuSvaT axla, rommaSin f : R(winaaRmdeg SemTxvevaSi, . RR 0(e. i. R aratrivialuri rgolia).homomorfizmis birTvi ver iqneba mTeli f da maSin R 01 1 0) da, amitom150

aris n saxis ideali, sadac n raime naturaluri ricxvia.amitom homomorfizmis Teoremis Tanaxmad, asaxva 1m n m Raris ineqciuri homomorfizmi. amitomR rgolis qvergolidan. n SegviZlia gavaigivoTw i n a d a d e b a 3 9 . zemoT aRweril situaciaSi, char R n .d a m t k i c e b a . Tu n 0 , maSin dasamtkicebeli araferia(radgan ar arsebobs iseTi naturaluri ricxvi m , rom m1 0 ).Tu n 0 da arsebobs iseTi naturaluri k ricxvi, rom k 1 0,maSin k n , anu k nm , m . amitom k n , rac niSnavs, romn aris is umciresi naturaluri ricxvi, rodesac sruldebatoloba 1 0n , anu sxva sityvebiT rom vTqvaT char R n .Tu n aris rgolis martivi ideali, maSin n 0an naris martivi ricxvi. rogorc viciT, pirvel SemTxvevaSi,char R 0 da amitom f aris ineqciuri asaxva, anu R rgoliSeicavs rgolis izomorful qvergols. meore SemTxvevaSi, naris martivi da R rgoli Seicavs n vels izomorfulqvevels. p veli aRiniSneba F puwodeben.simboloTi da mis m a r t i v v e l sw i n a d a d e b a 4 0 . sasrul rgols maxasiaTebeli aris amrgolis elementebis raodenobas gamyofi.d a m t k i c e b a . vTqvaT, n char Relementebis raodenobaa m . maSin m aris R, da R rgolisadiciuri jgufisrigi da lagranJis Teoremis Tanaxmad, mr 0 nebismieri r RelementisaTvis. axla, Tu m qn k , sadac 0 k n , maSinkr m qn r mr q nr 0 0 0 .magram, radgan k n da char R n , amitom k 0 . e. i. n arism -is gamyofi.151

winadadeba 31-is ZaliT, rodesac R aris mTelobis are, misimaxasiaTebeli an 0-ia, an martivi ricxvi. pirvel SemTxvevaSi rgoli Caidgmeba R rgolSi da amitom winadadeba 38-is ZaliT, esCadgma gagrZeldeba velis R mTelobis areSi Cadgmamde. anu Riqneba -algebra (gavixsenoT, rom Tu K raime komutaciurirgolia, maSin K -algebra ewodeba A,iwyvils, sadac A arisraime komutaciuri rgoli, xolo i ki homoorfizmi KA rgolSi).rgolidanw i n a d a d e b a 4 1 . Tu F sasruli velia, maSin F simravliselementebis raodenobanaturaluri ricxvi.Fn p , sadac p martivi ricxvia, n kid a m t k i c e b a . Tu char F 0 , maSin zemoT moyvaniliSeniSnvnis ZaliT, F Seicavs vels, rac SeuZlebelia. amitomarsebobs iseTi martivi p ricxvi, rom char Faris p . maSin FF p-algebra da amitom igi SeiZleba ganxiluli iqnas rogorcsasrulganzomilebiani veqtoruli sivrce Fp-velze. Tu esganzomileba aris n , maSin cxadia, romFn p .w i n a d a d e b a 4 2 . Tu F sasruli velia da char F p ,pmaSin asaxva a a aris F velis avtomorfizmi (e. i. bieqciurihomomorfizmi F velidan Tavis TavSi).d a m t k i c e b a . ganvixiloT asaxvamaSin cxadia, elementebia, maSin gveqneba pf : F F,f a a .pf 1 1 1. xolo Tu a,b Ff ab ab p a p b p f a f baxla ganvixiloT f a b, gveqneba:nebismieri ppk k p kpk0f a b a b c a b .152

adgan k 0, pmniSvnelobebisaTvis TiToeulikcpkoeficientiiyofa p -ze, amitom isini yvela aris 0. amitom0 0 p p p 0 p p f a b c a b c a b b a f a f b . amitom f arisFppvelis endomorfizmi. ganvixiloT axla misi birTvi:p f a F a ker : 0 0 .amitom f aris ineqciuri endomorfizmi da, radgan F veliselementebis raodenoba sasrulia, f unda iyos agreTvesureqciuli. amitom f aris bieqciuri da, maSasadame, f aris Fvelis avtomorfizmi.w i n a d a d e b a 4 3 . nebismieri martivi p da naturaluri nricxvebisaTvis arsebobs sasruli veliF n , romelic zustad pnpelements Seicavs. nebismieri ori aseTi veli izomorfulia.w i n a d a d e b a4 4 . nebismieri veli Seicavs umcires qvevels.d a m t k i c e b a . cxadia, rom mocemuli velis qvevelebsnebismieri TanakveTa isev qvevelia. kerZod, yvela qvevelisTanakveTa aris veli, romelic aris nebismieri sxva qvevelisqveveli.g a n m a r t e b aqvevels uwodeben.2 5 . velis umcires qvevels, am velis m a r t i vw i n a d a d e b a 4 5 . Tu F sasruli velia da char FmaSin F -is martivi qveveli izomorfuliaes izomorfizmi erTaderTia.d a m t k i c e b a . cxadia, rom f :F p p ,velis. garda amisa, F , n n1homomorfizmisanasaxi Im f aris F pvelis izomorfuli. axla, F velisnebismieri qveveli Seicavs 1-s da amitom n 1-sac, n , anuIm f qvevelia F velis nebismier qveveli, da amitom unda iyosmartivi.153

S e d e g i 4 . Tu p martivi ricvia, maSin F pizomorfizmamdesizustiT erTaderTi sasruli velia, romelic Seicavselements.analogiuri msjelobiT miiReba Semdegiw i n a d a d e b a 4 6 . Tu char F 0 , maSin F velis martiviqvevelia .Tu F velia da E F misi qvevelia, maSin F SeiZlebaganxilul iqnas rogorc veqtoruli sivrce E velze.w i n a d a d e b a 4 7 . Tu F sasruli velia, xolo E ki misiqveveli, maSinn dim F : E .E -s aqvsnq elementi, sadac q Ed a m t k i c e b a . radgan F aris veqtoruli sivrce E -ze daradgan F aris sasruli, F aris sasrulganzomilebiani E -veqtoruli sivrce. Tu n dim F : Epda, maSin F -s aqvs nelementisagan Semdgari bazisi. vTqvaT, es bazisia a 1, , an. maSinF -is nebismieri elementi erTaderTi formiT warmodgeniliarogoc jamin xi ai, sadac xii1 E . radgan TiToeuli xi-sSeuZlia miiRos zustad q gansxvavebuli mniSvneloba, amitom F -saqvs zustadqncali elementi.w i n a d a d e b a 4 8 . vTqvaT, F sasruli velia. maSin mas aqvsnp cali elementi, sadac p char Fganzomileba mis martiv qvevelze.d a m t k i c e b a . radgan char Fmartivi qveveli izomorfulia pSedegi gamomdinareobs wina winadadebidan., xolo n ki aris F -isF pvelis. radgan Fpmartivia, amitom F -is p , amitomw i n a d a d e b a 4 9 . vTqvaT, F sasruli velia daFk p .maSin mis nebismier qvevels aqvspmelementi, sadac m aris n -is154

gamyofi. piriqiT, n -is nebismieri m gamyofisaTvis arsebobs F -iserTaderTi qveveli, romelsac aqvs zustadpmelementi.w i n a d a d e b a 5 0 . nebismieri velis aranulovanis elementebissimravle aris cikluri jgufi gamravlebis mimarT.m a g a l i T i 9 7 . F 30 velis yvela qvevelis sapovnelad2saWiroa 30-is yvela gamyofis povna. radgan es gamyofebia 1, 2, 3,5, 6, 10, 15, amitom F 30 velis qvevelebs Soris damokidebuleba2gamosaxulia Semdeg diagramaze:sadacaRniSnavs qvevelis CarTvas.bulis algebrebivTqvaT, A raime simravlea. n a w i l o b r i v i d a l a g e b i sm i m a r T e b a A simravleze aris binaruli mimarTeba ` ~ A -ze,romelic aris1) refleqsuri, e. i., a a nebismieri a A elementisaTvis;2) antisimetriuli, e. i., Tu a b da b a , maSin a b ;3) tranzituli, e. i., Tu a b da a c , maSin a c .vTqvaT, A, nawilobriv dalagebuli simravlea (e. i.aris raime simravle, xolo ki masze gansazRvruli raime nawilobrividalagebis mimarTeba). amboben, rom a A elementi arisS A qvesimravlis zeda sazRvari (da amas ase weren a S),Tu155A

1. s a nebismieri s S elementisaTvis;2. Tu s b yvela sSelementisaTvis, maSin a b .antisimetriulobis gamo S simravlis umciresi zeda sazRvari,rodesac is arsebobs, erTaderTia. rodesac S aris orelementianis t simravle , , maSin s,t aRniSvnis navclad s t aRniSvnagamoiyeneba. rodesac S , maSin SaRiniSneba 0 simboloTi.cxadia, rom 0 aris A simravlis umciresi elementi (ratom?).ganvixiloT nawilobriv dalagebuli simravleiseTi,rom mis nebismier sasrul qvesimravles gaaCnia zusti zedasazRvari. advili sanaxavia, rom maSin binaruli operacia daelementi 0 akmayofilebs Semdeg pirobebs:(i) a a a ;(ii) a b b a ; (iii) a b c a b c ;(iv) a 0 a ,sadac a , b , c aris A simravlis nebismieri elementebi. amitom A, ,0aris komutaciuri monoidi, romlis nebismieri elementiaris idempotenti (Tu A, naxevarjgufia, maSin a A elementiaris idempotenti, Tu a a a ).piriqiT, gvaqvs Semdegiw i n a d a d e b a 5 1 . vTqvaT, A, ,0iseTi komutaciuri monoidia,romlis yvela elementi aris idempotenti. maSin A simravlezearsebobs erTaderTi nawilobriv dalagebis mimarTeba iseTi, roma b aris a,bsimravlis zusti zeda sazRvari, xolo 0 ki carielisimravlis zusti zeda sazRvari (anuti).A -s umciresi elemen-d a m t k i c e b a . davuSvaT, aris nawilobrivi dalagebismimarTeba A -ze, romelic akayofilebs winadadebis pirobebs. maSincxadia, rom nebismieri a A elementebisaTvis, a b maSin damxolod maSin, rodesac a b b . piriqiT, Tu a b b pirobiTganvsazRvreT a b mimarTebas, maSin aris refleqsuri (i)pirobis ZaliT, xolo misi antisimetriuloba gamomdinareobsA,156

ganmartebidan: Tu a b (e. i. Tu a b b ) da b a (e. i. Tub a a ), maSin b a b b a a . tranzitulobis saCvenebladdavuSvaT, rom a b da b c (e. i. rom a b b da b c c ).maSin gvaqvs:amitom a c .axla piriqiT, vTqvaT a,b A aris A simravlis nebismieriori elmenti. maSin a a b a a b a b (radgana a a (i) pirobiT), amitom a a b . analogiurad miviRebT,rom b a b . Tu a c da b c , maSinamitom a b c . e. i. a b aris a,b simravlis zusti zedasazRvari, da bolos, (iv)tolobis ZaliT, 0 arisnawilobriv dalagebuli simravlis umciresi elementi.a c a b c , radgan b c c a b c(iii)pirobis ZaliT= b c radgan a b b c radgan b c c. a b c a b c,radgan b c e. i. b c c a c c radgan a c e. i. a c c wina winadadebaSi gansazRvrul struqturas ewodeban a x e v a r m e s e r i .oradulad, zusti zeda sazRvris ganmartebaSi monawileutolobebis SebrunebiT miiReba z u s t i q v e d a s a z R v r i scneba. S, a b da 0 aRniSvnebis Sesabamisi aRniSvnebi axlaiqneba S,a b da 1.m e s e r i ewodeba nawilobriv dalagebul simravles, romlisnebismier sasrul qvesimravles gaaCnia zusti zeda da qvedasazRvrebi.winadadeba 51 da misi oradulidan gamomdinareobs, rom meseriaris simravle masze gansazRvruli ori komutaciuri monoidisstruqturiT ise, rom am simravlis nebismieri elementi aris idempotenturi orivebinaruli operaciis mimarT, da157.A,

am simravleze inducirebuli ori nawilobrivi dalagebismimarTeba erTmaneTis oradulia.w i n a d a d e b a 5 2 . vTqvaT, , ,0 danaxevarmeserebi. maSinmaSin, rodesac nebismieriSemdegi ori toloba: a a b a ;a a b a .d a m t k i c e b a .tolobidan gamomdinareobs, romaris meseri maSin da mxolodelementebisaTvis sruldebaTu es ori toloba sruldeba, maSinda piriqiT.amitom A simravleze inducirebuli orive nawilobrivi dalagebismimarTeba aris oraduli.im meserebSi, romlebic praqtikaSi xSirad gvxvdeba, sruldebaSemdegi e. w. d i s t r i b u c i u l o b i s kanoni:(1) a b c a b a c a b c A.w i n a d a d e b a 5 3 . Tu meserSi sruldeba distribuciulobiskanoni, maSin sruldeba agreTve misi oraduli kanoni:(2)d a m t k i c e b a . gvaqvs:A, ,0, ,1aq pirvel tolobaSi gamoviyeneT (1), meoreSi _ A A, ,1isev (1), meoTxeSi -is asociuroba, xolo mexuTeSi ki .a,b Aa b a a b a , , ,a b c a b a c, a, b,c A a a b ca a c b ca a cb ca b a c a b a a b c a b c. a b b, mesameSiw i n a d a d e b a 5 4 . vTqvaT, A aris distribuciuli meseri daa, b,c A . maSin arsebobs erTaderTi iseTi elementi xA , rom158

d a m t k i c e b a . Tu x da y iseTi elementebia, rom oriveakmayofilebs am pirobebs, maSinx a b da x a c x x x a x c x y a x y x a x y b x y ,radgan b x y y a aris x,y simravlis zusti qvedazRvari. analogiurad, y x y da amitom x y .g a n m a r t e b a 2 6 . vTqvaT, A meseria. xA elementsewodeba a A elements d a m a t e b a , Tu x a 0 da x a 1.wina winadadebis ZaliT, elementis damateba erTaderTia, rocais arsebobs. a elementis damateba a siboloTi aRiniSneba.g a n m a r t e b a 2 7 . distribuciul mesers ewodeba b u l i sa l g e b r a , Tu mis nebismier elements gaaCnia damateba.m a g a l i T i 9 8 . 0,1 simravle aris bulis algebra.m a g a l i T i 9 9 . Tu X raime simravlea, maSin ℘() arismeseri, sadac mimarTeba qvesimravleebis CarTviTaa gansazRvruli;e. i. Tu , ℘ (), maSin A B maSin da mxolod maSin, rodesacA aris B -s qvesimravle. maSin ∨ Seesabameba ori simravlisgaerTianebas, xolo ki maT TanakveTas. 0 Seesabameba simravles,da 1 Seesabameba X simravles. cnobilia, rom ℘() aris distribuciulimeseri. ufro metic, ℘() aris bulis algebra: Tu℘(), e. i. Tu A aris X simravlis qvesimravle, maSin misidamateba X aris X \ A sxvaoba.m a g a l i T i 1 0 0 . vTqvaT, L aris klasikur gamonaTqvamTalogikis yvela gamonaTqvamTa klasi. SemoviRoT am klasze mimarTeba` ~ Semdegnairad: p q maSin da mxolod maSin, rodesac p qda q p (sadac aRniSnavs implikaciis logikur operacias).advili sanaxavia, rom ` ~ aris ekvivalentobis mimarTeba L -ze. ekvivalentobisklasebis erToblioba aRvniSnoT L -iT. axla L -zeganvsazRvroT , da operaciebi Semdegnairad:.159

amis Semdeg, p p aRvniSnoT 1-iT da p p ki 0-iT (aqaris nebismieri gamonaTqvami). radgan nebismieri sxvagamonaTqvamebs, p p q q da p p q q , amitom 1 da0 ganzogadebulia koreqtulad. pirdapiri Semowmeba gviCvenebs, romL, , , ,0,1aris bulis algebra.m a g a l i T i 1 0 1 . ganvixiloT elementaruli xdomilobaTaraime sivrce da ganvsazRvroT masze , , operaciebi Semdegnairad: Tu A,B (e. i. Tu A da B aris ori nebismierixdomiloba sivrcidan), maSin A ∨ aris xdomileba,romelic Sesdgeba im elementaruli xdomilobebidan,romlebic an A -s ekuTvnis, an B -s. Tu A,B , maSin A B aris xdomiloba, romelicSedgeba im elementaruli xdomilobebisgan, romlebicerTdroulad ekuTvnis A -sac da B -sac. Tu A , maSin A aris misi sawinaaRmdego xdomiloba. 1 aris aucilebeli xdomiloba (e. i. ). 0 aris SeuZlebeli xdomiloba .maSin , , , ,1,0 aris bulis algebra.radgan elementis damateba erTaderTia, amitom gvaqvs Semdegiaw i n a d a d e b a 5 5 . Tu A bulis algebraa da a A , maSin a .w i n a d a d e b a 5 6 . Tu A bulis algebraa da a,b A , maSin ∧= ∨ da ∨= ∧ . d a m t k i c e b a . gvaqvs: p q p q, pq p q, p p. pq160

meores mxriv gvaqvs:a b a b a b a a b ba ab a b b 1 b a111 1.e. i. a b aris a b elementis damateba.S e n i S v n a 4 . wina winadadebaSi damtkicebul tolobebs d em o r g a n i s k a n o n e b i ewodeba.S e n i S v n a 5 . Tu gavaanalizebT wina winadadebis meoretolobis damtkicebas, davinaxavT, rom igi miiReba pirvelitolobis damtkicebidan masSi da da agreTve 1 da 0simboloebis urTierTSenacvlebiT. es aris kerZo SemTxveva e. w.o r a d u l o b i s p r i n c i p i s a b u l i s a l g e b r e b i s a T v i s ,romelic SemdgomSi mdebareobs: Tu bulis algebrebis raime klasSidamtkicebulia raime debulebis samarTlianoba, samarTliani iqnebaagreTve misi oraduli debulebac, romelic miiReba sawyisidebulebidan masSi da da agreTve 1 da 0 simboloebisurTierTSenacvlebiT.vTqvaT, A aris bulis algebra. A simravleze ganvsazRvroTSekrebis operacia Semdegnairad:w i n a d a d e b a 5 7 . sruldeba Semdegi distribuciulobiskanoni:a b c a b a c a b c A.d a m t k i c e b a . gvaqvs: . a b a b a ba a bb a a b a b b a b a 0 0 0 0.a b a b b a , , ,161

SevniSnoT, rom bolos wina tolobaSi gamoviyeneT de morganiskanoni.advili sanaxavia, rom asociurobis kanoniagreTve sruldeba. garda amisa, nebismierigvaqvs:daa 0 a 0 0 a a 1 0 1 a 0 a .elementisaTvissabolood miviReT, rom A, ,0aris jgufi (komutaciuri ` ~operaciis ganmartebis Tanaxmad) da, maSasadame,aris komutaciuri rgoli.piriqiT, vTqvaT, A iseTi rgolia, romlis yvela elementi2aris idempotenti (e. i. a a nebismieri a A elementisaTvis).aseT rgols b u l i s r g o l s uwodeben.gvaqvs a b c a c b 0 a b c 0a c b a b a a b c a c a a c b a b a c a c a b a b a ca c a ba b c a b c c b a b a c .a b c a b cw i n a d a d e b a 5 8 . Tu A bulis rgolia, maSin(i) A aris komutaciuri.(ii) nebismieri a A elementisaTvis, a a 0 .d a m t k i c e b a . vTqvaT, a,b A nebismieri elementebia. maSina b a b a ab ba a a ab ba b .a Aa a a a a a 0 0 0 2 2 2A A, , ,0,1162

2 2amitom ab ba 0. Tu a b maSin miviRebT, rom a a 0 , anua a 0 , maSin ab bada amitom ab ba .am winadadebis (ii)-dan gamomdinareobs, rom nebismieri bulisrgolis maxasiaTebeli aris 2.Tu bulis rgolia, maSin A, ,1aris naxevarmeseri,romelSic nawilobrivi dalagebis mimarTeba ganimartebaSemdegnairad: a b maSin da mxolod maSin, rodesac ab a (ix.winadadeba 51-is damtkiceba).ganvixiloT axla a b ab gamosaxuleba. gveqneba:da, amgvarad,a a b ab a ab a b a ab ab a 2ab a ,b a b ab b .amitom a b ab aris a,b simravlis zeda sazRvari. Tuaris agreTve am simravlis zeda sazRvari, maSina b ab c ac bc abc a b ab ,amitom a b ab c da, maSasadame, a b ab arissimravlis zusti zeda sazRvari. aRvniSnoT igi a b simboloTi.pirdapiri Semowmeba gviCvenebs, rom ` ~ aris distribuciuli ` ~ operaciis mimarT da rom 1a aris a -s damateba. amitomaris bulis algebra.#sainteresoa, ra iqneba A bulis algebraze gansazRvruliSekrebis operacia (ix. ganmarteba winadadeba 57-is win). gvaqvs:es niSnavs, rom Tu gvaqvs bulis algebraA bulis rgols, xolo misgan ki #da misgan avagebTbulis algebras, mivi-RebT isev sawyis bulis algebras. e. i. A A . analogiurad,#A A cA 2 2 , , ,1 ,0,1 a,ba b b a a 1 b b 1a a ab b ba A # a b 6 ab a b. a ab b ab a ab a ab a b ab ab ab ab ab ab A A . am faqts s t o u n i s o r a d o b a s uwodeben bulisAA #c163

algebrebsa da bulis rgolebs Soris. kategoriaTa Teoriis enazees niSnavs, rom bulis algebrebisa da bulis rgolebs kategoriebiizomorfulia.savarjiSoebi1. aCveneT, rom n elementian simravleze arsebobsgansxvavebulibinaruli operacia. ipoveT A a,b2nnsimravleze gansazRvruli16-ve operacia. maTgan ramdenia komutaciuri? asociuri?maTgan ramdens gaaCnia neitraluri elementi?2. simravleze ganvsazRvroT binaruli operacia Semdegnairad:x y xy 1. aCveneT, rom aris komutaciuri,magram araasociuri binaruli operacia.3. ganvsazRvroT -ze binaruli operacia Semdegnairad:n m n m .aris es operacia komutaciuri? asociuri? aaqvs am operaciasneitraluri elementi?4. aris magaliTi 7-Si gansazRvruli binaruli operacia komutaciuri?asociuri? aaqvs am operacias neitraluri elementi?5. ganvixiloT ori asaxva f , g : , romlebic gansazRvruliaSemdegnairad: f n 2nda g ndaamtkiceT, rom gf 1, magram fg 1.6. daamtkiceT, rom Tu G, ,en , Tu n luwia 2. 0, Tu n kentianaxevarjgufia, maSin misi orinebismieri x da y elementisaTvis, Tu x y e , maSinb a 2 b a .7. simravleze ganvsazRvroT binaruli operacia Semdegnairad:x y x y xydaamtkiceT, rom ,aris naxevarjgufi, romelsac gaaCnianeitraluri elementi.164

8. daamtkiceT, rom G jgufis neitraluri elementi aris iserTaderTi elementi, romlisTvisac sruldebatoloba.9. daamtkiceT, rom a bb a 2x xsaxis matricebis simravle, sadaca,b da a2 b2 0 , aris jgufi matricebis Cveulebrivigamravlebis mimarT.10. daamtkiceT, rom Tu sasrul jgufSi aris luwi raodenobiselementi, maSin arsebobs neitraluri elementisagan gansxvavebuliiseTi elementi, romelic aris Tavisi TavisSebrunebuli.11. daamtkiceT, rom Tu jgufi Seicavs xuTze nakleb elements,maSin es jgufi abeluria.12. daamtkiceT, rom Tu G jgufia da g G13. ipoveT 5-is rigi 25 ,14. ipoveT 2-is rigi jgufSi. 25 , jgufSi.1 , maSin 1g g .15. qvemoT CamoTvlili qvesimravleebdan, romeli warmoadgens qvejgufs?a) 1 , 3 8,;b) 1 , 2 , 3 , 4 5,;g) 0 , 2 , 4 , 6 8,;d) 0 , 2 , 4 , 6 , 8 10, .16. vTqvaT, X raime simravlea. ℘() simravleze ganvixiloTSemdegi binaruli operacia: \ X Y X Y X YdaamtkiceT, rom (℘(),∗) aris abeluri jgufi.17. vTqvaT, A a, b,c. aris ℘() simravle jgufi gaerTianebisbinaruli operaciis mimarT? TanakveTis binaruli operaciismimarT?165

18. daamtkiceT, rom G jgufi aris abeluri maSin da mxolodmaSin, rodesac misi nebismieri ori x da y elementisaTvis, xy x y .sruldeba toloba 1 1 119. ganvixiloT jgufi 16,. CamoTvaleT 16qvejgufisyvela elementi. ra rigi aqvs am elements?20. vTqvaT, xGelementis rigia 8. ipoveT Semdegi elementebisrigebi:a) x 2 ; b) x 3 ; g) x 4 ; d) x 5 ; e) x 6 .21. ipoveT Semdegi cikluri jgufebis yvela gansxvavebuliwarmomadgeneli:a) 8; b) 15; g) 16; d) 18.22. daamtkiceT, rom G jgufi aris abeluri maSin da mxolodmaSin, rodesac asaxva f : G G ,endomorfizmi.x x 1 aris G jgufis23. ganvixiloT asaxva f : 2 2, romelic gansazRvruliaformuliT 3f n n . daamtkiceT, rom f aris jgufebisendomorfizmi da ipoveT misi birTvi. aris f epimorfizmi?monomorfizmi?24. Tu G jgufia, maSin ra iqneba e Gqvejgufis mosazRvreobisklasebi G -Si? analogiuri kiTxva G GqvejgufisaTvis.25. daamtkiceT, rom Tu H da K aris G jgufis oriqvejgufi, maSin nebismieri xGelementisaTvis sruldebatoloba Hx Kx H K x .26. ganvixiloT ,jgufis normaluri qvejgufi ,(normalurobagamomdinareobs ,jgufis abelurobidan) daSesabamisi faqtor-jgufi . daamtkiceT, rom x elements, sadac x , aqvs sasruli rigi maSin da mxolodmmaSin, rodesac x aris racionaluri, e. i. rodesac x nraime mTeli m da naturaluri n ricxvebisaTvis.166

27. daamtkiceT, rom G da G qvejgufebi, maSin asaxvaf : G G , romelic mocemulia formuliT homomorfizmi.f x e , aris28. daamtkiceT, rom nebismieri mTeli n ricxvisaTvis, ,n,jgufebi izomorfulia.29. daamtkiceT, rom , da ,da jgufebi araaizomorfuli.30. ramdeni gansxvavebuli warmomqmneli SeiZleba hqondes me-6rigis ciklur jgufs?31. daamtkiceT, rom Tu raime abeluri jgufi Seicavs me-3 rigiselements, maSin es jgufi cikluria.32. Tu cikluri jgufis warmomqmnelia x elementi, romlisrigia m , maSinxkaris amave jgufis warmomqmneli maSin damxolod maSin, rodesac k da m urTierTmartivi ricxvebia.33. aCveneT, rom Tu K aris H jgufis qvejgufi, xolo H kiG jgufis qvejgufi, maSin K aris G jgufis qvejgufi.34. daamtkiceT, rom nebismieri abeluri jgufis sasruli rigismqone elementebis qvesimravle aris qvejgufi.35. daamtkiceT, rom jgufi, romlis rigia p m ( p martivia, mki naturaluri), Seicavs p rigis qvejgufs.36. qvemoT CamoTvlili asaxvebidan romelia , jgufisendomorfizmi?2 1a) x x ; b) x 2x; g) x x ; d) x ; e) x x ;x31v) x x ; z) x ; T) x x .x37. daamtkiceT, rom Tu jgufis rigia 3 an 4, maSin is cikluria.38. daamtkiceT, rom Tu G jgufia, xolo x ki misi nebismierielementi, maSinx x 1 .39. vTqvaT, R aris komutaciuri rgoli. ganvixiloT misi yvelaidempotentebis simravledaoperaciebi Semdegnairad:B R. SemoviRoT am simravleze167

a b ab,a b a b 2 ab.daamtkiceT, rom A, , ,0,1aris bulis algebra.40. daamtkiceT, rom orze meti elementis mqone bulis rgolsyovelTvis gaaCnia nulis gamyofebi.168

þàñòæ, îàëèæ, ãâèæ§ 1. ŽèàâIJîñèæ ïðîñóðñîâIJæ ᎠóãâïðîñóðñîâIJæà Ž ê é Ž î ð â IJ Ž . ïæéîŽãèâï éŽïäâ àŽêïŽäôãîñèæ øŽçâðæèæ ëìâîŽùæâIJæå,âûëáâIJŽ ŽèàâIJîñèæ ïðîñóðñîŽ.îëàëîù ûâïæ ëìâîŽùæâIJï àŽŽøêæŽå îŽæéâ éŽýŽïæŽåâIJâèæ åãæïâIJâIJæ, îëéèâIJæùöâïŽúèâIJâèæŽ øŽéëõŽèæIJáâï åâëîâéâIJæï ïŽýæå áŽ, îëéèâIJæù òŽîåëáàŽéëæõâêâIJŽ àŽéëåãèâIJöæ (ïðîñóðñîâIJï åŽãæïæ åâëîâéâIJæå, àŽéëåãèæï ûâïâIJæåᎠöâáâàâIJæå ýŽêáŽýŽê ñûëáâIJâê ŽèàâIJîñè ïæïðâéâIJï).õëãâèæ ïðîñóðñîæïåãæï ŽîïâIJëIJï óãâïðîñóðñîæï ùêâIJŽ. éŽàŽèæåŽá,àŽêãæýæèëå ßæìëåâðñîæ ïðîñóðñîŽ, â. û. éŽøãâêâIJâèæ. ãåóãŽå, A éŽøãâêâIJâèæŽ.áŽãñöãŽå, îëé A-äâ àŽêïŽäôãîñèæŽ éýëèëá âîåæ ⊛ ëìâîŽùæŽ.éŽöŽïŽáŽéâ, ñòîë äñïðŽá âï öâæúèâIJŽ øŽæûâîëï (A, ⊛) ïŽýæå, Žêñ éæéåæåâ-IJâèæ öâæùŽãï A ïæéîŽãèâï éŽïäâ àŽêïŽäôãîñèæ ⊛ ëìâîŽùææå. åñ B ⊆ A áŽ(B, ⊛) Žàîâåãâ éŽøãâêâIJâèæŽ (çâîúëá, ⊛ ëìâîŽùæŽ öâïŽúèâIJâèæŽ øŽçâðæèææõëï B-äâ), éŽöæê (B, ⊛)-ï âûëáâIJŽ óãâéŽøãâêâIJâèæ.àŽêãæýæèëå ïý㎠ïðîñóðñîŽ, (C, ⊕) (⊕ Ꭰ⊛ ëìâîŽùæâIJæ ñêᎠæõëïâîåæ Ꭰæàæãâ îæàæï; éŽàŽèæåŽá, åñ âîå-âîåæ éŽåàŽêæ IJæêŽîñèæŽ, ŽïâåæãâñêᎠæõëï éâëîâù. C-äâ öâïŽúèâIJâèæŽ àŽêãïŽäôãîëå ïý㎠ëìâîŽùæâIJæù,åñéùŽ ŽýèŽ éŽå Žî àŽêãæýæèŽãå). åñ ŽîïâIJëIJï ϕ : A → C ŽïŽý㎠æïâåæ,îëé ϕ(x ⊛ y) = ϕ(x) ⊕ ϕ(y) êâIJæïéæâîæ x Ꭰy-åãæï A-áŽê, éŽöæê ϕ-ï âûëáâIJŽßëéëéëîòæäéæ. åñ ŽîïâIJëIJï A-ïŽ áŽ C-ï öëîæï ßëéëéëîòæäéæ, éŽöæêàŽîçãâñèæ Žäîæå (A, ⊛) ïðîñóðñîæï (ϕ(A), ⊕) ßëéëéëîòñèæ ŽêŽïŽýæ æûãâãïûæêŽîâ ïŽýâï, îŽáàŽêŽù øãâê öâàãæúèæŽ öâãŽïîñèëå ⊛ ëìâîŽùæŽ A-äâᎠöâéáâà ŽãïŽýëå C-öæ (ϕ-ï ïŽöñŽèâIJæå), Žê þâî ŽãïŽýëå C-öæ, öâéáâàöâãŽïîñèëå ⊕ ëìâîŽùæŽ. ëîæãâ öâéåýãâãŽöæ öâáâàæ âîåæ ᎠæàæãâŽ. Žéæðëééëãæóùâãæå æïâ îëàëîù éëïŽýâîýâIJâèæŽ øãâêåãæï. Žé ïæðñŽùæŽöæ ñçâåàŽîçãâãŽöæ áŽàãâýéŽîâIJŽ 1.1 ïñîŽåäâ éëùâéñèæ çëéñðŽùæñîæ áæŽàîŽéŽ.1.1(Ž)-äâ éëùâéñèæŽ ïæéîŽãèââIJæ ᎠïðîñóðñîâIJæ, ýëèë 1.1(IJ)-äâ àŽêýæèñèæŽùŽèçâñèæ âèâéâêðâIJæïŽåãæï. 1.1(IJ)-äâ éŽîþãêæáŽê àŽéëïŽýñèæŽ âîåæᎠæàæãâ öâáâàæï ëîæ àŽêïýãŽãâIJñèæ òëîéŽ. áæŽàîŽéæï çëéñðŽùæñîëIJŽ=⇒ ëìâîŽùææï àŽêéŽîðâIJæáŽê.çâîúëá, éæãæôâIJå, îëé ϕ ◦ ⊛ = ⊕ ◦ ϕ, îŽù Žî êæöêŽãï çëéñðŽùæñîë-IJŽï éçŽùîæ Žäîæå, îŽáàŽê ⊛ Ꭰ⊕ àŽêïýãŽãâIJñèæ ëìâîŽùæâIJæŽ. ðëèëIJæïëîæãâ éýŽîâ ŽôêæöêŽãï âîåæ Ꭰæàæãâ îæàæï ëìâîŽùæâIJæï çëéIJæêŽùæŽï áŽ,169

170öâïŽIJŽéæïŽá, éæãáæãŽîå äñïðŽá àŽêéŽîðâIJŽéáâ:ŽïŽý㎠◦ ëìâîŽùæŽ = ëìâîŽùæŽ ◦ ŽïŽýãŽ,A × A ⊛ A (x, y)ϕC × C ⊕ Cϕ,(ϕ(x), ϕ(y))⊕x ⊛ y ϕ(x) ⊕ ϕ(y) = ϕ(x ⊛ y)ϕ.êŽý. 1.1(Ž)êŽý. 1.1(IJ)é Ž à Ž è æ å æ 1.1. ãåóãŽå, θ : Z → Z 10 Žîæï ŽïŽý㎠éåâè îæùýãåŽïæéîŽãèæï êŽöååŽ çèŽïâIJäâ éëáñèæå 10, éŽöæê θ(20) = 0, θ(17) = 7.àŽêãæýæèëå (Z, +) Ꭰ(Z 10 , +), øãâñèâIJîæãæ + ëìâîŽùææå Z-åãæï áŽZ 10 -åãæï. ùýŽáæŽ, θ Žîæï ßëéëéëîòæäéæ, îŽáàŽêθ(24 + 38) = θ(62) = 2,θ(24) + θ(38) = 4 + 8 = 2 (Z 10 -öæ).Žé öâéåýãâãŽöæ áæŽàîŽéŽ àŽéëæõñîâIJŽ öâéáâàêŽæîŽá:(Z, +) 2(Z, +)θ(Z 10 , +) 2 (Z 10 , +)êŽý. 1.1(à)ŽéîæàŽá, ïðîñóðñîâIJï öëîæï ßëéëéëîòæäéæ æêŽýŽãï ïðîñóðñîŽï.ïñîâóùæñèæ Žê æêâóùæñîæ ŽïŽýãâIJæï éæïŽôâIJŽá öâïŽúèâIJâèæŽ öâéëãæðŽêëåöâäôñá㎠ŽïŽýãæï îŽêàäâ, Žéæðëé, åñ ŽïŽý㎠ßëéëéëîòæäéæŽ, öâàãæúèæŽãæãŽîŽñáëå, îëé æï ñäîñêãâèõëòï ïðîñóðñîæáŽê ïðîñóðñîŽäâ àŽáŽïãèæïéâóŽêæäéï (ìæîæóæåŽù!). õëãâèàãŽîæ æêòëîéŽùææï áŽçŽîàãæï àŽîâöâ.àŽêéŽîðâIJŽ. æêâóùæñî ßëéëéëîòæäéï âûëáâIJŽ éëêëéëîòæäéæ, ïñîâóùæñèéëêëéëîòæäéï âûëáâIJŽ âìæéëîòæäéæ, IJæâóùæñî ßëéëéëîòæäéï çæ {æäëéëîòæäéæ. åñ ŽîïâIJëIJï æäéëéëîòæäéæ ïðîñóðñîâIJï öëîæï, ŽéëIJëIJâê,îëé æïæêæ æäëéëîòñèêæ ŽîæŽê.ïæðõ㎠\æäëéëîòñèæŽ" êæöêŽãï \æàæãâ òëîéæ" áŽ, Žéæðëé àëêæãîñèæŽãæòæóîëå, îëé æäëéëîòæäéæ àŽéëæûãâãï ŽèàâIJîñèæ ïðîñóðñîâIJæïïæéîŽãèæï âçãæãŽèâêðëIJæï çèŽïâIJŽá áŽõëòŽï (æý. ïŽãŽîþæöë 1.1.2).é Ž à Ž è æ å æ 1.2. ({∅, E}, ∩, ∪) Ꭰ({0, 1}, ∧, ∨) (æý. § 4, åŽãæ 4)æäëéëîòñèâIJæŽ.θ,

171á Ž é ð ç æ ù â IJ Ž . ãåóãŽå, φ(∅) = 0 Ꭰϕ(E) = 1. ùýŽáæŽ, φ ŽîæïIJæâóùæŽ, éŽöæêϕ(∅ ∩ ∅) = ϕ(∅) = 0 = 0 ∧ 0 = ϕ(∅) ∧ ϕ(∅),ϕ(∅ ∩ E) = ϕ(∅) = 0 = 0 ∧ 1 = ϕ(∅) ∧ ϕ(E),ϕ(E ∩ ∅) = ϕ(∅) = 0 = 1 ∧ 0 = ϕ(E) ∧ ϕ(∅),ϕ(E ∩ E) = ϕ(E) = 1 = 1 ∧ 1 = ϕ(E) ∧ ϕ(E),ϕ(∅ ∩ ∅) = ϕ(∅) = 0 = 0 ∨ 0 = ϕ(∅) ∨ ϕ(∅),ϕ(∅ ∩ E) = ϕ(E) = 1 = 0 ∨ 1 = ϕ(∅) ∨ ϕ(E),ϕ(E ∩ E) = ϕ(E) = 1 = 1 ∨ 0 = ϕ(E) ∨ ϕ(∅),ϕ(E ∩ E) = ϕ(E) = 1 = 1 ∨ 1 = ϕ(E) ∨ ϕ(E).ŽéîæàŽá, ϕ Žîæï ßëéëéëîòæäéæ áŽ, éŽöŽïŽáŽéâ, æäëéëîòñèæù. áŽIJëèëï,Žôãêæöêëå, îëé ïðîñóðñîŽ öâïŽúèâIJâèæŽ æäëéëîòñèæ æõëï åŽãæïæåŽãæï (æàñèæïýéâIJŽ ŽîŽðîæãæŽèñîæ æäëéëîòæäéæ) ᎠŽïâãâ, öâïŽúèâIJâèæŽæäëéëîòñèæ æõëï åŽãæïæ óãâïðîñóðñîæï (âï öâïŽúèâIJâèæŽ éýëèëá ñïŽïîñèëïæéîŽãèââIJöæ).à Ž ê é Ž î ð â IJ Ž . åñ ŽïŽýãæï àŽêïŽäôãîæïŽ áŽ éêæöãêâèëIJŽåŽŽîââIJæ âîåéŽêâåï âéåýãâãŽ, ßëéëéëîòæäéï âûëáâIJŽ âêáëéëîòæäéæ, ýëèëæäëéëîòæäéï { Žãðëéëîòæäéæ.é Ž à Ž è æ å æ 1.3. éëùâéñèæŽ A ïæéîŽãèæïåãæï (P(A), ∩, ∪) áŽ(P(A), ∪, ∩) æäëéëîòñèæŽ ϕ : X → X ′ ŽïŽýãæï éæéŽîå.á Ž é ð ç æ ù â IJ Ž . ùýŽáæŽ, ϕ Žîæï æêâóù掎 ᎠïñîâóùæŽù, åñB, C ∈ P(A), éŽöæêϕ(B ∩ C) = (B ∩ C) ′ = B ′ ∪ C ′ = ϕ(B) ∪ ϕ(C),ϕ(B ∪ C) = (B ∪ C) ′ = B ′ ∩ C ′ = ϕ(B) ∩ ϕ(C).éëàãæŽêâIJæå, öâãêæöêŽãå, îëé âï åŽêŽòŽîáëIJâIJæ êŽåèŽá àãæøãâêâIJï IJñèæïŽèàâIJîŽåŽ ïæéîŽãèæï ëîŽáñèëIJŽï Ꭰϕ Žîæï Žãðëéëîòæäéæ.ï Ž ã Ž î þ æ ö ë 1.1.1. ãŽøãâêëå, îëé ëîæ (Z 6 , ∗) ïðîñóðñîŽ àŽêýæèñè 4.1.2 éŽàŽèæåöææäëéëîòñèæŽ.2. ãåóãŽå, (A, ⊛), (B, ⊛) Ꭰ(C, ⊙) éŽøãâêâIJèâIJæŽ, ýëèë ϕ : A → B áŽθ : B → C æäëéëîòæäéâIJæŽ. ãŽøãâêëå, îëé θ ◦ ϕ : A → C, ϕ −1 : B → AŽïâãâ æäëéëîòæäéâIJæŽ.§ 2. ñéŽîðæãâïæ ëìâîŽùæñèæ ïðîñóðñîâIJæàŽêãæýæèëå ŽèàâIJîñèæ ïðîñóðñîâIJæ, îëéèâIJæù öâæùŽãâê éýëèëá âîåIJæêŽîñè ëìâîŽùæŽï, ïŽáŽù öâïŽúèâIJâèæŽ Žé Ꭰéëéáâãêë ìŽîŽàîŽòâIJöæïðîñóðñîâIJæ áŽèŽàâIJñè æóêâï \ïæúèæâîæï" éæýâáãæå. (ŽéIJëIJâê, îëé A ñòîëïñïðæŽ, ãæáîâ B, åñ A öâæúèâIJŽ àŽêãæýæèëå, îëàëîù B \êŽîøâêæ"ïðîñóðñîæå.)

172éëàãæŽêâIJæå ãêŽýŽãå, îëé äëàæâîåæ ïðîñóðñîŽ éëæùâéŽ ëîæ ñòîëïñïðæ ïðîñóðñîæï \öâîûõéæï" öâáâàŽá. áŽ, éŽöŽïŽáŽéâ, éçŽùîæ áŽèŽàâIJŽŽó öâñúèâIJâèæŽ. åæåëâñèæ ïðîñóðñîŽ öâæúèâIJŽ àŽêãéŽîðëå ŽîŽ éŽîðëúæîæåŽáæ ïðîñóðñîñèæ ðâîéæêâIJæå, ŽîŽéâá úæîæåŽáæ åãæïâIJâIJæåŽù.à Ž ê é Ž î ð â IJ Ž . S ïæéîŽãèâï ⊛ IJæêŽîñèæ ëìâîŽùææå, âûëáâIJŽêŽýâãŽîþàñòæ, åñ ïîñèáâIJŽ ŽïëùæŽùæñîëIJŽ:x ⊛ (y ⊛ z) = (x ⊛ y) ⊛ z, ∀ x, y, z ∈ S.à Ž ê é Ž î ð â IJ Ž . M ïæéîŽãèâï éŽïäâ àŽêïŽäôãîñèæ IJæêŽîñèæëìâîŽùææå, âûëáâIJŽ éëêëæáæ, åñ:I) ⊛ ŽïëùæŽùæñîæŽ;II) ŽîïâIJëIJï u ∈ M æïâåæ, îëéu ⊛ x = x = x ⊛ u, êâIJæïéæâîæ x ∈ M-åãæï(u-ï âûëáâIJŽ âîåâñèëãŽêæ âèâéâêðæ ⊛-æï éæéŽîå).êŽýâãŽîþàñòâIJæ ᎠéëêëæáâIJæ éêæöãêâèëãŽê îëèï åŽéŽöëIJâê ïæéIJëèëåŽïðîæóëêâIJæïŽ áŽ âêŽåŽ åâëîæâIJæï áŽéñöŽãâIJæïŽï.é Ž à Ž è æ å æ 2.1. ãåóãŽå, A = {x, y, z}. àŽêãæýæèëå x, y, z, îëàëîùïæéIJëèëâIJæ ᎠŽîŽ îëàëîù ëIJæâóðâIJæï Žê ‘ñùêëIJåŽ" ïŽýâèâIJæ, éæãæôâIJå,îëé A Žîæï ŽêIJŽêæ. A ∗ àŽêãïŽäôãîëå, îëàëîù ïæéîŽãèâ, A-ï ïæéIJëèëåŽïðîæóëêâIJæïŽ. éŽöæê A ∗ öâæùŽãï x, y, z, xx, xy, yx, xxyz, zyx ᎠŽ. ö. A ∗ -äâàŽêãéŽîðëå ⊙ ëìâîŽùæŽ öâéáâàæ ûâïæå:åñ α, β ∈ A ∗ , éŽöæê α ⊙ β = αβ, â. æ. ïðîæóëêâIJæï éæûâîŽ éæõëèâIJæå,Žêñ xyz ⊙ z = xyzz, xz ⊙ yz = xzyz ᎠŽ. ö.åæåëâñèæ α ïðîæóëêï Žóãï àŽîçãâñèæ ïæàîúâ, îëéâèæù Žîæï éŽïöæïæéIJëèëâIJæï îŽëáâêëIJŽ ᎠŽôæêæöêâIJŽ |α|-æå (àŽéâëîâIJŽ öâáæï îŽëáâêë-IJŽöæ). éŽàŽèæåŽá,|x| = 1, |xy| = 2, |xxxzy| = 5.âï îŽôŽù Žäîæå ßàŽãï ïæéîŽãèæï ïæéúèŽãîââIJæï ŽôêæöãêŽï. çâîúëá,éŽàîŽé|x| = 1, |{x}| = 1,|xy| = 2, |{x, y}| = 2,|yx| = 2, |{y, x}| = 2.|xyx| = 3, |{x, y, x}| = |{x, y}| = 2.Žéæðëé ïîñèæ ŽêŽèëàæŽ Žî Žîæï. ùŽîæâèæ ïæéîŽãèæï ŽêŽèëàæñîŽá ùŽîæâèæïðîæóëêæ ŽôæêæöêâIJŽ Λ-æå, Λ ∈ A α Ꭰ|Λ| = 0, Λ ⊙ α = α ⊙ Λ = αêâIJæïéæâîæ α ïðîæóëêæïåãæï.éŽöŽïŽáŽéâ, êâIJæïéæâîæ A ŽêIJŽêæïåãæï A ïðîñóðñîŽ (A ∗ , ⊙) Žîæï éëêëæáæ,Λ çæ { âîåâñèëãŽêæ âèâéâêðæŽ ⊙-æï éæéŽîå.

173äâéëå éëõãŽêæèæ éŽàŽèæåæ úŽèæŽê éêæöãêâèëãŽêæŽ. ïŽçéŽëá áæáæ AŽêIJŽêæïåãæï, îëéâèæù öâæùŽãï çëéìæñðâîæï ìâîæòâîæñèæ éëûõëIJæèëIJæïåãæïéæïŽûãáëé õãâèŽ ïæéîŽãèâï, çëéìæñðâîñè ïæïðâéâIJöæ àŽéëõâêâIJñèæ âêâIJæŽîæï A ∗ -æï óãâïæéîŽãèâ. âï ùêâIJŽ úæîæåŽáæŽ âêâIJæï òëîéŽèñîæ öâïûŽãèæïáîëï.éâïŽéâ (ᎠñçŽêŽïçêâèæ) ëìâîŽùæñèæ ïðîñóðñîŽ Žîæï éëêëæáæï ñöñ-Žèë ᎠIJñêâIJîæãæ àŽòŽîåëâIJŽ.à Ž ê é Ž î ð â IJ Ž . G ïæéîŽãèâï éŽïäâ àŽêïŽäôãîñèæ ⊛ IJæêŽîñèæëìâîŽùææå þàñòæ âûëáâIJŽ, åñ ïîñèáâIJŽ:Ž) ⊛ ŽïëùæŽùæñîæŽ;IJ) ŽîïâIJëIJï u ∈ G (âîåâñèëãŽêæ âèâéâðæ ⊛-æï éæéŽîå) æïâåæ, îëéu ⊛ x = x = x ⊛ u, ∀ x ∈ G.à) õëãâèæ x ∈ G-åãæï ŽîïâIJëIJï y ∈ G æïâåæ, îëé x ⊛ y = u = y ⊛ x(y-ï âûëáâIJŽ x-æï öâIJîñêâIJñèæ ⊛-æï éæéŽîå).åñ þàñòñîæ ëìâîŽùæŽ ŽôæêæöêâIJŽ ⊛ ïæéIJëèëåæ, éŽöæê u = 1 Ꭰy =x −1 , ýëèë, åñ ëìâîŽùæŽ ŽôæêæöêâIJŽ ⊕ ïæéIJëèëåæ, éŽöæê u = 0 Ꭰy = −x.þàñòâIJæ õãâèŽäâ éêæöãêâèëãŽêæŽ Žé ïŽé ïðîñóðñîŽï öëîæï, îŽáàŽê éŽïöæöâæúèâIJŽ ŽéëãýïêŽå a ⊛ x = b àŽêðëèâIJŽ. éŽîåèŽù, åñ a ⊛ x = b, éŽöæêa −1 ⊛ (a ⊛ x) = a −1 ⊛ b(a ∈ G =⇒ a −1 ∈ G),(a −1 ⊛ a) ⊛ x = a −1 ⊛ b (⊛ ŽïëùæŽùæñîëIJæï àŽéë),1 ⊛ x = a −1 ⊛ b (öâIJîñêâIJñèæï åãæïâIJŽ),x = a −1 ⊛ bâîåâñèëãêæï åãæïâIJŽ).\þàñòåŽê" Ꭰ\éëêëæáåŽê" âîåŽá ýöæîŽá ýéŽîëIJâê ïæðõãŽï { \çëéñðŽùæñîæ".âï êæöêŽãï, îëé þàñòñîæ ëìâîŽùæŽ Žîæï çëéñðŽùæñîæ, â. æ.y ⊛ x = x ∗ y ∀ x, y ∈ M Žê ∀ x, y ∈ G.þàñòæï ŽóïæëéâIJæáŽê àŽéëéáæêŽîâëIJï úŽèæŽê IJâãîæ ïŽïŽîàâIJèë öâáâàæ.àŽêãæýæèëå éŽîðæãæ éŽàŽèæåæ.é Ž à Ž è æ å æ 2.2. (G, ∗) þàñòöæá Ž é ð ç æ ù â IJ Ž .(a ∗ b) −1 = b −1 ⊛ a −1 .(a ∗ b) ∗ (b −1 ∗ a −1 ) = a ∗ (b ∗ b −1 ) ∗ a −1 = a ∗ 1 ∗ a −1 = a ∗ a −1 = 1.â. æ. a ∗ b-ï éŽîþãâêŽ öâIJîñêâIJñèæŽ b −1 ∗ a −1 . ŽêŽèëàæñîŽá, öâæúèâIJŽ ãŽøãâêëå,îëé æï éŽîùýâêŽ öâIJîñêâIJñèæùŽŽ, ïŽæáŽêŽù àŽéëéáæêŽîâëIJï ïŽIJëèëëöâáâàæ.þàñòâIJæ øãâêåãæï ìæîãâèæ éŽàŽèæåæŽ, ïŽáŽù òŽîåëá àŽéëæõâêâIJŽ æäëéëîòæäéâIJæ.àŽêãæýæèëå (R, +) Ꭰ(]0, ∞[ , ∗) þàñòâIJæï æäëéëîòæäéæ, îëéâèïŽùñûëáâIJâê èëàŽîæåéï. æï ŽéõŽîâIJï çŽãöæîï àŽéîŽãèâIJæïŽ áŽ öâçîâIJæï

174ëìâîŽùæâIJï öëîæï:ï Ž ã Ž î þ æ ö ë 2.1.a ∗ b = ϕ −1 (ϕ(a) + ϕ(b)),ϕ : x → hy p (x) îŽæéâ p ∈ ]0, +∞[ -åãæï.1. ᎎéðçæùâå âîåâñèëãŽêæ ᎠöâIJîñêâIJñèæ âèâéâêðâIJæï âîåŽáâîåëIJŽ(G, ∗) þàñòöæ;2. Žøãâêâå, îëé åñ (G, ⊛) þàñòöæ { a ⊛ b = a ⊛ c, éŽöæê b = c Ꭰåñx ∗ a = y ∗ a, éŽöæê x = y;3. öâŽéëûéâå, îëé ïŽïîñèæ ïæéîŽãèæï øŽïéæå ïæéîŽãèâ óéêæï þàñòïøŽïéŽåŽ àŽéîŽãèâIJæï éæéŽîå.3.1. îàëèâIJæ.§ 3. îàëèâIJæ ᎠãâèâIJæà Ž ê é Ž î ð â IJ Ž . ïæéîŽãèâï éŽïäâ àŽêïŽäôãîñèæ ëîæ IJæêŽîñèæŽèàâIJîñèæ ëìâîŽùææå, âûëáâIJŽ îàëèæ, åñ ïîñèáâIJŽ öâéáâàæ ŽóïæëéâIJæ:1) ⊛ ŽïëùæŽùæñîæŽ;2) ⊕ ŽïëùæŽùæñîæŽ;3) ⊕ çëéñðŽùæñîæŽ;4) ⊕ ëìâîŽùæŽï Žóãï âîåâñèëãŽêæ âèâéâðîæ, â. û. \êñèæ" ᎠŽôæêæöêâIJŽ\0"-æå;5) ŽîïâIJëIJï öâIJîñêâIJñèæ âèâéâêðâIJæ ⊕-æï éæéŽîå (â. û. éëìæîáŽìæîâ).éŽöŽïŽáŽéâ, (Z n , ⊛, +) õëãâèæ n ∈ N-åãæï Žîæï îàëèæ, îàëèæçëéñðŽùæñîæŽ, åñ ⊛ ëìâîŽùæŽ Žîæï çëéñðŽùæñîæ.6) áæïðîæIJñùæñèëIJæï Žóïæë鎎) X ⊛ (y ⊕ z) = (x ⊛ y) ⊕ (x ⊛ z);IJ) (x ⊕ y) ⊛ z = (x ⊛ z) ⊕ (y ⊛ z) õëãâèæ x, y, z ∈ R-åãæï.îàëèï âûëáâIJŽ âîåâñèëãŽêæ îàëèæ, åñ éŽïöæ ŽîïâIJëIJï âîåâñèëãŽêæâèâéâêðæ ⊛-æï éæéŽîå.Žáãæèæ ïŽêŽýŽãæŽ, îëé (R, ⊛, ⊕) îàëèöæ õëãâèæ a, b ∈ R-åãæï ïîñèáâIJŽ0 ⊛ a = a ⊛ 0 = 0,a ∗ (−b) = (−a) ⊛ b = −(a ⊛ b),(−a) ∗ (−b) = a ∗ b.−a Žîæï a-ï öâIJîñêâIJñèæ ⊕-æï éæéŽîå, a ⊕ (−b) øŽæûâîâIJŽ, îëàëîùa − b.

175é Ž à Ž è æ å æ 3.1. (Z n , ∗, +) Žîæï çëéñðŽùæñîæ âîåâñèëãŽêæ îàëèæn ∈ N-åãæï.äëàŽáŽá, (Z n , ∗, +) ïæïðâéŽöæ õëãâèåãæï Žî Žîæï öâïŽúèâIJâèæ \àŽõëòŽ"(öâIJîñêâIJñèæï ŽîïâIJëIJŽ ∗-æï éæéŽîå). âï Žîæï àŽêïýãŽãâIJŽ ãâèïŽ áŽ çëéñðŽùæñî,âîåâñèëãŽê îàëèï öëîæï. éŽàŽèæåŽá, àŽêãæýæèëå (Z 6 , ∗, +).Z 6 -öæ ŽîïâIJëIJï âèâéâêðâIJæ, îëéèâIJæù àŽêïýãŽãáâIJŽ êñèæïŽàŽê, éŽàîŽé êŽéîŽãèöæàãŽúèâãï êñèï; éŽàŽèæåŽá, (2; 3), (3; 4), (3; 2), (4; 3).(R, ⊛, ⊕)-öæ ŽîŽêñèëãŽê a Ꭰb âèâéâêðâIJï âûëáâIJŽ êñèæï àŽéõëòâIJæ,åñ ab = 0. åñ R Žî Žîæï çëéñðŽùæñîæ, éŽöæê a-ï âûëáâIJŽ éŽîùýâêŽ êñèæïàŽéõëòæ, b-ï çæ { éŽîþãâêŽ. Žáãæèæ ïŽøãâêâIJâèæŽ, îëé Z p Žî öâæùŽãï êñèæïàŽéõëòâIJï, åñ p éŽîðæãæ îæùýãæŽ.åñ (G, ⊛) þàñòöæ a ⊛ b = a ∗ c, éŽöæê b = c. éŽàîŽé îàëèæï öâéåýãâãŽöæâï õëãâèåãæï ŽîŽŽ ïŽéŽîåèæŽêæ.å â ë î â é Ž . äâéëå Žôêæöêñèæ ìæîëIJŽ K îàëèöæ ïŽéŽîåèæŽêæŽéŽöæê Ꭰéýëèëá éŽöæê, åñ îàëèæ Žî öâæùŽãï êñèæï àŽéõëò âèâéâêðâIJï.á Ž é ð ç æ ù â IJ Ž . ï Ž ç é Ž î æ ï ë IJ Ž. ãåóãŽå, R-ï ŽîŽ Žóãï êñèæïàŽéõëòâIJæ, åñ x ⊛ y = x ∗ z Ꭰx ≠ 0, éŽöæêx ⊛ y − x ⊛ z = x ⊛ y − x ⊛ y = 0,x ⊛ y − x ⊛ z = x ⊛ (y ⊕ (−z)) = x ∗ (y − z) = 0.éŽàîŽé, îŽáàŽê îàëèæ Žî öâæùŽãï êñèæï àŽéõëòâIJï Ꭰx ≠ 0, Žéæðëé y−z =0 =⇒ y = z. î.á.à.Ž ñ ù æ è â IJ è ë IJ Ž . ãåóãŽå, a⊛b = a⊛c ðëèëIJæáŽê àŽéëéáæêŽîâëIJï,îëé b = c. ãåóãŽå, x ⊛ y = 0, éŽöæê x ⊛ y = x ⊛ 0 Ꭰåñ x ≠ 0, éŽöæê y = 0.ìæîæóæå, åñ y = 0, éŽöæêx ⊛ y = 0 = 0 ⊛ y =⇒ x = 0,â. æ. åñ x ∗ y = 0, àŽéëéáæêŽîâëIJï, îëé Žê x = 0 Žê y = 0. î.á.à.à Ž ê é Ž î ð â IJ Ž . çëéñðŽùæñî, âîåâñèëãŽê, ñêñèàéõëòë îàëèïâûëáâIJŽ éåâèëIJæï Žîâ. â. æ. D ïæéîŽãèâ éŽïäâ àŽêïŽäôãîñèæ ëîæ IJæêŽîñèæëìâîŽùææå, îëéâèöæù ïîñèáâIJŽ ŽóïæëéâIJæ:1) ⊕ çëéñðŽùæñîæŽ;2) ⊕ ŽïëùæŽùæñîæŽ;3) ⊛ ŽïëùæŽùæñîæŽ;4) ⊛ çëéñðŽùæñîæŽ;5) âîåâñèëãŽêæï ŽîïâIJëIJŽ ⊕-æï éæéŽîå (0);6) öâIJîñêâIJñèæ âèâéâêðæï ŽîïâIJëIJŽ ⊕-æï éæéŽîå (−x).7) âîåâñèëãŽêæï ŽîïâIJëIJŽ ⊛-æï éæéŽîå (1).

1768) áæïðîæIJñùæñèëIJæï ŽóïæëéŽx ⊛ (y ⊕ z) = (x ⊛ y) ⊕ (x ⊛ z) ∀ x, y, z ∈ D.9) åñ x ≠ 0 Ꭰx ⊛ y = x ⊛ z, éŽöæê y = z. õëãâèæ ïŽïîñèæ éåâèëIJæïŽîâ ãâèæŽ, éŽàîŽé ŽîïâIJëIJï ñïŽïîñèë éåâèëIJæï ŽîââIJæ, îëéèâIJæùŽî ŽîæŽê ãâèâIJæ.3.2. ãâèâIJæ.à Ž ê é Ž î ð â IJ Ž . F ïæéîŽãèâï éŽïäâ àŽêïŽäôãîñèæ ëîæ IJæêŽîñèæŽèàâIJîñèæ ëìâîŽùææå, (F, ⊛, ⊕) ãâèæ âûëáâIJŽ, åñ ïîñèáâIJŽ öâéáâàæŽóïæëéâIJæ:1) ⊕ çëéñðŽùæñîæŽ;2) ⊕ ŽïëùæŽùæñîæŽ;3) ⊕-æï éæéŽîå âîåâñèëãŽêæ âèâéâêðæï ŽîïâIJëIJŽ (0);4) ⊕-æï éæéŽîå öâIJîñêâIJñè âèâéâêðæï ŽîïâIJëIJŽ (−x);5) ⊛ çëéñðŽùæñîæŽ;6) ⊛ ŽïëùæŽùæñîæŽ;7) ⊛-æï éæéŽîå âîåâñèëãŽêæï ŽîïâIJëIJŽ (1);8) õëãâèæ x ∈ F \ {0}-åãæï ŽîïâIJëIJï y ∈ F âèâéâêðæ æïâåæ, îëéx ⊛ y = 1 (öâIJîñêâIJñèæ x −1 );9) áæïðîæIJñùæñèëIJæï ŽóïæëéŽx ⊛ (y ⊕ z) = (x ⊛ y) ⊕ (x ⊛ z) ∀ x, y, z ∈ F.é Ž à Ž è æ å æ 3.2. (R; ∗, +) Žîæï ãâèæ, (R, +) Ꭰ(R \ {0}, ∗)çëéñðŽùæñîæ þàñòâIJæŽ. (N, ∗, +) Žî Žîæï ãâèæ, (B(A), ∩, ∪) Žî Žîæï ãâèæ,îŽáàŽê Žî Žóãï öâIJîñêâIJñèæ âèâéâêðæ.ûæêŽ àŽêéŽîðâIJâIJöæ àŽéëãæõâêâå ⊛ Ꭰ⊕ ïæéIJëèëâIJæ ëìâîŽùæâIJæï ŽôïŽêæöêŽãŽá,îŽáàŽê âï ëìâîŽùæâIJæöâæúèâIJŽ àŽêïýãŽãâIJñèæ æõëï øãâñèâIJîæãæ \·",\+" ëìâîŽùæâIJæïàŽê.û æ ê Ž á Ž á â IJ Ž . ãâèæï âîåâñèëãŽêæ âèâéâêðæ âîåùãèŽáæŽêæŽ.á Ž é ð ç æ ù â IJ Ž . ãåóãŽå, x ∗ e = x Ꭰx ∗ e ′ = x ∀ x ∈ F , éŽöæêe = e ∗ e ′ = e ′ ∗ e = e ′ =⇒ e = e ′ = 1.ŽêŽèëàæñîŽá éðçæùáâIJŽ öâçîâIJæï ëìâîŽùææï éæéŽîå.û æ ê Ž á Ž á â IJ Ž . öâIJîñêâIJñèæ âèâéâêðâIJæ ãâèöæ âîåŽáâîåæŽ.á Ž é ð ç æ ù â IJ Ž . ãåóãŽå, x ∈ F \ {0} ᎠáŽãñöãŽå, îëé ŽîïâIJëIJïëîæ x Ꭰy æïâåæ, îëé x ∗ y = 1, x ∗ z = 1. çëéñðŽùæñîëIJæáŽê

177àŽéëéáæêŽîâëIJï, îëéy ∗ x = 1 = z ∗ x =⇒=⇒ y = y ∗ 1 = y ∗ (x + z) = (y ∗ x) ∗ z = 1 ∗ z = z =⇒ y = z = x −1 .öâçîâIJæï ëìâîŽùææï éæéŽîå éðçæùáâIJŽ ŽêŽèëàæñîŽá.å â ë î â é Ž . (F, ∗, +)-öæ ïîñèáâIJŽ öâéáâàæ ŽóïæëéâIJæ:1) a ∗ 0 = 0;2) (−a) = a ∗ (−1), −a(+b) = (−a) + (−b);3) −(−a) = a, (−1) ∗ (−1) = 1;4) åñ a ≠ 0, éŽöæê a −1 ) −1 = a;5) a ∗ b = 0 =⇒ a = 0 Žê b = 0;6) (−a) ∗ (−b) = a ∗ b.á Ž é ð ç æ ù â IJ Ž .1) a ∗ 1 = 1 Ꭰa + 0 = a =⇒a + (a ∗ 0) = (a ∗ 1) + (a ∗ 0) = a ∗ (1 + 0) = a + 1 = a.â. æ. Žîæï öâçîâIJæï éæéŽîå âîåâñèëãŽêæ, =⇒ a ∗ 0 = 0.2) a + (a ∗ (−1)) = (a ∗ 1) + (a ∗ (−1)) = a ∗ (1 + (−1)) = a ∗ 0 = 0 =⇒−a = a ∗ (−1). Žéæï àŽéëõâêâIJæå−(a + b) = (a + b) ∗ (−1) = a ∗ (−1) + b ∗ (−1) = −a + (−6).3) (−a)+a = 0 Ꭰ(−a)+(−(−a)) = 0. îŽáàŽê öâIJîñêâIJñèæ âèâéâêðâIJæâîåŽáâîåæŽ, Žéæðëé a = −(−a) =⇒ 1 = −(−1). ãåóãŽå, x = −1, éŽöæê1 = −(x) = x ∗ (−1) = (−1) ∗ (−1).4) öâãêæöêëå, îëé a −1 ≠ 0. îŽáàŽê ûæꎎôéáâà öâéåýãâãŽöæ éæãæôâIJå:1 = a ∗ a −1 = a ∗ 0 = 0.îŽù âûæꎎôéáâàâIJŽ ãâèæï ŽóïæëéâIJï. áŽéðçæùâIJŽ ŽêŽèëàæñîæŽ 3)-æï áŽéðçæùâIJæï.5) ãåóãŽå, a ≠ 0, éŽöæê ŽîïâIJëIJï a −1 Ꭰb = 1 ∗ b = (a −1 ∗ a) ∗ b =a −1 ∗ (a ∗ b) = a −1 ∗ 0 = 0.6) 2)-áŽê àŽéëéáæêŽîâëIJï, îëé (−b) = b ∗ (−1) Ꭰ(−b) = b ∗ (−1).ŽóâáŽê àŽéëéáæêŽîâëIJï, îëé(−a)∗(−b) = (a∗(−1))∗(b∗(−1)) = a∗((−1)∗(−1))∗b = a∗(1)∗b = a∗b.øŽêŽûâîæï ïæéŽîðæãæïåãæï éæôâIJñèæ öâåŽêýéâIJŽ, îëé ãâèöæ âèâéâêðâ-IJæïåãæï, åñ Žî ûâîæŽ òîøýæèâIJæ, õëãâèåãæï ìæîãâèæ ïîñèáâIJŽ àŽéîŽãèâ-IJŽ, öâéáâà éæéŽðâIJŽ, éŽàŽèæåŽá,a + b ∗ c ≡ a + (∗b ∗ c).

178ãâèæï ŽéëýïêŽáëIJæáŽê àŽéëéáæêŽîâëIJï éŽïöæ ûîòæãæ àŽêðëèâIJæï ŽéëýïêŽáëIJŽ.âï úŽèæŽê éêæöãêâèëãŽêæŽ áŽ, òŽóðæñîŽá, ãâèâIJæïåãæï úæîæåŽáæåãæïâIJŽŽ.ãâèöæ ûîòæãæ àŽêðëèâIJŽ Žîæï a∗x+b = 0 ïŽýæï àŽéëïŽýñèâIJŽ, 0, a, b ∈F .å â ë î â é Ž . åñ a ≠ 0, éŽöæê a ∗ x + b = 0 àŽêðëèâIJŽï Žóãï âîåŽáâîåæŽéëêŽýïêæ F -öæ (Žêñ ŽîïâIJëIJï âîåŽáâîåæ ãâèæï âèâéâêðæ, îëéâèæùàŽêðëèâIJŽï Žóùâãï üâöéŽîæð ðëèëIJŽá, åñ x-ï öâãùãèæå Žé âèâéâêðæå).á Ž é ð ç æ ù â IJ Ž .a ∗ x + b = 0,a ∗ x + b + (−b) = 0 + (−b),a ∗ x + (b + (−b)) − b (ŽïëùæŽùæñîëIJŽ Ꭰ0-æï åãæïâIJŽ),a ∗ x + 0 = (−b) (éëìæîáŽìæîëIJæï åãæïâIJŽ),a ∗ x = (−b) (êñèæï åãæïâIJŽ),a −1 ∗ (a ∗ x) = a −1 ∗ (−b) (a ≠ 0),(a −1 ∗ a) + x = a −1 ∗ (−b) (ŽïëùæŽùæñîëIJŽ),1 ∗ x = a −1 ∗ (−b) (öâIJîñêâIJæåæ),x = a −1 ∗ (−b) (âîåâñèëãŽêæï åãæïâIJŽ).îŽáàŽê âîåâñèëãŽêæ ᎠéëìæîŽáŽìæîâ âèâéâêðâIJæ âîåŽáâîåæŽ ãâèöæ,Žéæðëé −b Ꭰa −1 âîåŽáâîåæ àäæå àŽêæïŽäôãîâIJŽ Žé àŽêðëèâIJæáŽê áŽ,öâïŽIJŽéæïŽá, öâáâàæù âîåŽáâîåæŽ.àŽêðëèâIJŽ a ∗ x ∗ x + b ∗ x + c = 0, a, b, c ∈ R, äëàŽáŽá ŽîŽŽ ŽéëýïêŽáæR-öæ, éŽåæ ŽéëýïêŽáëIJæïåãæï ïŽüæîëŽ R ãâèæï àŽòŽîåëâIJŽ çëéìèâóïñî îæùýãåŽãâèŽéáâ. çëéìèâóïñîçëâòæùæâêðâIJæŽêæ ìëèæêëéæŽèñîæ àŽêðëèâIJâIJæõëãâèåãæï ŽéëýïêŽáæŽ çëéìèâóïñî îæùýãåŽ ãâèöæ, ïý㎠ñòîë òŽîåë ãâèæŽî ŽîïâIJëIJï.3.3. ïŽïîñèæ ãâèâIJæ. ŽóŽéáâ êŽýïâêâIJæ õãâèŽ ãâèæ æõë ñïŽïîñèë.àŽêãæýæèëå ŽýŽèæ ãâèâIJæ, îëéâèæù öâæùŽãâê ïŽïîñè îŽëáâêëIJŽ âèâéâêðâ-IJï.ãåóãŽå, a ∈ F , éŽöæê a, a + a, a + a + a, . . . âèâéâêðâIJæ çãèŽã ãâèæïâèâéâêðâIJæŽ, a + a ≡ 2, a + a + a ≡ 3a, a + a + · · · + a ≡ na (n ŽîŽŽ} {{ }n-þâîŽñùæèâIJâèæ æõëï ãâèæï âèâéâêðæ), ŽêŽèëàæñîŽá, a∗a = a 2 , . . . ,}a · a{{· · · a}=n-þâîa n , a ≠ 0.à Ž ê é Ž î ð â IJ Ž . åñ ŽîïâIJëIJï æïâåæ ñéùæîâïæ êŽðñîŽèñîæ n, îëéna = 0, éŽöæê n-ï âûëáâIJŽ a âèâéâêðæï Žáæùæñîæ îæàæ, åñ ŽîïâIJëIJï æïâåæñéùæîâïæ îæùýãæ, îëé a m = 1, éŽöæê mï âûëáâIJŽ a-ï éñèðæìèæçŽùæñîæîæàæ.

179å â ë î â é Ž . ãâèæï ŽîŽêñèëãŽê âèâéâêðâIJï Žóãå âîåæ ᎠæàæãâŽáæùæñîæ îæàæ.á Ž é ð ç æ ù â IJ Ž . ãåóãŽå, a, b ∈ F \ {0} ᎠáŽãñöãŽå, îëé a áŽb-ï Žáæùæñîæ îæàâIJæŽ, öâïŽIJŽéæïŽá, n Ꭰm, éŽöæênb = n(a ∗ a −1 ) ∗ b = na ∗ (a −1 ∗ b) = 0 ∗ a −1 ∗ b = 0 =⇒ m ≤ n,ŽêŽèëàæñîŽá,ma = m(b ∗ b −1 ) ∗ a = (mb) ∗ (b −1 ∗ a) = 0 ∗ b −1 ∗ a = 0 =⇒=⇒ n ≤ m =⇒ m = n.à Ž ê é Ž î ð â IJ Ž . åñ ãâèæï õãâèŽ ŽîŽêñèëãŽêæ âèâéâêðæï îæàæŽn, ŽéIJëIJâê, îëé n Žîæï F ãâèæï éŽýŽïæŽåâIJâèæ, åñ Žïâåæ n Žî ŽîïâIJëIJï,ŽéIJëIJâê, îëé ãâèæ Žîæï êñèéŽýŽïæŽåâIJèæŽêæ.åñ |F | = m ∈ N, àŽéëéáæêŽîâëIJï, îëé F -ï Žóãï m âèâéâêðæ, Žêñ FïŽïîñèæŽ. åñ F êñèéŽýŽïæŽåâIJèæŽêæŽ, éŽöæê æï ŽñùæèâIJèŽá ñïŽïîñèëŽ.å â ë î â é Ž . õëãâèæ ïŽïîñèæ ãâèæï éŽýŽïæŽåâIJèæ éŽîðæãæ îæùýãæŽ.á Ž é ð ç æ ù â IJ Ž . áŽãñöãŽå, îëé ïŽïîñèæ F ãâèï Žóãï éŽýŽïæŽåâIJâèæn Ꭰn = p ∗ q, ïŽáŽù p, q < n, p, q ∈ N. ãåóãŽå, a ∈ F \ {0}. éŽöæê0 = na = (p ∗ q)a = p(qa), qa ∈ F . ŽóâáŽê àŽéëéáæêŽîâëIJï, îëé åñ qa = 0öâïîñèáâIJŽ n ≤ q, îŽáàŽê n Žîæï éŽýŽïæŽåâIJâèæ, ûæꎎôéáâà öâéåýãâãŽöæqa ∈ F \ {0}. ëîæãâ öâéåýãâãŽöæ éæãæôâå ûæꎎôéáâàëIJŽ. àŽéëéáæêŽîâëIJï,îëé n éŽîðæãæŽ.å â ë î â é Ž . ïŽïîñè ãâèï Žóãï éŽýŽïæŽåâIJâèæ p Ꭰ|F | = p n îŽæéân ∈ N-åãæï.á Ž é ð ç æ ù â IJ Ž . øãâê ñçãâ ãæùæå, îëé åñ p Žîæï F -æï éŽýŽïæŽåâIJâèæ,éŽöæê p éŽîðæãæŽ. ãåóãŽå, |F | = q, åñ p = q åâëîâéŽ ùýŽáæŽ, îŽáàŽê n = 1.ãåóãŽå, p ≠ q Ꭰa 1 ∈ F \ {0}. àŽêãæýæèëåF 1 = { y : y = na 1 , n ∈ N, 1 ≤ n ≤ p } , |F 1 | = p,Žãæôëå ŽýèŽ a 2 ∈ F \ F 1 ᎠãåóãŽå,F 2 = { y : y = na 1 + ma 2 : m, n ∈ N, 1 ≤ n ≤ p, 1 ≤ m ≤ p } .åñ F 2 = F ìîëùâïï ãŽéåŽãîâIJå. ûæꎎôéáâà öâéåýãâãŽöæ ãæýæèŽãå a 3 ∈F \ F 2 ᎠŽ. ö. IJëèëï îŽáàŽê F ïŽïîñèæŽ, ìîëùâïæ öâûõáâIJŽ ᎠéæãæôâIJåF 1 , F 2 , . . . , F n ïæéîŽãèââIJï îŽæéâ n ∈ N-åãæï.F -æï õëãâèæ âèâéâêðæ f âîåŽáâîåæ àäæå øŽæûâîâIJŽ f = m 1 a 1 +m 2 a 2 +· · ·+m n a n , ïŽáŽù 1 ≤ m i ≤ p õãâèŽ i = 1, . . . , n-åãæï. éŽöŽïŽáŽéâ, ŽîïâIJëIJïp n Žïâåæ ïŽýæï àŽéëïŽýñèâIJŽ, â. æ. |F | = p n . î.á.à.éŽöŽïŽáŽéâ, êâIJæïéæâî ïŽïîñè ãâèï Žóãï p n âèâéâêðæ îŽæéâ p, n ∈ N-åãæï (p éŽîæðæãæŽ). çâîúëá, õëãâèæ Žïâåæ p Ꭰn-åãæï ŽîïâIJëIJï p n îæàæïïŽïîñèæ ãâèæ, éŽàîŽé Žéæï áŽéðçæùâIJŽ Žîùåñ æïâ éŽîðæãæŽ.

180éŽàŽèæåæï ïŽýæå àŽêãæýæèëå (Z 3 , ∗, +), ïŽáŽù ∗, + êŽøãâêâIJæŽ 5.1 ùýîæèäâ.Žáãæèæ ïŽøãâêâIJâèæŽ, îëé ïîñèáâIJŽ 1){8) ŽóïæëéâIJæ.∗ 0 1 20 0 0 01 0 1 22 0 2 1ù ý î æ è æ 3.1.æàæãâ ùýîæèâIJæ àŽêãæýæèëå Z 4 -öæ.+ 0 1 2 30 0 1 2 31 1 2 3 02 2 3 0 13 3 0 1 2ù ý î æ è æ 3.2.ùýŽáæŽ, Z 4 -öæ 2-æŽêï Žî Žóãï öâIJîñêâIJñèæ, Žéæðëé Z 4 Žî Žîæï ãâèæ.ãâèæ 4 = 2 2 ŽîïâIJëIJï. éŽàîŽé Žî âéåýãâ㎠Z 4 -ï. éëùâéñèæ îæàæï ïŽïîñèæãâèâIJæ æàâIJŽ ïŽçéŽëá îåñèŽá { éîŽãŽèûâãîåŽ åâëîææï àŽéëõâêâIJæå. æïæêæéêæöãêâèëãŽê îëèï åŽéŽöëIJâê çëáæîâIJæï åâëîæŽöæ.3.4. áŽèŽàâIJñèæ ãâèâIJæ. øãâê ñçãâ ãêŽýâå, îëé R ïæéîŽãèâ øãâñèâ-IJîæãæ · Ꭰ+ ëìâîŽùæâIJæï éæéŽîå óéêæï ãâèï. åãæåëê ãâèæï ïðîñóðñî æûãâãï âèâéâêðåŽ îŽæéâ åãŽèïŽäîæïæå áŽèŽàâIJŽï, åñêáŽù R-æï IJñêâ-IJîæã áŽèŽàâIJŽï. õãâèŽ ãâèæ Žî Žîæï áŽèŽàâIJñèæ, Žéæðëé øãâê ñêᎠöâãŽéëûéëåáŽéŽðâIJæå îŽ ìæîëIJâIJæï öâïîñèâIJŽŽ ïŽüæîë, îëé ãæïŽñIJîëå âèâéâêðåŽáŽèŽàâIJŽäâ. áŽèŽàâIJŽ öâïŽúèâIJâèæŽ ïýãŽáŽïýãŽàãŽîŽá. áŽãæûõëåáŽáâIJæåëIJæï åãæïâIJæå. æï ñöñŽèëá éæàãæõãŽêï ãâèæï âèâéâêðâIJæï áŽèŽàâ-IJŽéáâ, öâéáâà çæ { ïæàîúæï ùêâIJŽéáâ.à Ž ê é Ž î ð â IJ Ž . ŽéIJëIJâê, îëé F ãâèæ áŽèŽàâIJñèæŽ, åñ æï öâæùŽãïŽîŽùŽîæâè P óãâïæéîŽãèâï, îëéâèæù øŽçâðæèæ · Ꭰ+ ëìâîŽùæâIJæï éæéŽîåᎠõëãâèæ x ∈ F -åãæï ïîñèáâIJŽ äñïðŽá âîåæ öâéáâàæ åŽêŽòŽîáëIJâIJæáŽêx ∈ P \ {0}, x = 0, −x ∈ P \ {0}.P Žîæï ïæéîŽãèâ f-æï õãâèŽ áŽáâIJæåæ âèâéâêðâIJæïŽ (0 öâæúèâIJŽ öâáæëáâïP -öæ, öâæúèâIJŽ ŽîŽ; àŽîçãâñèëIJæïŽåãæï öâãåŽêýéáâå, îëé öâáæï). åñx ∈ P , éŽöæê ãŽéIJëIJå, îëé x áŽáâIJæåæŽ áŽ ãûâîå: x ≥ 0 åñ −x ∈ P \ {0},ãŽéIJëIJå, îëé ñŽîõëòæåæŽ áŽ ãûâîå x < 0, ŽêŽèëàæñîŽá, åñ x Ꭰy F -æï âèâéâêðâIJæŽ, ãŽéIJëIJå, îëé x ≤ y (êŽçèâIJæŽ Žê ðëèæŽ) éŽöæê ᎠéýëèëáéŽöæê, îëùŽ y−x ∈ P Ꭰx < y éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ y−x ∈ P \{0}.ïæéIJëèëâIJæ

181å â ë î â é Ž . ãåóãŽå, F áŽèŽàâIJñèæ ãâèæŽ áŽ a, b, c, d ∈ F , éŽöæê1) åñ a ≤ b Ꭰb ≤ c, éŽöæê a ≤ c;2) a ≤ a;3) åñ a ≤ b Ꭰb ≤ a, éŽöæê a = b;4) åñ a ≠ 0, éŽöæê a 2 > 0;5) 1 > 0;6) åñ a ≤ b, éŽöæê a + c ≤ b + c;7) åñ a ≤ b Ꭰc ≤ d, éŽöæê a + c ≤ b + d;8) åñ a ≤ b, 0 < c Ꭰd < 0, éŽöæê a ∗ c ′ leqb ∗ c, b ∗ d ≤ a ∗ d;9) åñ 0 < a, éŽöæê 0 < a −1 Ꭰåñ b < 0, éŽöæê b −1 < 0.á Ž é ð ç æ ù â IJ Ž . Ž) a ≤ b Ꭰb ≤ c =⇒ b − a ∈ P Ꭰc − b ∈ P ,éŽöæê c − a = c − b + b − a ∈ P . îŽáàŽê P øŽçâðæèæŽ + ëìâîŽùææï éæéŽîå,â. æ. a ≤ c.2) ùýŽáæŽ, a − a = 0 ∈ P .3) a ≤ b Ꭰb ≤ a, åñ b − a = x, éŽöæê x ∈ P Ꭰ−x ∈ P , îŽùâûæꎎôéáâàâIJŽ P -ï àŽêéŽîðâIJŽï, â. æ. x = 0 Ꭰa = b.4) åñ a ≠ 0 ᎠŽê a ∈ P Žê −a ∈ P ; P øŽçâðæèæŽ, éŽöæê àŽéëéáæêŽîâëIJï,îëé a 2 ∈ P , (−a) · (−a) = a 2 . ŽóâáŽê, åñ −a ∈ P , éŽöæê a 2 ∈ P .5) 1 2 = 1 =⇒ (4)-æï åŽêŽýéŽá 1 > 0.6) ñöñŽèëá àŽéëéáæêŽîâëIJï b − a = (b + c) − (a + c) åŽêŽòŽîáëIJæáŽê.7) áŽïŽéðçæùâIJâèï éæãæôâIJå (b−a)+(d−c) = (b+d)−(a+c) åŽêŽòŽîáëIJæáŽêᎠæéæï àŽåãŽèæïûæêâIJæå, îëé P øŽçâðæèæŽ.8) éðçæùáâIJŽ ŽêŽèëàæñîŽá öâéáâàæ åŽêŽòŽîáëIJæï àŽåãŽèæïûæêâIJæå(b − a) ∗ c = b ∗ c − a ∗ c Ꭰ(b − a) ∗ (−d) = a ∗ d − b ∗ d.9) 0 < a =⇒ a ∈ P \ {0}, åñ a −1 = 0, éŽöæê 1 = a ∗ a −1 = a ∗ 0 = 0Ꭰåñ a −1 ∉ P , (8)-æï àŽéë 1 = a −1 ∗ a < 0 ∗ a = 0. ëîæãâ öâéåýãâãŽöæéæãæôâå ûæꎎôéáâàëIJŽ, éŽöŽïŽáŽéâ, 0 < a −1 . áŽêŽîøâêæ êŽûæèæù éðçæùáâIJŽŽêŽèëàæñîŽá.éæãæôâå éïàŽãïæ åŽêŽòŽîáëIJâIJæ ïæáæáæï ùêâIJæïåãæï áŽèŽàâIJñè ãâèâ-IJöæ.à Ž ê é Ž î ð â IJ Ž . åñ F Žîæï áŽèŽàâIJñèæ ãâèæ, éŽöæê{x åñ x ≥ 0X −→−x åñ x < 0òñêóùæŽï ãñûëáëå ŽIJïëèñðñîæ éêæöãêâèëIJŽ (ïæáæáâ, ïæàîúâ Žê éëáñèæ).âï òñêóùæŽ ŽôæêæöêâIJŽ |x| ïæéIJëèëåæ ᎠæçæåýâIJŽ, îëàëîù \x-æï éëáñèæ".øŽéëãŽõŽèæIJëå åâëîâéâIJæ áŽéðçæùâIJæï àŽîâöâ (ᎎéðçæùâå åŽãŽá ïŽãŽîþæöëïïŽýæå!).

182å â ë î â é Ž . åñ F áŽèŽàâIJñèæ ãâèæŽ áŽ a, b ∈ F , éŽöæê1) |a| = 0 éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ a = 0;2) | − a| = |a|;3) |a ∗ b| = |a| ∗ |b|;4) åñ 0 ≤ b, éŽöæê |a| ≤ b éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ −b ≤ a ≤ b;5) −|a| ≤ a ≤ |a|;6) ||a| − |b|| ≤ |a ± b| ≤ |a| + |b| ïŽéçñåýâáæï ñðëèëIJŽ.ï Ž ã Ž î þ æ ö ë 3.1.1. ᎎéðçæùâå, îëé (R, ∗, +)-öæ ïîñèáâIJŽŽ) 0 ∗ a = a ∗ 0 = 0;IJ) a ∗ (−b) = (−a) ∗ b = −(a ∗ b);à) (−a) ∗ (−b) = a ∗ b.2. ãŽøãâêëå, îëé åñ (R, ∗, +) îæàâIJöæ õëãâèæ a ∈ R-åãæï ïîñèáâIJŽa ∗ a = 0, éŽöæê R çëéñðŽùæñîæŽ.3. Žøãâêâå, îëé Z n -öæ êñèæï àŽéõëòâIJæ éýëèëá æï âèâéâêðâIJæŽ, îëéâèåŽùŽóãå ïŽâîåë ŽîŽðîæãæŽèñîæ àŽéõëòæ n-åŽê. éŽöŽïŽáŽéâ, Z p , ïŽáŽùp éŽîðæãæŽ Žîæï ãâèæ.4. Žøãâêâå, îëé õëãâèæ ïŽïîñèæ éåâèëIJæï Žîâ Žîæï ãâèæ.5. Žøãâêâå, îëé (Z, ∗, +) Žîæï éåâèëIJæï Žîâ, éŽàîŽé Žî Žîæï ãâèæ.6. ãåóãŽå, p < q, ýëèë (Z p , ∗ p , + p ) Ꭰ(Z q , ∗ q , + q ) øãâñèâIJîæãæ êŽöååŽçèŽïâIJæŽ éëáñèæå p Ꭰq. ᎎéðçæùâå, îëé Z p ⊂ Z q , éŽàîŽé (Z p , ∗ p , + p )Žî Žîæï óãâîàëèæ (Z q , ∗ q , + q ) îàëèöæ.Žøãâêâå, îëé ∗q Ꭰ+q Žî Žîæï Z p -öæ.7. ᎎéðçæùâå, îëé (Z 6 , ∗, +) Žî Žîæï ãâèæ.8. Žøãâêâå, îëé ïŽïîñè ãâèï Žóãï ŽîŽêñèëãŽêæ éŽýŽïæŽåâIJâèæ ᎠîëéêñèéŽýŽïæŽåâIJèæŽêæ ãâèæ ñïŽïîñèëŽ.9. ãåóãŽå, a 1 , a 2 , . . . , z n 3 Žîæï àŽêéŽîðâIJñèæ 3.2-æï ñçŽêŽïçêâè åâëîâéŽöæ.Žøãâêâå, îëé õëãâèæ m 1 a 1 + m 2 a 2 + · · · + m n a n àŽéëïŽýñèâIJŽ àŽêïŽäôãîŽãïãâèæï âîåŽáâîå âèâéâêðï.10. (F, ∗, +) ãâèöæ Žéëýïâêæå àŽêðëèâIJŽa + d ∗ y = c,x ∗ d + y = b.

∗ a b c da a a a ab a b c dc a c d bd a d b c+ a b c da a b c db b a d vc c d a bd d a b a11. ᎎéðçæùâå, îëé åñ a ∗ b > 0, a, b ∈ F , éŽöæê Žê a, b > 0 Žê a, b < 0.12. Žøãâêâå, îëé áŽèŽàâIJñè ãâèöæ a 2 + b 2 = 0 éŽöæê Ꭰéýëèëá éŽöæê,îëùŽ a = b = 0.13. ãåóãŽå, F áŽèŽàâIJñèæŽ áŽ a, b ∈ F , 0 ≤ a ≤ b. Žøãâêâå, îëéa 2 ≤ b 2 .14. ᎎéðçæùâå, îëé õëãâèæ ãâèæ Žîæï éåâèëIJæï Žîâ.15. áŽèŽàâIJñè ãâèöæ −|a| ≤ a ≤ |a| Ꭰ−|a| ≤ −a ≤ |a| ñðëèëIJâIJæïàŽåãŽèæïûæêâIJæå ᎎéðçæùâå, îëéŽ) |a ± b| ≤ |a| + |b|;IJ) ||a| − |b|| ≤ |a ± b|.183

ûîòæãæ ŽèàâIJ 1. ûîòæãæ ŽèàâIJîŽñéâðâï ïŽýâèéúôãŽêâèëöæ ãâóðëîâIJï Žôûâîâê îëàëîù ëIJæâóðâIJï, îëéâèåŽùŽóãå \ïæàîúâ" Ꭰ\éæéŽîåñèâIJŽ". øãâê àŽêãéŽîðŽãå ãâóðëîâIJï ñòîëäëàŽáŽá, ïŽáŽù ïæàîúæïŽ áŽ éæéŽîåñèâIJæï ùêâIJŽ ŽîïâIJæåæŽ.1.1. ãâóðëîñèæ ïæãîùââIJæ Ꭰûîòæãæ àŽîáŽóéêâIJæ.à Ž ê é Ž î ð â IJ Ž . ãåóãŽå, F ãâèæŽ, V óãâïæéîŽãèâ \+" IJæêŽîñèæëìâîŽùææå. áŽãñöãŽå, îëé õëãâèæ a ∈ F Ꭰx ∈ V -åãæï àŽêïŽäôãîñèæŽax ∈ V âèâéâêðæ. åñ ïîñèáâIJŽ ŽóïæëéâIJæ:Ž) (V, +) çëéñðŽùæñîæ þàñòæŽ;IJ) ∀ x, y ∈ V Ꭰa, b ∈ F -åãæï(a + b)x = ax + bx,a(x + y) = ax + by,(a − b)x = a(bx),1 F x = x,ïŽáŽù 1 F Žîæï F -æï éñèðæìèæçŽùæñîæ âîåâñèæ,éŽöæê ŽéIJëIJâê, îëé V Žîæï ãâóðëîñèæ ïæãîùâ F -äâ. V -ï âèâéâêðâIJï ñûëáâIJâêãâóðëîâIJï, \+" ëìâîŽùæâIJï âûëáâIJŽ ãâóðëîâIJæï öâçîâIJŽ áŽ Λ : F ×V → V ŽïŽýãŽï öâéáâàæ ûâïæå: Λ(a, x) = ax âûëáâIJŽ ãâóðëîæï àŽéîŽãèâIJŽïçŽèŽîäâ.ãâóðëîñèæ ïæãîùâ F -äâ àŽêæéŽîðâIJŽ îëàëîù (V, +, Λ) ïŽéâñèæ, îëéâèæùŽçéŽõëòæèâIJï äâéëŽôêæöêñè ŽóïæëéâIJï. ãâóðëîñè ïæãîùæï êñèæ ŽôæêæöêâIJŽ0-æå, ŽóïæëéâIJæáŽê àŽéëéáæêŽîâëIJï, îëé 0 F x = 0 ∀ x ∈ V -åãæï áŽa0 = 0 ∀ a ∈ F -åãæï.öâéáâà éŽàŽèæåâIJöæ êŽøãâêâIJæ æóêâIJŽ, îëé æóêâIJŽ ïæéîŽãèâåŽ ïýãŽáŽïýãŽçèŽïâIJï Žóãå ãâóðëîñèæ ïæãîùæï ïðîñóðñîŽ.é Ž à Ž è æ å æ 1.1.1. F n (n ∈ N) ãâóðëîñèæ ïæãîù⎠F -äâ öâéáâàæ ëìâîŽùæâIJæå:(a 1 , a 2 , . . . , a n ) + (b 1 , b 2 , . . . , b n ) = (a 1 + b 1 , a 2 + b 2 , . . . , a n + b n ),a(a 1 , a 2 , . . . , a n ) = (aa 1 , aa 2 , . . . , aa n ).185

186F n -æï êñèæ Žîæï (0 F , . . . , 0 F ), a 1 , a 2 , . . . , a n âèâéâêðâIJï âûëáâIJŽ a ãâóðëîæïçëéìëêâêðâIJæ.2. ãåóãŽå, B õãâèŽ ŽïŽýãŽåŽ ïæéîŽãèâŽ, f : [a, b] → R. éŽöæê B Žîæïãâóðëîñèæ ïæãîùâ R-äâ öâéáâàæ ëìâîŽùæâIJæå:(f + g)(x) = f(x) + g(x), ∀ f, g ∈ B,(af)(x) = af(x), ∀ a ∈ R.3. ãåóãŽå, B, B ⋐ B Žîæï õãâèŽ ñûõãâð ŽïŽýãŽåŽ ïæéîŽãèâ B-áŽê.éŽöæê B Žîæï ãâóðëîñèæ ïæãîùâ B-äâ àŽêïŽäôãîñèæ ëìâîŽùæâIJæï éæéŽîå.U ïæéîŽãèâï U ⊆ V âûëáâIJŽ V -ï óãâïæãîùâ, åñ æï åãæåëê Žîæï ãâóðëîñèæ{ïæãîùâ V -äâ àŽêïŽäôãñèæ ëìâîŽùæâIJæï éæéŽîå.(a1 , . . . , a n−1 , 0 F ) : a i ∈ F } ïæéîŽãèâ Žîæï F n ãâóðëîñèæ ïæãîùæïóãâïæãîùâ. B Žîæï B-ï óãâïæãîùâ. åñ U ãâóðëîñè ïæãîù⎠V -öæ, éŽöæê0 ∈ U.R n ãâóðëîñèæ ïæãîùâ (1 ≤ n ≤ 3) ûŽîéëæóéêâIJŽ IJñêâIJîæãŽá. R n -æïëìâîŽùæâIJï Žóãå àŽîçãâñèæ àâëéâðîæñèæ æêðâîìîâðŽùæŽ. R 2 -åãæï êŽøãâêâ-IJæŽ 1.1 ïñîŽåäâ. åñ r = (x, y) ∈ R 2 , éŽöæê x Ꭰy çëéìëêâêðâIJæ àŽêæïŽäôãîâIJŽo ûâîðæèöæ ñîåæâîåàŽáŽéçãâåæ ëîæàæêŽèñîæ ûîòââIJæï àŽïûãîæã. çñåýâŽé ûîòââIJï öëîæï ŽæåãèâIJŽ åæï æïîæï ïŽûæꎎôéáâàë éæéŽîåñèâIJæåox ôâîúæáŽê. Žïâå ïæïðâéŽï âûëáâIJŽ éŽîåçñåýŽ çëëîáæêŽðåŽ ïæïðâéŽ R 2 -öæ. ãâóðëîâIJæï öâçîâIJŽ R 2 -öæ àâëéâðîæñèŽá öââïŽIJŽéâIJŽ ìŽîŽèâèëàîŽéæïûâïï, îëàëîù âï êŽøãâêâIJæŽ 1.1(à) êŽýŽääâ.R n ïæãîùæï àâëéâðîæŽ àŽêýæèñèæ æóêâIJŽ óãâãæå, ŽýèŽ çæ öâéëãæðŽêëåIJŽäæïæïŽ áŽ àŽêäëéæèâIJæï ùêâIJŽ.∑åñ V ãâóðëîñèæ ïæãîù⎠F -äâ ᎠS ⊆ V , éŽöæê n a i x i , a i ∈ F , x i ∈ Séëùâéñè þŽéï âûëáâIJŽ S-æï ãâóðëîñèæ ûîæòæãæ çëéIJæêŽùæŽ. ŽéIJëIJâê, îëé{x i : 1 ≤ i ≤ k} ãâóðëîñèæ ïŽïîñèæ ïæéîŽãèâ ûîòæãŽá áŽéëñçæáâIJâèæŽ,åñk∑a i x i = 0 =⇒ a 1 = a 2 = · · · = a k = 0.i=1ûæꎎôéáâà öâéåýãâãŽöæ ïæéîŽãèâï âûëáâIJŽ ûîòæãŽá áŽéëçæáâIJñèæ. S óãâïæéîŽãèâïS ≤ V âûëáâIJŽ V ïæãîùæï ûŽîéëéóéêâèæ. V -ï êâIJæïéæâîæ âèâéâêðæŽîæï S-æï âèâéâêðâIJæï ûîòæãæ çëéIJæêŽùæŽ. V ïæãîùæï áŽèŽàâIJñè ûîòæãŽááŽéëñçæáâIJâè ïæéîŽãèâï âûëáâIJŽ V -ï IJŽäæïæ.é Ž à Ž è æ å æ 1.2. R 3 -öæ (5; 5; √ 2) ãâóðëîæŽîæï (1; 1; 0) Ꭰ(0; 0; 3)ãâóðëîâIJæï ûîòæãæ çëéIJæêŽùæŽ, îŽáàŽê(5; 5; √ √22) = 5(1; 1; 0) + (0; 0; 3).3L = { (1; 1; 0), (0; 0; 3) } ïæïðâéŽ ûîòæãŽá áŽéëñçæáâIJâèæŽ R 3 -öæ. îŽáàŽêa(1; 1; 0) + b(0; 0; 3) = (a; a; 3b) = 0 éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ a = 0i=1

187êŽý. 1.1Ꭰb = 0. éŽàîŽé L Žî Žîæï IJŽäæïæ, îŽáàŽê æï àŽêïŽäôãîŽãï éýëèëá{(x; x; y) : x, y ∈ R} ⊂ R 3 óãâïæãîùâï. B = L ∪ {1; 0; 0} IJŽäæïæ Žîæï L-æïàŽòŽîåëâIJŽ R 3 -æï IJŽäæïŽéáâ. Žáãæèæ ïŽøãâêâIJâèæŽ, îëé ãóðëîñèæ ïæãîùæïõëãâèæ âèâéâêðæ ùŽèïŽýŽá ûŽîéëæáàæêâIJŽ B = {e 1 , e 2 , . . . , e n } IJŽäæïæïâèâéâêðâIJæå. îŽáàŽê åñx =n∑a i e i =i=1n∑b i e i =⇒ 0 = x − x =i=1n∑(b i − a i )e i =⇒i=1=⇒ b i − a + i = 0 =⇒ b i = a i , 1 ≤ i ≤ n.û æ ê Ž á Ž á â IJ Ž . ãåóãŽå, S = {x 1 , . . . , x m } Žîæï V -ï ûŽîéëéóéêâèæïæéîŽãèâ ᎠL = {y 1 , . . . , y l } Žîæï V -ï ûîòæãŽá áŽéëñçæáâIJâè âèâéâêðåŽïæéîŽãèâ. éŽöæê m ≥ l.

188á Ž é ð ç æ ù â IJ Ž . ãåóãŽå, m < l. îŽáàŽê S ûŽîéëóéêæï V -ï, Žéæðëé ŽîïâIJëIJï a 1 , a 2 , . . . , a m ∈ F æïâåæ, îëé y 1 = a 1 x 1 + a 2 x 2 +· · · + a m x m . ŽéŽïåŽê, y 1 ≠ 0. îŽáàŽê L ûîòæãŽá áŽéëñçæáâIJâèæ ïæéîŽãèâŽV -öæ (æý. ïŽãŽîþæöë 4.4). éŽöŽïŽáŽéâ, a 1 , a 2 , . . . , a m -áŽê îëéâèæôŽù ≠ 0.ãåóãŽå, a 1 ≠ 0, éŽöæê x 1 = a −11 y 1 − a −11 a 2x 2 − · · · − a −11 a mx m . â. æ. x 1Žîæï {y 1 , y 2 , . . . , x m } ûîòæãæ çëéIJæêŽùæŽ. îŽáàŽê S ûŽîéëóéêæï V -ï, Žéæðëé{y 1 , x 2 , . . . , x m }-æù ûŽîéëóéêæï V -ï, Žéæðëé {y 1 , x 2 , . . . , x m }-æù ûŽîéëóéêæïV -ï; ŽêŽèëàæñîŽá, éæãæôâIJå, îëé {y 1 , y 2 , x 3 , . . . , x m } ûŽîéëóéêæï V -ï. åñàŽãæéâëîâIJå Žé ìîëùâïï m-þâî, éæãæôâIJå, îëé {y 1 , y 2 , . . . , y m } ûŽîéëóéêæïV -ï, y m+1 = p 1 y 1 +p 2 y 2 +· · ·+p m y m , ïŽáŽù p 1 , . . . , p m ∈ F ᎠõãâèŽ âîåáîëñèŽáŽîŽŽ êñèæ, îŽáàŽê y m+1 ≠ 0 =⇒ y m+1 −p 1 y 1 −p 2 y 2 −· · ·−p m y m =0. âï çæ öâñúèâIJâèæŽ, îŽáàŽê {y 1 , y 2 , . . . , y l } ûîòæãŽá áŽéëñçæáâIJâèæŽ. ŽóâáŽêàŽéëéáæêŽîâëIJï, îëé m > l.û æ ê Ž á Ž á â IJ Ž . ãåóãŽå, B ᎠB ′ Žîæï V ãâóðëîñèæ ïæãîùæïIJŽäæïâIJæ. éŽöæê |B| = |B ′ |.á Ž é ð ç æ ù â IJ Ž . ãåóãŽå, B ={e 1 , e 2 , . . . , e n } ᎠB ′ ={e ′ 1, e ′ 2, . . . , e ′ n}.ûæêŽ ûæêŽáŽáâIJæï úŽèæå n ≥ m Ꭰm ≥ n =⇒ m = n. ãâóðëîñèæïæãîùæï IJŽäæïæï ïæéúèŽãîâï âûëáâIJŽ V ïæãîùæï àŽêäëéæèâIJŽ ᎠòŽôæêæöêâIJŽdim(V )-æå.û æ ê Ž á Ž á â IJ Ž . dim(F n ) = n.á Ž é ð ç æ ù â IJ Ž . àŽêãéŽîðëå B = {e 1 , e 2 , . . . , e n }, ïŽáŽùl i = (0, . . . , 0, 1}{{} F , 0, . . . , 0)i-ñî ŽáàæèŽïᎠãŽøãâêëå, îëé B Žîæï F n -æï IJŽäæïæ. ùýŽáæŽ,n∑(a 1 , a 2 , . . . , a n ) = a i e i .Žéæðëé B ûŽîéëóéêæï F n -ï áŽn∑b i e i = 0 =⇒ (b 1 , b 2 , . . . , b n ) = 0 =⇒ b 1 = b 2 = · · · = b n = 0 = 0 F .i=1â. æ. B Žîæï IJŽäæïæ Ꭰdim(V ) = |B| = n.ûæêŽ àŽêéŽîðâIJæáŽê àŽéëéáæêŽîâëIJï, îëé IJŽäæïæ õëãâèåãæï öâæùŽãïïŽïîñè îŽëáâêëIJŽ âèâéâêðâIJï. ŽéŽïåŽê, õãâèŽ ïæãîùâöæ öâñúèâIJâèæŽ IJŽäæïæàŽéëõëòŽ. éŽàŽèæåŽá, B ᎠB-öæ Žî Žîæï IJŽäæïæ.IJŽäæïæïŽ áŽ àŽêäëéæèâIJæï ùêâIJŽ öâæúèâIJŽ àŽêäëàŽááâï éåèæŽêŽá ãâóðëîñèïæãîùââIJäâ, éýëèëá Žïâåæ àŽêäëàŽáâIJŽ øãâê Žî àãüæîáâIJŽ. åñ Vïæãîùâï Žóãï IJŽäæïæ (äâéëå êŽýïâêâIJæ), éŽöæê ŽéëIJëIJâê, îëé ïæãîùâï ŽóãïïŽïîñèæ àŽêäëéæèâIJŽ ᎠéŽï ñûëáâIJâê ïŽïîñèàŽêäëéæèâIJæŽê ãâóðëîñè ïæãîùâï.àŽêãæýæèëå ŽýèŽ ãâóðëîñèæ ïæãîùæï ßëéëîòæäéâIJæ.i=1

189à Ž ê é Ž î ð â IJ Ž . ãåóãŽå, V 1 ᎠV 2 ãâóðëîñèæ ïæãîùââIJæŽ F -äâ.ŽéIJëIJâê, îëé T : V 1 → V 2 ŽïŽý㎠ûîòæãæŽ, åñT (x + y) = T x + T y,T (ax) = a(T x).åñ V 2 = V 1 , éŽöæê T -ï âûëáâIJŽ V 1 ïæãîùæï ûîòæãæ àŽîáŽóéêŽ.öâáàëéöæ øãâêåãæï ïŽæêðâîâïë æóêâIJŽ ïŽïîñèàŽêäëéæèâIJæŽêæ ãâóðëîñèæïæãîùâ R-äâ Ꭰéæïæ ûîòæãæ àŽîáŽóéêŽ. Žé åŽãæï áŽêŽîøâê êŽûæèöæ V -æåŽôêæöêñèæ æóêâIJŽ ãâóðëîñèæ ïæãîùâ, ýëèë End(V ) { V ïæãîùæï õãâèŽûîòæã àŽîáŽóéêŽåŽ ïæéîŽãèâ (âêáëéëîòæäéåŽ ïæéîŽãèâ).öâãêæöêëå, îëé äâéëåóéñèæ ûæêŽáŽáâIJâIJæï ñéâðâïëIJŽ öâïŽúèâIJâèæŽ ûŽîéëãŽáàæêëåñòîë äëàŽáæ ïŽýæå.àŽáŽãæáâå V -ï ŽèàâIJîæáŽê End(V )-æï ŽèàâIJîŽäâ ᎠãŽøãâêëå, îëéEnd(V ) øŽçâðæèæŽ öâçîâIJæï, àŽéîŽãèâIJæïŽ áŽ ïçŽèŽîäâ àŽéîŽãèâIJæï ëìâîŽùæâIJæïéæéŽîå. åŽãáŽìæîãâèŽá öâãêæöêëå, îëé I V æàæãñîæ ŽïŽý㎠ᎠO VêñèëãŽêæ ŽïŽý㎠ûîòæãæŽ V -äâ, îŽáàŽê, àŽêéŽîðâIJæï åŽêŽýéŽá,I V x = x∀ ∈ V, O V x = 0 ∀ x ∈ V.éŽöŽïŽáŽéâ, ∀ x, y ∈ V Ꭰλ ∈ R àãŽóãïI V (x + y) = x + y = I V x + I V y,I V (λx) = λx = λ(I V x),O V (x + y) = 0 = 0 + 0 = O V x + O V y,O V (λx) = 0 = λ0 = λO V x.åñ S, T ∈ End(V ), éŽöæê S + T ᎠS ◦ T àŽêæéŽîðâIJŽ òëîéñèâIJæå:(S + T )x = Sx + T x∀ x ∈ V,(S · T )x = S(T x)∀ x ∈ V.ŽôïŽêæöêŽãæŽ End(V )-æï öâéáâàæ åãæïâIJâIJæ.û æ ê Ž á Ž á â IJ Ž . (End(V ), ◦, +) Žîæï âîåâñèëãŽêæ îàëèæ.á Ž é ð ç æ ù â IJ Ž . éæãñåæåëå áŽéðçæùâIJæï úæîæåŽáæ âðŽìâIJæ. ñêáŽãŽøãâêëå, îëé(I) S, T ∈ End(V ) =⇒ S + T ∈ End(V ) ᎠS ◦ T ∈ End(V );(II) (End(V ), +) çëéñðŽùæñîæ þàñòæŽ.åñ S, T, Y ∈ End(V ), éŽöæê(III) S ◦ (T ◦ U) = (S ◦ T ) ◦ U;(IV) S ◦ (T + U) = S ◦ T + S ◦ U;(V) I V ◦ T = T ◦ I V = T .àãŽóãï:(I) (S + T )(x + y) = S(x + y) + T (x + y) = Sx S y + T x + T y =(Sx + T x) + (Sy + T y) = (S + T )x + (S + T )y.

190ŽêŽèëàæñîŽá(S + T )(λx) = Sλx + T λx = λSx + λT x = λ(Sx + T x) = λ(S + T )x.S ◦ T ∈ End(V )-æï áŽéðçæùâIJŽá ãðëãâIJå ïŽãŽîþæöëï ïŽýæå(II) (S + (T + U))x = Sx + (T + U)x = Sx + (T x + Ux) == (Sx + T x) + Ux = (S + T )x + Ux = ((S + T ) + U)x.â. æ. + ëìâîŽùæŽ ŽïëùæŽùæñîæŽ áŽ O V ∈ End(V ) âèâéâêðæ ŽçéŽõëòæèâIJïìæîëIJŽï:T + O V = O V + T = T ∀ T ∈ End(V )ᎠŽîæï Žáæùæñîæ âîåâñèæ End(V )-öæ. T ∈ End(V )-åãæï àŽêãïŽäôãîëåT : V → V ŽïŽý㎠öâéáâàæ åŽêŽòŽîáëIJæå(−T )x = −(T x) ∀ x ∈ V.Žáãæèæ ïŽøãâêâIJâèæŽ, îëé −T ∈ End(V ) Ꭰ(−T + T ) = T + (−T ) =O V , Žéæðëé −T ŽïŽý㎠Žîæï Žáæùæñîæ öâIJîñêâIJñèæ T -åãæï. (End(V ); +)-æïçëéñðŽùæñîëIJŽ àŽéëéáæêŽîâëIJï (V ; +)-æï çëéñðŽùæñîëIJæáŽê.(III) éðçæùâIJŽ àŽéëéáæêŽîâëIJï III åŽãæï öâáâàâIJæáŽê.(IV) õëãâèæ x ∈ V -åãæï àãŽóãï:(S ◦ (T + U))x = S((T + U)x) = S(T x + Ux) = S(T x) + S(Ux) == (S ◦ T )x + (S ◦ U)x = (S ◦ T + S ◦ U)x.(V) áŽéðçæùâIJŽ ðîæãæŽèñîæŽ.ãåóãŽå, T ∈ End(V ) Ꭰλ ∈ R. àŽêãïŽäôãîëå λT : V → V ŽïŽýãŽöâéáâàæ ûâïæå(λT )x = λ(T x) ∀ x ∈ V.Žáãæèæ ïŽøãâêâIJâèæŽ, îëé λT ∈ End(V ). Λ : R × End(V ) → End(V )ŽïŽýãŽï, îëéâèæù àŽêæïŽäôãîâIJŽ Λ(λ, T ) = λT åŽêŽòŽîáëIJæå, âûëáâIJŽïçŽèŽîäâ àŽéîŽãèâIJŽ.äâ.û æ ê Ž á Ž á â IJ Ž . (End(V ); +; Λ) ãâóðëîñèæ ïæãîù⎠(End(V ); +)-á Ž é ð ç æ ù â IJ Ž . ûæêŽ ûæêŽáŽáâIJæáŽê àŽéëéáæêŽîâëIJï, îëé(End(V ); +) Žîæï çëéñðŽùæñîæ þàñòæ. éŽöŽïŽáŽéâ, øãâê ñêᎠãŽøãâêëå, îëéïçŽèŽîäâ àŽéîŽãèâIJŽ ŽçéŽõëòæèâIJï ìæîëIJâIJï:(λ + µ)T = λT + µT, λ(S + T ) = λS + λT,(λµ)T = λ(µT ), 1 R T = T,ïŽáŽù λ, µ ∈ R ᎠS, T ∈ End(V ). àãŽóãï öâéáâàæ åŽêŽòŽîáëIJâIJæï þŽüãæ:((λ + µ)T )x = (λ + µ)(T x) = λ(T x) + µ(T x) = (λT )x + (µT )x.áŽêŽîøâêæ åŽêŽòŽîáëIJâIJæ éðçæùáâIJŽ ŽêŽèëàæñîŽá.

191û æ ê Ž á Ž á â IJ Ž . àŽéîŽãèâIJæï ëìâîŽùæŽ îàëèöæ áŽ Λ ïçŽèŽîäâàŽéîŽãèâIJŽ End(V )-öæ ŽçéŽõëòæèâIJï öâéáâà ìæîëIJâIJï:ïŽáŽù λ ∈ R ᎠS, T ∈ End(V ).á Ž é ð ç æ ù â IJ Ž .λ(S ◦ T ) = (λS) ◦ T = S ◦ (λT ),(λ(S ◦ T ))x = λ((S ◦ T )x) = λ(S(T x)) = (λS)(T x) = ((λS) ◦ T )x,(λ(S ◦ T ))x = λ((S ◦ T )x) = λ(S(T x)) = S(λ(T x)) = (S ◦ (λT ))x.ŽèàâIJîñè ïðîñóðñîâIJï, îëéâèåŽù Žóãå æàæãâ åãæïâIJâIJæ, îŽù End(V )-ï, ñûëáâIJâê ûîòæã ŽèàâIJîâIJï.à Ž ê é Ž î ð â IJ Ž . (X; +, ◦A) ëåýâñèï âûëáâIJŽ ûîòæãæ ŽèàâIJîŽR-äâ, åñ éëùâéñèæŽ Λ : R × X → X ŽïŽý㎠ᎠïîñèáâIJŽ:(I) (X; +, Λ) ãâóðëîñèæ ïæãîù⎠R-äâ;(II) (X; ◦, +) îàëèæŽ;(III) Λ áŽ ◦ ŽçéŽõëòæèâIJâê öâéáâà ìæîëIJâIJï:λ(x 1 · x 2 ) = (λx 1 ) ◦ x 1 = x 1 ◦ (λx 2 )∀ λ ∈ R Ꭰx 1 , x 2 ∈ X.End(V )-öæ éæôâIJñèæ öâáâàâIJæ öâàãæúèæŽ øŽéëãŽõŽèæIJëå öâéáâàêŽæîŽá.û æ ê Ž á Ž á â IJ Ž . End(V ) äâéëŽôêæöêñèæ ëìâîŽùæâIJæå Žîæï ûîòæãæŽèàâIJîŽ éñèðæìèæçŽùæñîæ âîåâñèæå.åñ T ∈ End(V ) Ꭰ∃ S : V → V æïâåæ, îëéS ◦ T = T ◦ S = I V ,éŽöæê S ∈ End(V ). Žïâå öâéåýãâãŽöæ T -ï âûëáâIJŽ öâIJîñêâIJŽáæ, ýëèë S =T −1 -ï T àŽîáŽóéêæï öâIJîñêâIJñèæ.Aut(V )-åæ Žôãêæöêëå õãâèŽ öâIJîñêâIJñèæ àŽîáŽóéêâIJæï ïæéîŽãèâ End(V )-áŽê, â. æ. V -ï ŽãðëéëîòæäéåŽ ïæéîŽãèâ.û æ ê Ž á Ž á â IJ Ž . (Aut(V ), ◦) Žîæï þàñòæ.á Ž é ð ç æ ù â IJ Ž . îŽáàŽê I V ∈ Aut(V ) ᎠI V ◦I V = I V , öâïŽIJŽéæïŽáŽîïâIJëIJï I −1V, îëéâèæù I V -æï ðëèæŽ.ãåóãŽå, S ∈ Aut(V ), éŽöæêS −1 ◦ S = S ◦ S −1 = I V .Žéæðëé (S −1 ) −1 ŽîïâIJëIJï Ꭰâéåýãâ㎠S-ï. â. æ. S −1 ∈ Aut(V ), åñ S ′ , T ∈Aut(V ), éŽöæê(S ◦ T ) ◦ (T −1 ◦ S −1 ) = S ◦ (T ◦ T −1 ) ◦ S −1 = S ◦ S −1 = I)v.

192êŽý. 1.2ŽêŽèëàæñîŽá, (T −1 ◦ S −1 ) ◦ (S ◦ T ) = I V , Žéæðëé (S ◦ T ) −1 ŽîïâIJëIJï áŽîŽáàŽê S, T ∈ Aut(V ) =⇒ S ◦ T ∈ Aut(V ). ëìâîŽùææï ŽïëùæŽùæñîëIJŽñçãâ áŽéðçæùâIJñèæŽ.1.2. ïðîñóðñîñèæ àŽéëïŽýñèâIJâIJæ R n -öæ. àŽêãæýæèëå R n -æï àâëéâðîæñèææêðâîìîâðŽùæŽ, ïŽáŽù ãâóðëîæï \ïæàîúâï" Ꭰ\éæéŽîåñèâIJŽï"Žóãï àâëéâðîæñèæŽäîæ. àŽêãæýæèëå R 2 . åñ r(x, y) ∈ R 2 , éŽöæê éŽêúæèæ(x, y) ûâîðæèæáŽê (0, 0) ûâîðæèŽéáâ Žîæï (x 2 + y 2 ) 1/2 . âï éŽêúæèæ Žôãêæöêëå‖r‖-æå, âï òŽóðæñîŽá Žîæï ‖ · ‖ : R 2 → R ŽïŽýãŽ. éŽï âûëáâIJŽ ïæàîúâ(æàæãâŽ, îŽù éëáñèæ Žê êëîéŽ). àŽêãæýæèëå r 1 (x 1 , y 1 ) Ꭰr 2 (x 2 , y 2 ) ûâîðæèâIJæ(ïñî. 1.4)ãåóãŽå, θ 1 Ꭰθ 2 [0, π] öñŽèâáæáŽáê ŽôâIJñèæ çñåýââIJæŽ, îëéâèïŽù óéêæïr 1 Ꭰr 2 ãâóðëîâIJæ ox ôâîúæï áŽáâIJæå éæéŽîåñèâIJŽïåŽê. éŽöæê r 1 Ꭰr 2ãâóðëîâIJï öëîæï éŽêúæèæŽ ‖r 2 − r 1 ‖, ýëèë θ = θ 2 − θ 1 { éŽå öëîæï çñåýâŽ,cos θ = cos(θ 2 − θ 1 ) = cos θ 1 cos θ 2 + sin θ 1 θ 2 == x 1 x 2‖r 1 ‖ ‖r 2 ‖ + y 1 y 2‖r 1 ‖ ‖r 2 ‖ = x 1x 2 + y 1 y 2‖r 1 ‖ ‖r 2 ‖ .x 1 x 2 + y 1 y 2 àŽéëïŽýñèâIJŽ öâàãæúèæŽ àŽéëãæõâêëå éŽêúæèâIJæï ᎠçñåýââIJæïáŽïŽåãèâèŽá r 2 -öæ. àŽêãïŽäôãîëå Φ : R 2 × R 2 → R ŽïŽý㎠öâéáâàêŽæîŽáéŽöæêΦ(r 1 , r 2 ) = x 1 x 2 + y 1 y 2 ,‖r‖ = (Φ(r 1 , r 2 )) 1/2 , cos θ =Φ(r 1 , r 2 )[Φ(r 1 , r 1 )Φ(r 2 , r 2 )] 1/2 .

193θ çñåýâ R 2 -æï ëî ãâóðëîï öëîæï àŽêæéŽîðâIJŽ ùŽèïŽýŽá æé ìæîëIJæå, îëé0 ≤ θ ≤ π. îëùŽ θ = 0, Žê θ = π, ŽéIJëIJâê, îëé r 1 Ꭰr 2 ŽîŽèâèâñîâIJæŽ(çëèæêâŽîñèæŽ).àŽéëõâêâIJæå éŽåâéŽðæçŽöæ Φ(r 1 , r 2 )-ï ŽôêæöêŽãâê r 1 · r 2 -æå Ꭰñûëáâ-IJâê ïçŽèŽîñè êŽéîŽãèï. åñ r 1 ≠ 0 Ꭰr 2 ≠ 0, éŽöæê r 1 · r 2 = 0. éŽöæê áŽéýëèëá éŽöæê, îëùŽ θ = π/2. Žé öâéåýãâãŽöæ ŽéIJëIJâê, îëé r 1 Ꭰr 2 ñîåæâîåéŽîåëIJñèæŽ(ëîåëàëêŽèñîæŽ). éëãæõãŽêëå ïçŽèŽîñèæ êŽéîŽãèæïåãæïâIJâIJæ.û æ ê Ž á Ž á â IJ Ž .1) r · r ≥ 0 Ꭰr · r = 0 éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ r = (0, 0);2) r 1 · r 2 = r 2 · r 1 ∀ r 1 , r 2 ∈ R 2 ;3) r 1 · (r 2 + r 2 ) = r 1 · r 2 + r 1 · r 3 ∀ r 1 , r 2 , r 3 ∈ R 2 ;4) λ(r 1 · r 2 ) = (λr 1 ) · r 2 = r 1 · (λr 2 ), ∀ r 1 , r 2 ∈ R 2 , λ ∈ R.á Ž é ð ç æ ù â IJ Ž .1) åñ r = (x, y) = R 2 , éŽöæê r · r = (x 2 + y 2 ) ≥ 0 ∀ x, y ∈ R Ꭰr · r = 0éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ x = 0 Ꭰy = 0.2) r 1 · r 2 = x 1 x 2 + y 1 y 2 = x 2 x 1 + y 2 y 1 = r 2 · r 1 .3) Ꭰ(4) éðçæùáâIJŽ ŽêŽèëàæñîŽá (ᎎéðçæùâå!).äëàŽáŽá, åñ V ãâóðëîñèæ ïæãîù⎠R-äâ áŽ Φ : V × V → R ŽïŽý㎎çéŽõëòæèâIJï (1){(4) ìæîëIJâIJï, éŽöæê (V ; Φ) ãâóðëîñè ïæãîùâöæ àŽêæýæèâIJŽçñåýæïŽ áŽ ïæàîúæï ùêâIJâIJæ. Φ ŽïŽýãŽï âûëáâIJŽ V -ï öæàŽ êŽéîŽããèæ, ýëèë(V ; Φ)-ï { ãâóðëîñèæ ïæãîùæï öæàŽ êŽéîŽãèæ. çâîúëá, åñ àŽêãéŽîðŽãåΦ : R n × R n → R (n ∈ N) ŽïŽýãŽï öâéáâàæ ûâïæå:Φ(a, b) ≡ a · b =n∑a i b i ,ïŽáŽù a = (a 1 , a 2 , . . . , a n ), b = (b 1 , b 2 , . . . , b n ), éŽöæê âï ŽïŽý㎠æóêâIJŽ ŽôêæöêñèæåãæïâIJâIJæï éóëêâ. àŽêãæýæèëå a ãâóðëîæï ïæàîúâ, îëàëîù ‖a‖ =(a · a) 1/2 . ýëèëi=2cos(â, b) =a · b‖a‖ ‖b‖(çñåýâ ∈ [0, π]).R b öâïŽúèâIJâèæŽ àŽêæéŽîðëï ïý㎠öæàŽ êŽéîŽãèâIJæù.äâéëéëõãŽêæè àŽéîŽãèâIJŽï ñûëáâIJâê øãâñèâIJîæãï Žê âãçèæáñî êŽéîŽãèï.îëùŽ n = 1, âï êŽéîŽãèæ Žîæï øãâñèâIJîæãæ àŽéîŽãèâIJŽ R-öæ, ýëèëxyarccos|x| |y|çñåýâ Žîæï 0 Žê π, xy-æï êæöêæï öâïŽIJŽéæïŽá. ‖ · ‖ êëîéŽ ŽîæïR-æï | · | éëáñèæï àŽêäëàŽáâIJŽ ᎠŽóãï ŽêŽèëàæñîæ åãæïâIJâIJæ. éŽàŽèæåŽá,

194öâàãæúèæŽ ãŽøãâêëå, îëé‖a‖ ≥ 0 ∀ a ∈ R n ,‖a‖ = 0 éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ a = 0,‖λa‖ = |λ| ‖a‖ ∀ a ∈ R n Ꭰλ ∈ R,‖a + b‖ ≤ ‖a‖ + ‖b‖ ∀ a, b ∈ R n .a ∈ R n ãâóðëîï âûëáâIJŽ âîåâñèëãŽêæ, åñ ‖a‖ = 1 (âï æàæãâŽ, îëé a·a = 1).åñ a ∈ R n \{0}, éŽöæêa‖a‖Žîæï âîåâñèëãŽêæ ãâóðëîæ, îëéâèæù a ãâóðëîæïåŽêŽéæéŽîåñèæŽ. âîåâñèëãŽê ãâóðëîï ŽôêæöêŽãâê â-æå.åñ B = { }ê 1 , ê2, . . . , ê n Žîæï R n -æï IJŽäæïæ æïâåæ, îëée i e j ={0, åñ i ≠ j1 åñ i = j ,éŽöæê IJŽäæïï âûëáâIJŽ ëîåëêëîéæîâIJñèæ.R n -æï ëîåëêëîéæîâIJñè IJŽäæïï, îëéâèæù ŽàâIJñèæŽ öâéáâàæ ûâïæåê i = (0, 0, . . . , 0,}{{}1 , 0, . . . , 0), 1 ≤ i ≤ n.i-ñî ŽáàæèŽïñûëáâIJâê ïðŽêáŽðñè IJŽäæïï R n -öæ. R 2 -æï ᎠR 3 -æï ïðŽêáŽîðñèæ IJŽäæïâIJæŽ(î, ĵ) Ꭰ(î, ĵ, ̂k). àŽêãæýæèëå Žé IJŽäæïåŽ àâëéâðîæñè ææêðâîìîâðŽùæŽ. i áŽj ãâóðëîâIJæ R 2 -öæ óéêæŽê éŽîþãâêŽ ïæïðâéŽï, ýëèë éâïŽéâ oz ôâîúæ éŽîåëIJæŽî Ꭰĵ ãâóðëîâIJæï öâéùãâèæ ïæIJîðõæï, ŽéŽïåŽê éæéŽîåñèæŽ æïâ, îëé ïŽéâñèæ(î, ĵ, ̂k) æõëï éŽîçãâêŽ (æý. ïñî 1.5).à Ž ê é Ž î ð â IJ Ž . ãåóãŽå, a = (a 1 , a 2 , a 3 ) ∈ R 3 Ꭰb = (b 1 , b 2 , b 3 ) ∈R 3 , a Ꭰb ãâóðëîâIJæï a × b ãâóðëîñèæ êŽéîŽãèæ Žîæïa × b = (a 2 b 3 − a 3 b 2 , a 3 b 1 − a 1 b 3 , a 1 b 2 − a 2 b 1 )ãâóðëîæ. \×" ëìâîŽùæŽ öâàãæúèæŽ àŽêãæýæèëå, îëàëîù R 3 × R 3 → R 3ŽïŽýãŽ.û æ ê Ž á Ž á â IJ Ž . åñ a, b ∈ R 3 , éŽöæê1) ‖a × b‖ = ‖a‖ ‖b‖ sin θ, ïŽáŽù θ Žîæï çñåýâ a-ïŽ áŽ b-ï öëîæï;2) a × b ãâóðëîæ éŽîåëIJñèæŽ a Ꭰb ãâóðëîâIJæï;

195êŽý. 1.3á Ž é ð ç æ ù â IJ Ž .1) ‖a × b‖ 2 = (a 2 b 3 − a 3 b 2 ) 2 + (a 3 b 1 − a 1 b 3 ) 2 + (a 1 b 2 − a 2 b 1 ) 2 == a 2 2b 2 3 + a 2 3b 2 2 − 2a 2 b 3 a 3 b 2 + a 2 3b 2 1 + a 2 1b 2 3−−2a 3 b 1 a 1 b 3 + a 2 1b 2 2 + a 2 2b 2 1 − 2a 1 b 2 a 2 b 1 == (a 2 1 + a 2 2 + a 2 3)(b 2 1 + b 2 2 + b 2 3) − (a 1 b 1 + a 2 b 1 + a 3 b 3 b) 2 == ‖a‖ 2 ‖b‖ 2 − (a · b) 2 = ‖a‖ 2 ‖b‖ 2( (a ·)b)21 −‖a‖ 2 ‖b‖ 2 == ‖a‖ 2 ‖b‖ 2 (1 − cos 2 θ) = ‖a‖ 2 ‖b‖ 2 sin 2 θ.2) Žáãæèæ ïŽøãâêâIJâèæŽ, îëé a · (a × b) = 0 Ꭰb · (a × b) = 0, =⇒áŽïŽéðçæùâIJâèæ.

196a × b ãâóðëîæï àâëéâðîæñèæ æêðâîìîâðŽùææïåãæï öâãêæöêëå, îëé åña = (a 1 , 0, 0), b = (b 1 , b 2 , 0),éŽöæê a × b = (0, 0, a 1 b 2 ) Žéæðëé, îëùŽ a 1 > 0, éŽöæê{|a 1 b 2 |̂k, b 2 > 0a × b =−|a 1 b 2 |̂k, b 2 < 0 ,a × b ãâóðëîæï éæéŽîåñèâIJŽ ñêᎠŽãæîøæëå æïâ, îëé ïŽéâñèæ (a, b, a × b)æõëï éŽîþãâêŽ. ãâóðëîñèæ êŽéîŽãèæ ûŽîéëæáàæêâIJŽ öâéáâàæ ïŽýæå:a × b = ‖a‖ · ‖b‖ sin θ · ̂n,ïŽáŽù ̂n âîåâñèëãŽêæ ãâóðëîæŽ, îëéâèæù éŽîåëIJñèæ a Ꭰb ãâóðëîâIJæï.ŽéŽïåŽê, (a, b, n) éŽîþãâꎎ. åñ a × b = 0, éŽöæê a Ꭰb ûîòæãŽá áŽéëçæáâ-IJñèæŽ, Ꭰåñ ‖a‖ > 0 Ꭰ‖b‖ > 0, éŽöæê a × b = 0 ðëèëIJæáŽê àŽéëéáæêŽîâëIJï,îëé a Ꭰb çëèæêâŽîñèæŽ, ãåóãŽå, éëùâéñèæŽ a Ꭰb ãâóðëîâIJæ(æý. ïñî 5.6) éŽöæê ‖a × b‖ { âï Žîæï OACB ìŽîŽèâèëàîŽéæï òŽîåëIJæ áŽa × b-ï àŽêæýæèŽãâê, îëàëîù òŽîåëIJæï ãâóðëîï.êŽý. 1.4éëãæõãŽêëå äëàæâîåæ åãæïâIJŽ (ᎎéðçæùâå!).û æ ê Ž á Ž á â IJ Ž .1) a × b = −b × a;2) a × (b + c) = a × b = a × c;3) (λa) × b = a × (λb) = λ(a × b);4) î × ĵ = ̂k, ĵ × ̂k = î, ̂k × î = ĵ;5) a × (b × c) = (a · c)b − (a · b)c;6) a × (b × c) = b · (c × a) = c · (a × b) = −a · (c × b) = −b · (a × c) =−c(b × a).

197a × (b × c) àŽéëïŽýñèâIJŽï ýöæîŽá ñûëáâIJâê ëîéŽà ãâóðëîñè êŽéîŽãèï,ýëèë a · (b × c)-ï { öâîâñè êŽéîŽãèï. àâëéâðîæñèŽá a · (b × c) Žîæï a, b,c ãâóðëîâIJäâ áŽüæéñèæ ìŽîŽèâèâìæìâáæï éëùñèëIJŽ.û æ ê Ž á Ž á â IJ Ž . {a, b, c} ⊂ R 3 ûîòæãŽá áŽéëçæáâIJñèæŽ éŽöæê áŽéýëèëá éŽöæê, îëùŽ a · (b × c) = 0.á Ž é ð ç æ ù â IJ Ž . áŽãñöãŽå, îëé a, b Ꭰc ûîòæãŽá áŽéëçæáâIJñèæŽ,éŽöæê ∃ λ, µ, σ ∈ R, îëéèâIJæù âîåáëñèŽá ŽîŽŽ êñèæ ᎠïîñèáâIJŽλs + µb + σc = 0.äëàŽáëIJæï áŽñîôãâãèŽá ãæàñèæïýéëå, îëé λ ≠ 0, éŽöæêa = −λ −1 (µ + σc),a · (b × c) = −λ −1 (µ + σc) · (b × c) == −λ −1[ µb · (b × c) + σc · (b × c) ] = 0.ìæîæóæå, åñ a · (b × c) = 0, éŽöæê Žê Ž) a, b Ꭰc ãâóðëîâIJæáŽê âîå-âîåæêñèæŽ, Žê IJ) a ëîåëàëêŽèñîæŽ b × c ãâóðëîæï. b Ꭰc ëîåëàëêŽèñîæŽb×c ãâóðëîæï, Žéæðëé a = λ ′ b+µ ′ c îŽæéâ λ ′ , µ ′ ∈ R. â. æ. a, b Ꭰc ûîòæãŽááŽéëçæáâIJñèæŽ.ᎠIJëèëï, éëçèâá àŽêãæýæèëå \ãâóðëîñèæ" òñêóùææï áæòâîâêùæîâ-IJŽáëIJæï ïŽçæåýæ. â. æ. R n -äâ éëùâéñèæŽ øãâñèâIJîæãæ êëîéŽ. àŽêãïŽäôãîëf : R → R n ûŽîéëâIJñèæ òñêóùæŽ.âîåàŽêäëéæèâIJæŽêæ öâéåýãâãæï àŽêäëàŽáâIJæå ãæðõãæå, îëé f áæòâîâêùæîâIJŽáæŽt ûâîðæèöæ, åñ ∃ F (t) = (F 1 (t), . . . , F n (t)) ∈ R n æïâåæ, îëéf(t + h) − f(t)∥ − F (t) ∥ −→ 0, îëùŽ t → 0,hâ. æ. åñ f-æï çëéìëêâêðâIJæŽ f 1 , . . . , f n æïâåæ, îëé∥ f 1(t + h) − f 1 (t)− F 1 (t), . . . , f n(t + h) − f n (t)− F n (t) ∥ −→ 0.hhîëùŽ h → 0. ùýŽáæŽ, åæåëâñèæ çëéìëêâêðæ éææïûîŽòãæï êñèæïçâê, îëùŽh → 0, Žéæðëé dfdtŽîïâIJëIJï éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ ŽîïâIJëIJïdf 1dt , . . . , df ndtáŽdf(dt = df1dt , . . . , df )n.dtïýãŽàãŽîŽá, \ãâóðëîñèæ" òñêóùææï áæòâîâêùæîâIJŽáëIJæïåãæï àãüæîáâIJŽéæïæ çëéìëêâêðâIJæï áæòâîâêùæîâIJŽáëIJŽ. éŽàŽèæåŽá, åñ f : R → R 3àŽêïŽäôãîñèæŽ öâéáâàæ ûâïæåéŽöæêf(t) = (2t 2 , ln t, sin 2 t),dfdt = (4t, 1 t , 2 sin t cos t ).

198ãåóãŽå, éëùâéñèæŽ f : R → R 3 Ꭰg : R → R 3 . àŽêãïŽäôãîëå f · g :R → R 3 Ꭰf × g : R → R 3 . áŽãñöŽå, îëééŽöæê(ᎎéðçæùâå!)(f · g)(t) = f(t) · g(t) Ꭰ(f × g)(t) = f(t) × g(t),ï Ž ã Ž î þ æ ö ë 5.4.ddg(f · g) = f ·dt dt + dfdt · g,ddg(f × g) = f ×dt dt + dfdt × g.1. åñ V Žîæï ãâóðëîñèæ ïæãîùâ F ãâèäâ, éŽöæêO F x = 0 ∀ x ∈ V,a0 = 0 ∀ a ∈ F.2. (a; 1; 3) ∈ R 3 ãâóðëîæ (a ∈ R) ûŽîéëŽáàæêâåS { (1; 1; 0), (0; 2; 0), (0; 0; 4) }ãâóðëîâIJæï ûîòæãæ çëéIJæêŽùææï ïŽýæå. Žøãâêâå, îëé S Žîæï ûîòæãŽá áŽéëñçæáâIJâèæïæïðâéŽ. Žîæï åñ ŽîŽ S ïæïðâéŽ R 3 -æï IJŽäæïæ?3. Žøãâêâå, îëé åñ {x 1 , . . . , x m } Žîæï V ïæãîùæï ûîòæãŽá áŽéëñçæáâ-IJâèæ ïæïðâéŽ, éŽöæê x i ≠ 0 ∀ i-åãæï, 1 ≤ i ≤ m.4. Ž) îëéâèæŽ ûîòæãæ àŽîáŽóéêŽ?T 1 (x, y) = (a, y), a ∈ R \ {0},T 2 (x, y) = (λx + y, σy), λ, σ ∈ R \ {0},T 3 (x, y) = (x 2 , 0),T 4 (x, y) = (x, 0).IJ) àŽêéŽîðâå T 2 ◦ T 4 ᎠT 4 ◦ T 2 êŽéîŽãèæ.à) ᎎéðçæùâå, îëé åñ T ∈ End(V ), éŽöæê T 0 = 0.5. Ž) åñ V Žîæï ãâóðëîñèæ ïæãîùâ, éŽöæê p : V → V àŽîáŽóéêŽïâûëáâIJŽ ìîëâóùæŽ, åñ (p◦p)x = P x, ∀ x ∈ V . ᎎéðçæùâå, îëé p : R 2 → R 2àŽîáŽóéêŽ, ïŽáŽù( 1b)p(x, y) = (ax − y + c);a − b a − b (ax − y + c) + c ,a, b, c ∈ R Ꭰa ≠ b, Žîæï ìîëâóùæŽ R 2 -öæ. îŽ ìæîëIJâIJöæ ïîñèáâIJŽ p ∈End(R 2 )?IJ) (4Ž) ïŽãŽîþæöëï îëéâèæ àŽîáŽóéꎎ ìîëâóùæŽ?

1996. ãåóãŽå, T ∈ End(V ). N(T ) = {x ∈ V : T x = 0} óãâïæéîŽãèâïâûëáâIJŽ êñèëãŽêæ óãâïæãîùâ (ïæîåãæ). Žøãâêâå, îëé N(T ) óãâïæãîù⎠V -öæ. Žøãâêâå, îëé T -ï ŽêŽïŽýæ Žïâãâ óãâïæãîù⎠V -öæ.7. ãåóãŽå, V ãâóðëîñèæ ïæãîù⎠R-äâ ᎠT ∈ End(V ). ŽéIJëIJâê, îëéT -ï Žóãï ïŽçñåîæãæ éêæöãêâèëIJŽ λ ∈ R, åñ ∃ ŽîŽêñèëãŽêæ x ∈ V ãâóðëîæ,îëé T x = λx. x-ï âûëáâIJŽ T -ï ïŽçñåîæãæ ãâóðëîæ, îëéâèæù öââïŽIJŽéâIJŽλ-ï.Ž) Žøãâêâå, îëé åñ T ∈ End(V ) æïâåæŽ, îëéT (x, y) = (x + ay; y),éŽöæê ∀ (p, 0) ïŽýæï ãâóðëîæ (p ≠ 0) Žîæï T -ï ïŽçñåîæãæ ãâóðëîæ. îëéâèæŽéæïæ ïŽçñåîæãæ éêæöãêâèëIJŽ?IJ) ãåóãŽå, T ∈ End(V ). V λ -æå Žôãêæöêëå λ ïŽçñåîæãæ éêæöãêâèëIJâIJæïöâïŽIJŽéæïæ ïŽçñåîæãæ ãâóðëîâIJæï ïæéîŽãèâ. Žøãâêâå, îëé V λ ∪ {0} îæï ãâóðëîñèæóãâïæãîùâ V -öæ. ᎎéðçæùâå ŽêŽèëàæñîæ áâIJñèâIJŽ N(T )-åãæï.8. æìëãâå T 1 , T 2 ∈ End(R 2 )-æï ïŽçñåîæãæ ãâóðëîâIJæ ᎠïŽçñåîæãæéêæöãêâèëIJâIJæ:T 1 (x, y) = (−y, x),îŽ àâëéâðîæñèæ Žäîæ Žóãï T 1 -ï ᎠT 2 -ï?T 2 (x, y) = (x, −y).9. ᎎéðçæùâåŽ) åñ S, T ∈ End(V ), éŽöæê S ◦ T ∈ End(V );IJ) åñ T ∈ End(V )-åãæï ŽîïâIJëIJï S : V → V àŽîáŽóéêŽ æïâåæ, îëéS ◦ T = T ◦ S = I V ,éŽöæê S ∈ End(V );à) åñ T ∈ Aut(V ), éŽöæê Λ 0 (T ) = {0}.10. ᎎéðçæùâå, îëé åñ r 1 , r 2 , r 3 ∈ R 2 Ꭰλ ∈ R, éŽöæêŽ) r 1 · (r 2 + r 3 ) = r 1 · r 2 + r 1 · r 3 ;IJ) λ(r 1 · r 2 ) = (λr 1 ) · r 2 = r 1 · (λr 2 );à) |r 1 · r 2 | ≤ ‖r 1 ‖ ‖r 2 ‖;á) ‖r 1 − r 2 ‖ 2 = ‖r 1 ‖ 2 + ‖r 2 ‖ 2 − 2‖r 1 ‖ ‖r 2 ‖ cos θ, ïŽáŽù θ Žîæï çñåýâr 1 -ïŽ áŽ r 2 -ï öëîæï.â) ∣ ∣ ‖r1 ‖ − ‖r 2 ‖ ∣ ∣ ≤ ‖r1 + r 2 ‖ ≤ ‖r 1 ‖ + ‖r 2 ‖. (IJëèë ñðëèëIJŽ ùêëIJæèæŽïŽéçñåýâáæï ñðëèëIJæï ïŽýâèæå.)Žýïâêæå éŽåæ àâëéâðîæñèæ Žäîæ.11. àŽéëåãŽèâå a = (1, 1, 1), b = (1, p, 0), p ∈ R ãâóðëîâIJæï ìŽîŽèâèñîæâîåâñèëãŽêæ ãâóðëîâIJæ. æìëãâå âîåâñèëãŽêæ ãâóðëîæ, îëéâèæù éŽîåëIJñèæŽa Ꭰb ãâóðëîâIJæï âîåáîëñèŽá.12. ãåóãŽå, a, b, c ∈ R 3 Ꭰλ ∈ R. ᎎéðçæùâå, îëé

2001) a × b = −b × a;2) a × (b + c) = a × b = a × c;3) (λa) × b = a × (λb) = λ(a × b);4) î × ĵ = ̂k, ĵ × ̂k = î, ̂k × î = ĵ;5) a × (b × c) = (a · c)b − (a · b)c;6) a(b×c) = b·(c×a) = c·(a×b) = −a·(c×b) = −b·(a×c) = −c(b×a).13. àŽéëæõâêâå 5.4.12-æï öâáâàâIJæ ᎠŽøãâêâå, îëé X : R 3 → R 3 ŽîŽŽïëùæñîæŽ (â. æ. a × (b × c) ≠ (a × b) × c, äëàŽáŽá).14. ãåóãŽå, g : R → R 3 . ᎎéðçæùâå, îëéŽ) d dtIJ) d dt(f · g) = f · dgdt + dfdt · g;(f × g) = f ×dgdt + dfdt × g.à) åñ ‖f(t)‖ = a ∀ t-åãæï, ïŽáŽù a (a ∈ R) éñáéæãæŽ, éŽöæê dfdt ëîåëàëêŽèñîæŽf-æï õãâèŽ t-åãæï. àŽéëåãŽèâåf(t) = (a cos t, a sin t, 0),g(t) = (0, 1, t).

éâïâîâIJæ ᎠIJñèæï ŽèàâIJîâIJæ§ 1. éâïâîâIJæ ᎠIJñèæï ŽèàâIJîâIJæIJñèæï ŽèàâIJîŽï øãâê ñçãâ àŽãâùŽêæå äâéëå. ãŽøãâêâIJå, îëé IJñèæï ŽîŽâîåæŽèàâIJîŽ ŽîïâIJëIJï. áŽãæûõëå ñòîë äëàŽáæ ïðîñóðñîâIJæï, â. û. éâïâîâ-IJæï öâïûŽãèŽ. äëàæâîåæ éâïâîæ úŽèæŽê éêæöãêâèëãŽêæŽ àŽéëåãèâIJæï åâëîæ-Žöæ ŽìîëóïæéŽùææï åãŽèïŽäîæïæå. âîåæ ìîëàîŽéŽ Žýáâêï éâëîæï ŽìîëóïæóéŽùæŽï,åñçæ éëùñèëIJŽåŽ ïæãîùâöæ âîåæ Ꭰæàæãâ àŽéëåãèâIJæ ðŽîáâIJŽ.àŽéëåãèŽ Žîæï îŽæéâ éëùâéñèëIJâIJäâ ìîëàîŽéæï éëóéâáâIJæï öâáâàæ.ŽéŽïåŽê, âï ìîëàîŽéŽ áŽûîæèæŽ îŽæéâ âêŽäâ. ïŽäëàŽáëá, àŽéëåãèæèæ åãæïâIJâIJæøŽûâîæèæ ñêᎠæõëï æé âêæï ðâîéæêâIJöæ, îëéèæåŽù âï ìîëàîŽéŽŽáŽûâîæèæ.1.1. éâïâîâIJæ. öâàŽýïâêâIJå, îëé ρ IJæêŽîñèæ éæéŽîåâIJŽ S ïæéîŽãèâäâŽîæï êŽûæèëIJîæãæ áŽèŽàâIJæï éæéŽîåâIJŽ, åñ ïîñèáâIJŽ îâòèâóïñîëIJŽ,ðîŽêäæðñèëIJŽ ᎠŽêðæïæéâðîæñèëIJŽ. â. æ. (S, ρ) Žîæï êŽûæèëIJîæã áŽèŽàâ-IJñèæ ïæéîŽãèâ, öâïŽúèâIJâèæŽ ρ øŽæûâîëï, îëàëîù \≤", ýëèë (S, ≤) {ñIJîŽèëá S. êŽûæèëIJîæã áŽèŽàâIJñè ïæéîŽãèâï âûëáâIJŽ ûîòæãŽá áŽèŽàâ-IJñèæ (þŽüãæ), åñ ∀ x, y ∈ S Žê x ≤ y Žê y ≤ x, Žê ïîñèáâIJŽ ëîæãââîåŽá. ïŽäëàŽáëá, êâIJæïéæâîæ êŽûæèëIJîæãæ áŽèŽàâIJŽ öâæúèâIJŽ ûŽîéëáàæêáâï,îëàëîù ûîòæãæ áŽèŽàâIJæï àŽâîåæŽêâIJŽ.öâãêæöêëå, îëé õëãâèæ ïŽïîñèæ ûîòæãæ áŽèŽàâIJŽ (A, ≤) öâæúèâIJŽ ûŽîëãŽáàæêëå,îëàëîùa 1 ≤ a 2 ≤ · ≤ a n .Žïâå öŽâéåýãâãŽöæ îâòèâóïñîëIJæïŽ áŽ ðîŽêäæðñèëIJæï áŽéðçæùâIJŽ ŽîŽŽïŽüæîë. 1.1(Ž) êŽýŽääâ êŽøãâêâIJæŽ (A, ≤)-ï ûŽîéëáàâêŽ. A-ï øŽãûâîå, îëàëîù(a 1 , a 2 , . . . , a n ).û æ ê Ž á Ž á â IJ Ž . ïŽïîñè ïæéîŽãèâï êŽûæèëIJîæã áŽèŽàâIJŽ öâæúèâIJŽûŽîéëãŽáàæêëå, îëàëîù îŽôŽù óãâïæéîŽãèâå ûîòæãæ áŽèŽàâIJâIJæïàŽâîåæŽêâIJŽ. (ᎎéðçæùâå!)Žé òŽóðæï àŽéëõâêâIJæå, õëãâèæ êŽûæèëIJîæãæ áŽèŽàâIJŽ öâæúèâIJŽ ûŽîéëãŽáàæêëåþŽüãâIJæï ïŽöñŽèâIJæå. éæôâIJñè áæŽàîŽéŽï ñûëáâIJâê ߎïæï áæ-ŽàîŽéŽï.é Ž à Ž è æ å æ 1.1. ρ = { (x, y · x) Žîæï y-æï àŽéõëòæ } éæéŽîåâIJŽ,îëéâèæù àŽêïŽäôãñèæŽ {1, 2, 3, 4, 6, 10, 12, 20} ïæéîŽãèâäâ, àãŽúèâãï ߎïæï201

202êŽý. 1.1áæŽàîŽéŽï (æý. êŽý. 1.1(IJ)) ᎠáŽõëòæèæŽ þŽüãâIJŽá:{(1, 2, 4, 12), (1, 3, 6, 12), (2, 6), (4, 20), (2, 10, 20)}âï áŽõëòŽ âçãæãŽèâêðñîæŽ { (2, 6, 12), (1, 3, 6), (1, 2, 10, 20), (2, 4, 20), (4, 12) }ïæïðâéæï.öâãêæöêëå, îëé éæñýâáŽãŽá æéæïŽ, îëé 1ρx ∀ x-åãæï éùâéñèæ ïæéîŽãèæáŽê,áæŽàîŽéŽäâ Žî Žîæï 1-áŽê àŽéŽãŽèæ 8 æïŽîæ. âï éææôûâ㎠îâòèâóïñîë-IJæïŽ áŽ ðîŽêäæðñèëIJæï åãæïâIJæï àŽéë.ãåóãŽå, éëùâéñèæŽ (A, ≤) ᎠB ⊆ A. IJñêâIJîæãæŽ æïéæï çæåýãŽ: B Žîæïåñ ŽîŽ öâéëïŽäôãîñèæ äâãæáŽê (óãâãæáŽê) A-ï âèâéâêðâIJæå? öâéáâàöæ éëãúâIJêæåñéùæîâï äâᎠïŽäôãŽîï (ñáæáâï óãâᎠïŽäôãŽîï), îëéâèæù ŽôæêæöêâIJŽsup (inf). âï àŽêéŽîðâIJâIJæ éëùâéñèæ æõë åŽãæ 2-æï éâ-5 ìŽîŽàîŽòöæ.éŽå àŽéëãæõâêâIJå éâïâîâIJæï àŽêýæèãæïŽï.à Ž ê é Ž î ð â IJ Ž . êŽûæèëIJîæã áŽèŽàâIJñè (A, ≤) ïæéîŽãèâï âûëáâIJŽéâïâîæ, åñ éæï õëãâè ûõãæèï Žóãï ïñìîâéñéæ Ꭰæêòæéñéæ. ŽéŽï ŽôãêæöêŽãåöâéáâàêŽæîŽá:X ∧ Y − inf({x, y}),X ∨ Y = sup({x, y}).õëãâèæ êŽûæèëIJîæã áŽèŽàâIJñèæ ïæéîŽãèâ éâïâîæ Žî Žîæï. éŽàŽèæåŽá,1.1. éŽàŽèæåæï êŽûæèëIJîæã áŽèŽàâIJñæèæ ïæéîŽãèâ Žî Žîæï éâïâîæ, îŽáàŽê12 ∨ 20 Žî Žîæï àŽêïŽäôãîñèæ.

203∧ Ꭰ∨ ëìâîŽùæâIJæ êŽûæèëIJîæã áŽèŽàâIJñè ïæéîŽãèæï ûõãæèâIJï öëîæïöâæúèâIJŽ çæáâã ñòîë àŽêãŽäëàŽáëå.ãåóãŽå, ∧X =∧ x = inf X, ∨X =∨ x = sup X âï Žîæï ïŽïîñèæx∈XŽîŽùŽîæâèæ X ïæéîŽãèæï sup X Ꭰinf X.Žáãæèæ ïŽêŽýŽãæŽ, îëé ŽîïâIJëIJï éîŽãŽèæ ïìâùæŽèñîæ ðæìæï éâïâîâIJæ,îëéâèöæù öâïŽúèâIJâèæŽ ïýãŽáŽïý㎠ëìâîŽùæâIJæï øŽðŽîâIJŽ. àŽêãæýæèëå ïéæéŽåàŽêæ.x∈Xà Ž ê é Ž î ð â IJ Ž . L éâïâîï ãñûëáëå áæïðîæIJñùæñèæ, åñ æïŽçéŽõëòæèâIJï ìæîëIJâIJï:∀ x, y, z ∈ L.x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z),x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z),õãâèŽ éâïâîæ Žî Žîæï áæïðîæIJñùæñèæ.é Ž à Ž è æ å æ 1.2. 1.9 êŽýŽääâ àŽéëïŽýñèæ éâïâîæ Žî Žîæï áæïðîæ-IJñùæñèæ, îŽáàŽêb ∧ (d ∨ c) = b ∧ e = b,éŽöæê(b ∧ d) ∨ (b ∧ c) = a ∨ a = a.û æ ê Ž á Ž á â IJ Ž . ãåóãŽå, (L, ∧, ∨) áæïðîæIJñùæñè éâïâîöæ ïîñèáâIJŽåŽêŽòŽîáëIJâIJæx ∨ y = x ∨ z 1 , x ∧ y = x ∧ z.éŽöæê y = z.êŽý. 1.2

204á Ž é ð ç æ ù â IJ Ž. öâãêæöêëå, îëéŽ) a ∧ b = b ∧ a, a ∨ b = b ∨ a;IJ) a ∧ b ≤ a ≤ a ∨ b;à) (a ∧ b) ∨ a = a, a ∧ (a ∨ b) = a.Žéæðëéy = y ∨ (y ∧ x) = y ∨ (z ∧ x) = (y ∨ z) ∧ (y ∨ x) == (z ∨ y) ∧ (z ∨ x) = z ∨ (y ∧ x) = z ∨ (z ∧ x) = z.à Ž ê é Ž î ð â IJ Ž . áŽãñöãŽå, îëé (L, ∧, ∨) éâïâîöæ 0, 1 ∈ L áŽ0 ≤ x ≤ 1, ∀ x ∈ L, éŽöæêx ∨ 1 = 1, x ∧ 1 = x, x ∧ 0 = 0, x ∨ 0 = 0, ∀ x ∈ L.Žïâå éâïâîï ñûëáâIJâê éâïâîï à Ž ï î ñ è â IJ æ å, åñ ∀ x ∈ L-åãæï ∃ x ∈ Læïâåæ, îëé x ∧ x = 0, x ∨ x = 1 (x-ï âûëáâIJŽ x-æï àŽïîñèâIJŽ).û æ ê Ž á Ž á â IJ Ž . åñ (L, ∧, ∨) Žîæï áæïðîæIJñùæñèæ éâïâîæàŽïîñèâIJæå, éŽöæê àŽïîñèâIJŽ âîåŽáâîåæŽ.á Ž é ð ç æ ù â IJ Ž. áŽãñöãŽå, îëé x, y, z ∈ L áŽx ∨ y = x ∨ z(= 1), x ∧ y = x ∧ z(= 0).éŽöæê ûæêŽ ûæêŽáŽáâIJæï úŽèæå y = z.éâïâîæï éâïŽéâ, ïìâùæŽèñîæ ðæìæ ñøãâñèëIJŽ æé åãŽèïŽäîæïæå, îëé ïŽïîñèæéâïâîâIJæï öâïŽýâIJ Žî ŽéIJëIJï îŽæéâ ŽýŽèï.øŽéëãŽõŽèæIJæëå éâïâîåŽ ïýãŽàãŽîæ àŽêéŽîðâIJŽ ûæêŽáŽáâIJæï ïŽýæå.û æ ê Ž á Ž á â IJ Ž . L Žîæï éŽîðæãæ éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ ∨xᎠ∧x ŽîïâIJëIJï L-æï õëãâèæ ŽîŽùŽîæâèæ, ïŽïîñèæ X óãâïæéîŽãèæïåãæï.(éðçæùáâIJŽ æêáñóùææå, ᎎéðçæùâå!)åñ L-öæ àŽêïŽäôãîñèæ ∧L, æàæ ŽôæêæöêâIJŽ O-æå ᎠâûëáâIJŽ ñéùæîâïæâèâéâêðæ L-öæ. ŽêŽèëàæñîŽá, åñ L-öæ ŽîïâIJëIJï ∨L, ŽôæêæöêâIJŽ 1-æå áŽâûëáâIJŽ L-æï ñáæáâïæ âèâéâêðæ. àŽêéŽîðâIJæï úŽèæå ∨∅ = 0.à Ž ê é Ž î ð â IJ Ž . L éâïâîï âûëáâIJŽ ïîñèæ, åñ ∨x Ꭰ∧x ŽîïâIJëIJïõãâèŽ X óãâïæéîŽãèâåŽåãæï L-áŽê.õãâèŽ ïŽïîñèæ éâïâîæ ïîñèæŽ. àŽêãæýæèëå Q ïæéîŽãèâ øãâñèâIJîæãæ ≤áŽèŽàâIJæå ᎠñïŽïîñèë ïæéîŽãèâ π-îæùýãæï éæŽýèëâIJæïŽï, ïŽáŽù îæùýãâ-IJæ âîåéŽêâåæïàŽê àŽêïýãŽãáâIJŽ ŽåëIJæåæ åŽêîæàæï éýëèëá âîåæ ùæòîæå.Žé ŽìîëóïæéŽùææï äâᎠäôãŽîæ, ùýŽáæŽ, Žîæï π, éŽàîŽé πR \ Q, éŽöŽïŽáŽéâ,(Q, ≤) Žî Žîæï ïîñèæ éâïâîæ. (R, ≤) éâïâîæ ïîñèæŽ áŽ Q ⊆ R.öâàãæúèæŽ ãŽøãâêëå, îëé öâïŽúèâIJâèæŽ õëãâèæ éâïâîæï àŽïîñèâIJŽ ïîñèéâïâîŽéáâ, åñéùŽ Žé ïŽçæåýï øãâê Žî öâãâýâIJæå.

2051.2. IJñèæï ŽèàâIJîâIJæ. éëãæõãŽêëå òëîéŽèñîæ àŽêéŽîðâIJŽ äëàŽáæIJñèæï ŽèàâIJîâIJæï (Žïâ âûëáâIJŽ XIX ïŽñçñêæï éŽåâéŽðæçëï þ. IJñèæï ìŽðæãïŽùâéŽá).ïæéîŽãèâåŽ ŽèàâIJîŽ IJñèæï ŽèàâIJîæï çâîúë öâéåýãâ㎎ ᎠåñéùŽàŽêïýãŽãâIJñèæ IJñèæï ŽèàâIJîâIJæ ïðîñóðñîñèŽá ,ïàŽãïæŽ, ñêᎠöâãêæöêëå,îëé æïæêæ öâæúèâIJŽ Žî æõëï âîåéŽêâåöæ øŽáàéñèæ.à Ž ê é Ž î ð â IJ Ž . B ïæéîŽãèâï ∨, ∧ Ꭰ− (∨ { \Žê" ëìâîŽùæŽ,∧ { \áŽ" ëìâîŽùæŽ, − àŽïîñèâIJæï ëìâîŽù掎, ýöæîŽá ∨-ï âûëáâIJŽ áæäñêóùæŽï,ýëèë ∧-ï çëêæñêóùæŽï). ëìâîŽùæâIJæå âûëáâIJŽ IJñèæï ŽèàâIJîŽ.IJæêŽîñèæ ëìâîŽðëîâIJæ: ∧, ∨ ᎠñêŽîñèæ ëìâîŽðëîæ B-ï ëî àŽêïýãŽãâ-IJñè âèâéâêðåŽê 0 Ꭰ1 ŽçéŽõëòæèâIJâê öâéáâà ŽóïæëéâIJï: õëãâèæ a, b, áŽC-åãæï B-áŽê1) a ∨ b = b ∨ a;2) a ∨ (b ∨ c) = (a ∨ b) ∨ c;3) a ∨ 0 = a;4) a ∨ a = 1;5) a ∧ b = b ∧ a;6) a ∧ (b ∧ c) = (a ∧ b) ∧ c;7) a ∧ 1 = a;8) a ∧ a = 0;9) a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c);10) a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c).ŽéîæàŽá, \áŽ" Ꭰ\Žê" çëéñðŽùæñîæ, ŽïëùæŽùæñîæ ᎠáæïðîæIJñùæñèæŽ.åãæåâñè éŽåàŽêï Žóãï âîåâñèëãŽêæ âèâéâêðæ ᎠîëáâïŽù àãŽóãï âèâéâêðæïçëéIJæêŽùæŽ åãæï áŽéŽðâIJŽïåŽê \áŽ/Žê"-æï ïŽöñŽèâIJæå, öâáâàŽá ãæôâIJå\Žê/áŽ"-ï éæéŽîå âîåâñèëãŽê âèâéâêðï; äâéëéëõãŽêæèæ ëìâîŽðëîâIJæáŽçŽãöæîâIJñèæŽ çëéìæñðâîñè èëàæçŽïåŽê, îëéèæï ïŽöñŽèâIJæåŽù ýáâIJŽïóâéâIJæï ŽàâIJŽ. ïý㎠IJñèæï ŽèàâIJîâIJöæ öâïŽúèâIJâèæŽ ñòîë éëïŽýâîýâIJâèææõëï ∨ øŽãåãŽèëå îëàëîù àŽâîåæŽêâIJŽ, ýëèë ∧ { îëàëîù åŽêŽçãâåŽ. aëìâîŽùæŽï øŽûâîâê îëàëîù a ′ Žê ⌉a.IJñèæï ŽèàâIJîŽ öâàãæúèæŽ àŽêãéŽîðëå îëàëîù áæïðîæIJñùæñèæ éâïâîæïàŽïîñèâIJŽ. Žéæðëé IJñèæï (B(x), ∩, ∪ ′ ) ŽèàâIJîæï ŽêŽèëàæñîŽá ŽîŽùŽîæâèæX ïæéîŽãèæïåãæï öâàãæúèæŽ éæãæôëå éåâèæ îæàæ åãæïâIJâIJæ.æêãëèñùææï çŽêëêæ (ëîéŽàæ ñŽîõëòæï çŽêëêæ). x-æï áŽéŽðâIJæïáŽéŽðâIJŽ Žîæï x, x ∈ B.öåŽêåóéæï çŽêëêæ. ∀ a, b ∈ B-åãæï ïŽéŽîåèæŽêæŽa ∧ (a ∨ b) = a, a ∨ (a ∧ b) = a.

206æáâéìëðâêðñîëIJæï çŽêëêæ. õëãâèæ âèâéâêðæ B-áŽê æáâéìëðâêðñîæŽ∧ Ꭰ∨-æï éæéŽîå.éëîàŽêæï çŽêëêæ. ∀ a, b ∈ B-åãæï ïŽéŽîåèæŽêæŽa ∧ b = a ∨ b, a ∨ b = a ∧ b.æêãëèñùæñîëIJæïŽ áŽ éëîàŽêæï çŽêëêâIJæáŽê àŽéëéáæêŽîâëIJï, îëé åñŽãæôâIJå B IJñèæï ŽèàâIJîŽï (B, ∨, ∧, −) ᎠàŽîáŽãóéêæå ŽýŽè ïæïðâéŽá Žêx-æï ŽïŽýãæå x, Žê ∧ Ꭰ∨ ëìâîŽùæâIJæï áŽýéŽîâIJæå, éŽöæê éæôâIJñèæ ïæïðâéŽçãèŽã IJñèæï ŽèàâIJ. âï òŽóðæ ùêëIJæèæŽ ëîŽáñèëIJæï ìîæêùæìæï ïŽýâèæå.éŽîåèŽù, éëîàŽêæï çŽêëêâIJæ öâæúèâIJŽ éæãæôëå ïýãŽàãŽîŽáŽù, õëãâèæâèâéâêðæï ŽïŽýãæå åŽãæï áŽéŽðâIJŽöæ.ŽýèŽ øãâê éëãæõãŽêå åâëîâéŽï, îëéâèæù àãæøãâêâIJï, îëàëîæ ïðŽêáŽîðñèæòëîéæå øŽæûâîâIJŽ IJñèæï àŽéëïŽýñèâIJâIJæ. Žéæï ñöñŽèë öâáâàæŽ, îëéëîæ ëìâîŽðëîæ ïŽçéŽîæïæŽ õãâèŽ IJæêŽîñèæ òñêóùææï ŽôïŽûâîŽá. öâéëãæðŽêëåŽñùæèâIJâèæ ðâîéæêëèëàæŽ.ŽéIJëIJâê, îëé IJñèæï A ᎠB òëîéñèâIJæ âçãæãŽèâêðñîæŽ (A ≡ B ŽêA ←→ B), åñ æïæêæ àŽéëïŽýŽãâê âîåæ Ꭰæàæãâ òñêóùæŽï. ñéŽîðæãâï öâéåýãâãŽöæâï Žîæï âçãæãŽèâêðëIJæï éæéŽîåâIJŽ.ù ý î æ è æ 1.1.A B A ←→ B0 0 10 1 11 0 01 1 1ŽêŽèëàæñîŽá, ŽéIJëIJâê, îëé A æûãâãï B-ï æéìèæçŽùæŽï (A → B), åñ ïîñèáâIJŽ1.3 ùýîæèæï àŽéëïŽýèâIJâIJæ (üâöéŽîæðëIJæï ùýîæèæ).æïîâIJæ ýöæîŽá ïýãŽáŽïý㎠Žäîï ŽðŽîâIJâê. Žó A → B Žîæï äñïðŽá æàæãâ,îŽù (⌉A) ∨ B, ýëèë A ≡ B æàæãâŽ, îŽù (A → B) ∧ (B → A). èëàæçñîŽáA → B ŽôêæöêŽãï \åñ A, éŽöæê B" (â. æ. åñ A ü-æŽ, éŽöæê B-ù ü-æŽ). ýöæîŽá−→ Ꭰ←→ ïæéIJëèëâIJï ñûëáâIJâê ìæîëIJæå ᎠñìæîëIJë ëìâîŽðëîâIJï,öâïŽIJŽéæïŽá.øãâñèâIJîæã ñŽîõëòæï Žôéêæöãêâè öâéáâà ïæéIJëèëâIJï æõâêâIJâê.A ≢ B =⌉(A ≡ B),A ̸−→ B =⌉(A −→ B),A ↑ B =⌉(A ∧ B),A ↓ B =⌉(A ∨ B).ŽýèŽ öâàãæúèæŽ Žôãûâîëå õãâèŽ IJæêŽîñèæ ëìâîŽùæŽ {0; 1} 2 −→ {0; 1} (ŽóæïŽîæ éæñåæåâIJï ŽïŽýãŽï). 1.4 ùýîæèäâ éæåæåâIJñèæ òëîéæå.

207ù ý î æ è æ 1.2.A 0 1 0 1B 0 0 1 1òñêóùæŽA 0 1 0 1B 0 0 1 1òñêóùæŽf 0 0 0 0 0 0f 1 0 0 0 1 A ∧ Bf 2 0 0 1 0 B ̸−→ Af 3 0 0 1 1 Bf 4 0 1 0 0 A ̸−→ Bf 5 0 1 0 1 Af 6 0 1 1 0 A ≢ Bf 7 0 1 1 1 A ∨ Bf 8 1 0 0 0 A ↓ Bf 9 1 0 0 1 A ≡ Bf 10 1 0 1 0 ⌉Af 11 1 0 1 1 A −→ Bf 12 1 1 0 0 ⌉Bf 13 1 1 0 1 B −→ Af 14 1 1 1 0 A ↑ Bf 15 1 1 1 1 1ñòîë éâðæù, øãâê öâàãæúèæŽ Žôãûâîëå õãâèŽ ŽïŽý㎠{0; 1} n -áŽê{0; 1}-öæ éýëèëá ↑ Žê ↓ ëìâîŽùæâIJæï ïŽöñŽèâIJæå (↑ âûëáâIJŽ öâòâîæï öðîæýæᎠŽôæêæöêâIJŽ \|", èëàæçñîŽá êæöêŽãï \ŽîŽ áŽ", ýëèë ↓ { \ŽîŽ Žê"). ↑ Žê↓ ŽïëùæŽùæñîæŽ. àŽêéŽîðâIJæå A ↑ B ↑ Žîæï (A∧B∧C) ᎠŽîŽ ((A∧B ′ ∧C).ŽéIJëIJâê, îëé {↑} Žêá {↓} ëìâîŽðëîåŽ ïæéîŽãèâ ŽáâçãŽðñîæŽ.å â ë î â é Ž . êâIJæïéæâîæ {0; 1} n −→ {0; 1} ŽïŽý㎠öâæúèâIJŽ ûŽîéëæáàæêëï↑ Žê ↓ ëìâîŽðëîâIJæï òëîéñèâIJæå.á Ž é ð ç æ ù â IJ Ž . àŽêãæýæèëå f : {0; 1} n −→ {0; 1} ŽïŽýãŽp 1 , p 2 , . . . , p n (n ∈ N) ùãèŽáâIJæïåãæï. òñêóùæŽ ïîñèŽá æóêâIJŽ àŽêïŽäôãîñèæ2 n ïðîæóëêæï ∗ éóëêâ üâöéŽîæðëIJæï ùýîæèæå. éðçæùâIJŽ êŽåâèæŽ, åñùýîæèæï õëãâè ïðîæóëêöæ Žîæï 1 (éŽöæê f = 1) Žê õëãâè ïðîæóëêöæ öâáâàæŽ0 (éŽöæê f = 0). ûæꎎôéáâà öâéåýãâãŽöæ ŽîïâIJëIJï ùýîæèæï n ïðîæëêæ,ïŽáŽù öâáâàæ Žîæï { 1.àŽêãæýæèëå ∗∗ B 1 , B 2 , . . . , B m àŽéëïŽýñèâIJâIJæ, åãæåâñèæ B i öââïŽIJŽéâIJŽ(B i1 , B i2 , . . . , B in ) áŽèŽàâIJñè n-ï, ïŽáŽù B ij = p ij Žê ⌉p ij , æéæïᎠéæýâáãæåp s = 1 åñ 0-ï, ùýîæèæï éëùâéñè ïðîæóëêöæ. éŽöæêf = B 1 ∨ B 2 ∨ · · · ∨ B m = (B 1 ∨ B 2 ∨ · · · ∨ B m ) ′′ == (B ′ 1 ∧ B ′ 2 ∧ · · · ∧ B ′ m) ′ = (B ′ 1 ↑ B ′ 2 ↑ · · · ↑ B ′ m),B ′ i = (B i1 ∧ B i2 ∧ · · · ∧ B in ) ′ = B i1 ↑ B i2 ↑ · · · ↑ B in .ñòîë éâðæù, åñ îëéâèæéâ B ij ùýîæèöæ öââïŽIJŽéâIJŽ 0-ï, éŽöæê ñêᎠŽãæôëå⌉p j àŽéëïŽýñèâIJŽ, îëéâèæù éææôâIJŽ öâéáâàêŽæîŽá:(⌉a) ≡ (a ↑ a).∗ õëãâè ïðîæóëêöæ öââïŽIJŽéâIJŽ îŽæéâ n ïæàîúæï ëîâñèï ᎠöâæùŽãï Žé ûõãæèæïåãæï òñêóùææïéêæöãêâèëIJŽï.

208ŽéîæàŽá, ûæêŽáŽáâIJŽ áŽéðçæùâIJñèæŽ. çëéìŽóðñîŽá âï øŽæûâîâIJŽ:Ž)m∨ ( ∧nf = B ij),f =i=1m↑ ⏐i=1j=1( n↑ ⏐j=1B ij).IJ) ŽêŽèëàæñîŽá, éëîàŽêæï çŽêëêâIJæï úŽèæå Ꭰêñèæï öâéùãâèæ ïðîæóëêâIJæïàŽêýæèãæå, éæãæôâIJåf =2 n ⏐−m↓i=1(∗)( n⏐ ↓ ⌉B ij). (∗∗)Žé îâäñèðŽðæáŽê àŽéëéáæêŽîâëIJï, îëé õëãâèæ àŽéëïŽýñèâIJŽ àŽéëõãŽêŽáæŽ,îŽù æéŽï êæöêŽãï, îëé p 1 , p 2 , . . . , p n -æå ûŽîéëóéêæèæ IJñèæï ŽèàâIJîæï âèâéâêðåŽîæùýãæŽ { 2 n .é Ž à Ž è æ å æ 1.3. éýëèëá ↑-æï àŽéëõâêâIJæå éæãæôâå f òñêóùææïûŽîéëáàâêŽ. üâöéŽîæðëIJæï ùýîæèæ æý. 1.5.Žôûâîæèæ éâåëáæï àŽéëõâêâIJæåj=1x y z t x y z t0 0 0 0 1 0 0 10 0 1 1 1 0 1 10 1 0 1 1 1 0 00 1 1 0 1 1 1 1ù ý î æ è æ 1.3.f = (x ′ ∧ y ′ ∧ z) ∨ (x ′ ∧ y ∧ z ′ ) ∨ (x ∧ y ′ ∧ z ′ ) ∨ (x ∧ y ′ ∧ z) ∨ (x ∧ y ∧ z).ŽéîæàŽá,f ′ = (x ′ ∧ y ′ ∧ z) ′ ∧ (x ′ ∧ y ∧ z ′ ) ′ ∧ (x ∧ y ′ ∧ z ′ ) ′ ∧ (x ∧ y ′ ∧ z) ′ ∧ (x ∧ y ∧ z) ′ ,Žéæðëéf = (x ′ ↑ y ′ ↑ z) ↑ (x ′ ↑ y ↑ z ′ ) ↑ (x ↑ y ′ ↑ z ′ ) ↑ (x ↑ y ′ ↑ z) ↑ (x ↑ y ↑ z),ïŽáŽùx ′ = x ↑ x, y = y ↑ y, z ′ = z ↑ z.∗∗ ãåóãŽå, (α 11 , . . . , α 1n ), . . . , (α m1 , . . . , α mn) Žîæï õãâèŽ êŽçîâIJæ, îëéâèäâáŽù f = 1,1 ≤ m < 2 n , õëãâèæ Žïâåæ ïæïðâéæïŽåãæï æàâIJŽ çëêæñêóùæŽ B i = B i1 ∧ · · · ∧ B in , ïŽáŽùB ik = p k , åñ α ik = 1 ᎠB ik =⌉p k , åñ α ik = 0 (k = 1, . . . , n). Žáãæèæ ïŽêŽýŽãæŽ, îëéf = B 1 ∨ · · · ∨ B m .

209(∗) àŽéëïŽýñèâIJŽï âûëáâIJŽ êëîéŽèñîæ òëîéæï áæäæñêóùæŽ. æï Žîæïçëêæñêóùææï áæäæñêóùæŽ (Žê Ꭰ{ ᎠŽê). (∗∗) àŽéëïŽýñèâIJŽï âûëáâIJŽ êëîéŽèñîæòëîéæï çëêæñêóùæŽ ∗ .áŽIJëèëï, Žôãêæöêëå, îëé éðçæùâIJŽï IJñèæï ŽèàâIJîŽöæ îëéâèïŽù Žóãïõëãâèåãæï 1 éêæöãêâèëIJŽ âûëáâIJŽ ðŽãðëèëàæŽ, ýëèë { àŽéëïŽýñèâIJŽï,îëéâèæù õëãâèåãæï æôâIJï 0 éêæöãêâèëIJŽï, âûëáâIJŽ { ûæꎎôéáâàëIJŽ.1.3. IJñèæï ŽèàâIJîæï äëàæâîåæ àŽéëõâêâIJŽ. éîŽãŽèæ ìîëIJèâéŽéŽåâéŽðæçŽöæ, çâîúëá, çëéIJæêŽðëîæçŽöæ, õãâèŽäâ ñçâå æýïêâIJŽ IJñèæï ŽèàâIJîâIJæïïŽöñŽèâIJæå. IJñèæï ŽèàâIJîâIJæ, Žàîâåãâ, àŽéëæõâêâIJŽ îâŽèñîæ ïæðñŽùææïéëáâèæîâIJæïŽï. âï éâåëáæ öâïŽúèâIJâèæŽ àŽéëõâêâIJñè æóêŽï òŽîåëáàŽãîùâèâIJñè ŽîââIJöæ, îëéâèåŽ çèŽïæòæçŽùæŽ öâñúèâIJâèæŽ. àŽêãæýæèëåâîå-âîåæ éŽåàŽêæ (àŽáŽéîåãâèæ ïóâéâIJæ).êŽý. 1.3ŽñùæèâIJèŽá ñêᎠŽôãêæöêëå, îëé øãâê öâéëãæòŽîàèâIJæå ïóâéâIJæï Žàâ-IJæïŽ áŽ àŽîáŽóéêæï éýëèëá àŽîçãâñèæ ïŽçæåýâIJæå. Žé ûæàêæï éæäŽêæŽ òëîéŽèñîæàŽéëçãèâãŽ. åŽãæáŽê ãêŽýëå îëàëî àŽéëæõâêâIJŽIJñèæï ŽèàâIJîŽ ïŽáâêâIJæïŽ áŽ øŽéîåãâèâIJæïŽàŽê öâáàâêæèæ ïóâéæï ŽôûâîŽàŽêïŽäôãîŽöæ.1.10 êŽýŽääâ àŽéëïŽýñèæ ïóâéŽ öâáàâIJŽ 5 øŽéîåãâèæïàŽê, çãâ-IJæï ûõŽîëïàŽê, êŽåñîŽïŽ ᎠöâéŽâîåâIJâèæ ïŽáâêâIJæïàŽê. áæŽàîŽéæáŽê øŽêï,îëé áâêæ éëúîŽëIJï øŽçâðæè þŽüãöæ, éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ A øŽéîåãâèæøŽçâðæèæ ᎠB ᎠC ëîæãâ øŽçâðæèæŽ, Žê D ᎠE ëîæãâ øŽçâðæèæŽ.âï ŽèàâIJîñèŽá Žïâ àŽéëæõñîâIJŽ:A ∧ ( (B ∧ C) ∨ (D ∧ E) ) .ãæáîâ àŽáŽãŽèå åâéæï àŽêýæèãŽäâ, àŽãŽçâåëå îŽéáâêæéâ öâêæöãêŽ àŽáŽéîåãâèâIJæŽêæâèâóðîñèæ ïóâéæï âèâéâêðâIJæï öâïŽýâIJ áæŽàîŽéŽöæ. ïóâéæïéæýâáãæå éýëèëá äâáŽ, øŽéîåãâèâIJæŽêæ êŽûæèæ àŽêïýãŽãáâIJŽ àŽêýæèñèæéŽàŽèæåâIJæïŽàŽê. éŽöŽïŽáŽéâ, öâïŽúèâIJâèæŽ óãâᎠéýŽîâ Žîù àŽêãæýæèëå.∗ êëîéŽèñîæ òëîéæï çëêæñêóùæŽ Žîæïf =2 n ∧−mi=1( ∨ n )⌉B ij (æý. (∗∗)).j=1

210ïóâéæï öæàêæå äëàæâîåæ øŽéîåãâèæ öâïŽúèâIJâèæ æõëï ñîåæâîåáŽéëçæáâ-IJñèæ (ïñî. 1.11a) ᎠöâæúèâIJŽ æéñöŽëê æïâ, îëé îëáâïŽù âîåæ øŽîåñèæŽ,éâëîâ ŽñùæèâIJèŽá ñêᎠæõëï àŽéëîåñèæ (ïñî. 1.11b).êŽý. 1.4úæîæåŽáæ éâåëáæ øŽéîåãâèæŽ àŽêèŽàâIJæï, Žîæï éæéáâãîëIJæå öââîåâIJŽ(ïñî. 1.11c), îŽïŽù öââïŽIJŽéâIJŽ \áŽ" (â. æ. A ∧ B) ᎠìŽîŽèâèñîæ öââîåâIJŽ(ïñî. 1.11d) \Žê"-æï öâïŽIJŽéæïæ (â. æ. A ∨ B). öâãêæöêëå IJëèëï, îëéáæŽàîŽéŽäâ Žîæï äâáéâðæ øŽéîåãâèâIJæù ᎠéëýâîýâIJñèëIJæïåãæï æïæêæ Žôãêæöêëåæàæãâ \ïŽýâèâIJæå" (ïñî. 1.11e). ŽéîæàŽá, êâIJæïéæâîæ ëìâîŽùæŽ IJñèæï

211ŽèàâIJîŽöæ öâæúèâIJŽ ûŽîéëãŽáàæêëå ïóâéæï ïŽýæå Ꭰìæîæóæå. ŽéîæàŽá, âçãæãŽèâêðñîæïóâéæï éæïŽôâIJŽá ïŽüæîëŽ ïŽûõæïæ ïóâéæáŽê àŽáŽãæáâå IJñèæïŽèàâIJîæï öâïŽIJŽéæï àŽéëïŽýñèâIJŽäâ, öâéáâà ñòîë éŽîðæã àŽéëïŽýñèâIJŽäâ,áŽIJëèëï, éæï öâïŽIJŽéæï ïóâéŽäâ.éŽàŽèæåæ 1.4. 1.12 ïóâéŽ àŽãŽéŽîðæãëå. éŽï öââïŽIJŽéâIJŽ öâéáâàæ àŽéëïŽýñèâIJŽ:X ∧ ((( (X ′ ∧ Y ) ∨ (Y ′ ∧ Z) ) ∧ W ′) ∨ (Z ∧ X ∧ W ) ) == X ∧ ( (X ′ ∧ Y ∧ W ′ ) ∨ (Y ′ ∧ Z ∧ W ′ ) ∨ (Z ∧ X ∧ W ) ) == (X ∧ X ′ ∧ Y ∧ W ′ )(X ∧ Y ′ ∧ Z ∧ w ′ )(X ∧ Z ∧ Z ∧ W ) == (X ∧ Y ′ ∧ Z ∧ W ′ ) ∨ (X ∧ Z ∧ W ) = (X ∧ Z) ∧ ((Y ′ ∧ W ′ ) ∨ W ).IJëèë àŽéëïŽýñèâIJŽ öââïŽIJŽéâIJŽ 1.13-ï.êŽý. 1.5êŽý. 1.6ŽýèŽ àŽáŽãæáâå çëéIJæêæîâIJñè ïóâéâIJäâ. Žïâå ïóâéŽöæ éåŽãŽîæ çëéìëêâêðâIJæŽèâéâêðâIJæŽ. õãâèŽäâ ñéŽîðæãâïæ âèâéâêðæ Žîæï n öâïŽïãèâèæïŽ áŽ màŽéëïŽïãèâèæï éóëêâ éëûõëIJæèëIJŽ (m, n ∈ N). õëãâè öâïãèŽäâ àŽŽóðæñîáâIJŽâîå-âîåæ ëîæ ïæàêŽèæáŽê. æïæêæ ŽôêæöêŽãï, îëàëîù 0 Ꭰ1, éŽàîŽéïŽâîåëá õëãâèæ ëîæ àŽêïýãŽãâIJñè éêæöãêâèëIJŽï ïæïðâéŽù àŽéëáàâIJŽ. îâ-ŽèñîŽá øãâê ïæàêŽèæ àãŽóãï 0 ãëèðæï éŽýèëIJèëIJŽöæ (éŽàŽèæåŽá, 0-áŽê0,8 ãëèðŽéáâ), îëéâèïŽù éæãæøêâãå 0-Žá, Ꭰéâëîâ { 4 ãëèðæï éæáŽéëöæ(3-áŽê 4-éáâ), îëéâèïŽù éæãæøêâãå 1-Žá. ŽóâáŽê àŽêæéŽîðâIJŽ Žéëïñèæ éêæöãêâèëIJâIJæ(0 Ꭰ1). öâæúèâIJŽ éëàãâøãâêëï, îëé {0, 1} n −→ {0, 1} õãâèŽ òñêóùææïûŽîéëïŽáàâêŽá áŽàãüæîáâIJŽ ïŽçéŽëá IJâãîæ àŽêïýãŽãâIJñèæ âèâéâêðæ,

212éŽàîŽé âï Žï⠎. áîëæï áŽõëãêâIJæï Žîéóëêâ ïóâéâIJï öââïŽIJŽéâIJŽ âîåŽáâîåæðæìæïâèâéâêðâIJæ. âï ìæîáŽìæî Žî øŽêï, éŽàîŽé ïóâéæï ïðîñóðñîŽ ñòîëùýŽáæ àŽýáâIJŽ åñ àŽêãæýæèŽãå 1.14-äâ àŽéëïŽýñè ïŽéæ ðæìæï âèâéâêðâIJï.êŽý. 1.7éŽîðæãæŽ âèâéâêðæ \ŽîŽ" éæïæ éëóéâáâIJŽ ýáâIJŽ öâéáâàêŽæîŽác =⌉a (c = ŽîŽ a)1.6. ùýîæèæï ïŽöñŽèâIJæå.ùýîæèæ 1.4.INPUT(a)OUTPUT(c)0 11 0ŽêŽèëàæñîŽá, \áŽ" Ꭰ\Žê"-æï âèâéâêðâIJæ àŽêæïŽäôãîâIJŽ 1.7. ùýîæèæå.a b ᎠŽê0 0 0 00 1 0 11 0 0 11 1 1 1ù ý î æ è æ 1.5.\áŽ" Ꭰ\Žê" âèâéâêðâIJï öâæúèâIJŽ ßóëêáâï îŽéáâêæéâ öâïŽïãèâèæ. Žé öâéåýãâãŽöææïæêæ àŽêæéŽîðâIJŽ ŽêŽèëàæñîŽá, Žêñ \áŽ" âèâéâêðæï àŽéëïãèæïïæàêŽèæ 1-æŽ éŽöæê Ꭰéýëèëá éŽöæê, åñ îëéâèæéâ öâéëïñèæ ïæàêŽèæ Žîæï1. ŽèàâIJîñèŽá \Žî" âèâéâêðæï àŽéëïãèæï ïæàêŽèï Žóãï ïŽýâ: ⌉ (öâïãèæïïæàêŽèæ), ýëèë \áŽ" âèâéâêðæï àŽéëïãèæï ïæàêŽèï Žóãï ïŽýâ: (ìæîãâèæ öâïãèæïïæàêŽèæ)∧(éâëîâ öâïãèæï ïæàêŽèæ)∧ . . . áŽIJëèëï, \Žê"-æï àŽéëïãèæïïæàêŽèï Žóãï ïŽýâ: (ìæîãâèæ öâïãèæï ïæàêŽèæ)∨(éâëîâ öâïãèæï ïæàêŽèæ)∨ . . .ñòîë îåñèæ òñêóùæâIJæï ûŽîéëïŽáàâêŽá, âèâéâêðâIJæ öâæúèâIJŽ \àŽãŽâîåæ-Žêëå" ïýãŽáŽïý㎠àäâIJæå. éŽàŽèæåŽá, àŽêãæýæèëå ïóâéŽ:

213êŽý. 1.8ŽéæðëéêŽý. 1.9ïóâéæï àŽïŽïãèâèæáŽê \ñçŽê" áŽIJîñêâIJæïŽï éæãæôâIJåx = (( a ∨ (a ∧ b ′ ) ) ∧ (y) ) ′,y = (b ∧ c) ′ ,x = (( (a ∨ a) ∧ (a ∨ b ′ ) ) ∧ (b ∧ c) ′) ′=((a ∧ (a ∨ b ′ )) ∧ (b ∧ c) ′) ′= (a ∧ (b ∧ c) ′ ) ′(öåŽêåóéæï çŽêëêæ).ŽéîæàŽá, âîå-âîåæ õãâèŽäâ àŽéŽîðæãâIJñèæ ïóâéŽ àŽéëïŽýñèæŽ 1.9 êŽýŽääâ.éŽöŽïŽáŽéâ, IJñèæï ŽèàâIJîâIJæ öâàãæúèæŽ àŽéëãæõâêëå îåñèæ ïóâéâIJæïàŽïŽéŽðæãâIJèŽá ᎠéŽåæ ŽêŽèæäæïŽåãæï. àŽêýæèñèæ éŽàŽèæåæ àãæøãêâIJï îëàëîñêᎠøŽãŽðŽîëå âï ìîëùâïæ.ŽîïâIJëIJï æïâåæ éëûõëIJæèëIJâIJæ, îëéèâIJæù Žýáâêâê ïý㎠èëàæçñîæ ëìâîŽùæâIJæïîâŽèæäâIJŽï. õãâèŽäâ éêæöãêâèëãŽêæŽ \ŽîŽ áŽ" Ꭰ\ŽîŽ Žê" âèâéâêðâIJæ(æý. êŽý. 1.10).é Ž à Ž è æ å æ 1.5. öâãêæöêëå, îëé f-æï IJëèë òëîéŽöæ òîøýæèâIJöæéëåŽãïâIJñèæ õëãâèæ àŽéëïŽýñèâIJŽ Žîæï ñçŽêŽïçêâèæ âèâéâêðæï öâïŽïãèâèæ1.11 êŽýŽääâ ᎠöâïŽúèâIJâèæŽ àŽéëåãèæè æóêŽï ûæêŽéáâIJŽîâ âèâéâêðâIJæïïŽöñŽèâIJæå. ûõãâðæèæå àŽéëïŽýñèæ ïâéæï êŽûæèæìîŽóðæçñèŽá ŽîŽŽ ïŽüæîë,îŽáàŽê IJâãîæ âèâóðîëêñèæ âèâéâêðæïŽåãæï àŽïŽïãèâèæï áŽéŽðâIJŽ éææôûâãŽáŽéýéŽîâ àŽïŽïãèâèæå. Žôãêæöêëå, îëé éæôâIJñèæ ïóâéŽ äëàŽáŽá

214êŽý. 1.10àŽéëõâêâIJñèæ âèâéâêðâIJæï îŽëáòâêëIJæå Žî Žîæï éæêæéŽèñîæ. IJëèëï àŽêãæýæèëåëîæ éŽàŽèæåæ, îëéâèæù ŽøãâêâIJï, åñ îëàëî àŽéëæõâêâIJŽ IJñèæïŽèàâIJîŽ ïý㎠ŽéëùŽêâIJöæ.é Ž à Ž è æ å æ 1.6. áŽãñöãŽå, îëé ïŽéŽîåèæŽêæŽ öâéáâàæ ûæêŽáŽáâ-IJâIJæ:1. éëêŽùâéåŽ ïðîñóðñîæï ùëáêŽ ŽñùæèâIJâèæŽ àŽûãîåêæèæ ðãæêæïïîñèõëòæïŽåãæï;2. éýëèëá ìîëàîŽéæîâIJæï àŽéëùáæèâIJŽï öâñúèæŽ öâóéêŽïàŽûãîåêæèæ ðãæêæ.3. çëéìæèŽðëîæï áŽïŽûâîŽá ïŽüæîëŽ ŽéëùŽêæï ŽêŽèæäæï ñêŽîæ.4. àŽñûãîåêâè ðãæêï Žî öâñúèæŽ ŽéëùŽêæï ŽêŽèæäæ.5. õãâèŽ, ãæïŽù áŽñûâîæŽ ïðîñóðñîñèæ ìîëàîŽéâIJæ, öâæúèâIJŽ øŽæåãŽèëïîëàëîù àŽéëùáæèæ ìîëàîŽéæïðæ.6. öâïŽúèâIJâèæŽ åñ ŽîŽ Žé ûæêŽéôãîâIJæáŽê áŽãŽáàæêëå óãâéëå øŽéëåãèæèæáâIJñèâIJâIJæï ïŽéŽîåèæŽêëIJŽ:1 ′ . ïðîñóðñîñèæ ìîëàîŽéâIJæï ûâïæï àŽéëùáæèâIJŽ ŽñùæèâIJâèæŽæéæïŽåãæï, îëé öâúèëå çëéìæèŽðëîæï áŽûâîŽ.2 ′ . éëêŽùâéåŽ ïðîóñðñîæï ùëáêŽ êŽûæèæŽ ìîëàîŽéæîâIJæï àŽéëùáæèâIJŽ.3 ′ . ŽéëùŽêŽåŽ ŽêŽèæäæ öâñúèâIJâèæŽ éŽååãæï, ãæêù ñàñèâIJâèßõëòïéëêŽùâéåŽ ïðîñóðñîâIJï.4 ′ . àŽéëùáæè ìîëàîŽéæïðï, îëéâèæù ûâîᎠïðîñóðñîñè ìîëàîŽéâIJï,öâñúèæŽ ŽéëùŽêâIJæï ŽêŽèæäæ ᎠŽóãï àŽûãîåêæèæ ðãæêæ,öâñúèæŽ áŽûâîëï çëéìæèŽðëîæ?Žé çæåýãâIJäâ ìŽïñýâIJæï àŽïŽùâéŽá, øãâê öâàãâúèë àŽéëàãâçãèæŽ éðçæùâIJŽåŽèëàæçñîæ öâáâàâIJæ, éŽàîŽé ñòîë îåñè ïæðñŽùæâIJöæ ñéþëIJâïæŽãæïŽîàâIJèëå øãâêæ ùëáêæå IJñèæï ŽèàâIJîâIJöæ. åŽãáŽìæîãâèŽá éëãŽýáæêëåøãâêæ éðçæùâIJñèâIJŽåŽ çëáæîâIJŽ. ãåóãŽå,E Žîæï õãâèŽ ìîëàîŽéæïðåŽ ïæéîŽãèâ.

215êŽý. 1.11U æïæêæ éŽå öëîæï, ãæêù æùæï éëêŽùâéåŽ ïðîñóðñîŽ.V æïæêæ éŽå öëîæï, ãæïŽù Žóãï àŽûãîåêæèæ ðãæêæ.W æïæêæ éŽå öëîæï, ãëêù Žîæï àŽéëùáæèæ ìîëàîŽéæïðæ.X æïæêæ éŽå öëîæï, ãæïŽù öââúèâIJŽ çëéìæèŽðëîæï áŽûâîŽ.Y æïæêæ éŽå öëîæï, ãæïŽù öâñúèæŽ ŽéëùŽêŽåŽ ŽêŽèæäæ.Z æïæêæ éŽå öëîæï, ãæïŽù öâñúèæŽ ïðîñóðñîñèæ ìîëàîŽéâIJæï ûâîŽ,éŽöæê:1) U ⊇ V ;

2162) W ⊇ V ;3) X ⊆ Y ;4) V ⊇ Y ;5) W ⊆ Z;1 ′ ) Z ⊇ X;2 ′ ) U ⊇ W ;3 ′ ) Y ⊆ U;4 ′ ) W ∩ Z ∩ Y ∩ V 4 ⊇ X.Žáãæèæ ïŽêŽýŽãæŽ, îëé (1 ′ ) ñöñŽèëá àŽéëéáæêŽîâëIJï (3), (4), (2) áŽ(5)-áŽê. ŽêŽèëàæñîŽá (3 ′ ) àŽéëéáæêŽîâëIJï (40) Ꭰ(1)-áŽê. Žïâãâ,W ∩ Z ∩ Y ∩ V = W ∩ Y ∩ V = Y ∩ V = Y ⊇ X.ŽéîæàŽá, (4 ′ ) Žïâãâ ïŽéŽîåèæŽêæŽ. (2 ′ ) Žî àŽéëéáæêŽîâëIJï (1){(5)-áŽê,îŽáàŽê øãâê ãæùæå éýëèëá W ⊆ Z.é Ž à Ž è æ å æ 1.7. ŽèæïŽ ŽéIJëIJï: IJŽîIJŽîŽ ᎠçèŽîŽ ŽéIJëIJâê ïæéŽîåèâï.çèŽîŽ ŽéIJëIJï: âèäŽ áŽ òèëîŽ ëîæãâ Žê ïæéŽîåèâï ŽéIJëIJï Žê ëîæãâðõñæï. éâëîâï éýîæã, áâIJæ òæóîëIJâê, îëé ñçæáñîâï öâéåýãâãŽöæ Žê çèŽîŽ ŽêIJŽîIJŽîŽ ŽéIJëIJï ïæéŽîåèâï, îŽáàŽê IJŽîIJŽîŽ æîûéñêâIJŽ, îëé éýëèëá âîåâîåæŽéŽîåŽèæ { ŽèæïŽ Žê òèëîŽ. âèäŽ òæóîëIJï, îëé ŽèæïŽ áŽ IJŽîIJŽîŽõëãâèåãæï ŽéIJëIJâê ïæéŽîåèâï, éŽàîŽé, òèëîŽ áŽîûéñêâIJñèæŽ, îëé IJŽîIJŽîŽáŽ çèŽîŽ ëîæãâ âîåŽá Žî ŽéIJëIJï ïæéŽîåèâï. ãæï ñêᎠãâêáëå?àŽêãæýæèëå éðçæùâIJŽåŽ ïæéîŽãèâ:\ŽèæïŽ (Žî) ŽéIJëIJï ïæéŽîåèâï".\IJŽîIJŽîŽ (Žî) ŽéIJëIJï ïæéŽîåèâï". . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ᎠãåóãŽå, A öâæùŽãï éýëèëá æé éðçæùâIJâIJï, ïŽáŽù ŽèæïŽ ŽéIJëIJï ïæéŽîåèâï,ŽêŽèëàæñîŽá àŽêãéŽîðëå B, C, D, E ᎠF ïæéîŽãèââIJæ; åñS ⊆ A ∩ B ′′ ∩ C Žîæï ŽîŽùŽîæâèæ ïæéîŽãèâ, éŽöæê S øŽîåñèæŽ æïâå éðçæùâIJâIJöæ,îëàëîæùŽŽ\ŽèæïŽ ŽéIJëIJï ïæéŽîåèâï",\IJŽîIJŽîŽ ðõñæï",\çèŽîŽ ŽéIJëIJï ïæéŽîåèâï".ìæîãâèæ éðçæùâIJæáŽê àŽéëéáæêŽîâëIJï, îëé ŽèæïŽ ŽéIJëIJï ïæéŽîåèâï,éŽöæê ïæéŽîåèâï ŽéIJëIJï IJŽîIJŽîŽ ᎠçèŽðîŽù.Žêñ åñ x ∈ A, éŽöæê x ∈ B ∩ C, â. æ.A ⊆ B ∩ C. (1)

217ïý㎠éðçæùâIJñèâèIJâIJæù øŽæûâîâIJŽ ŽêŽèëàæñîŽáC ⊆ (E ∩ F ) ∪ (E ′ ∩ F ′ ), (2)D ⊆ A ∪ B, (3)B ⊆ (E ∪ F ) \ (E ∩ F ) = (E ∩ F ′ ) ∪ (E ′ ∩ F ), (4)E ⊆ A ∩ B, (5)F ⊆ (B ∩ C) ′ . (6)(2), (4) Ꭰ(1)-áŽê àŽéëéáæêŽîâëIJï, îëé B ∩ C = ∅, Žéæðëé A = ∅, éŽöæê(5)-áŽê éæãæôâIJå, îëé E = ∅. ŽóŽéáâ øãâêæ éïþâèëIJŽ âòâóðñîæ æõë, îŽáàŽêãàñèæïýéëIJáæå, îëé éŽîåŽèæ ŽáŽéæŽêæ ŽéIJëIJï ïæéŽîåèâï. éŽàîŽé ŽîŽòâîæãæùæå æéæï öâïŽýâIJ, åñ îŽï ŽéIJëIJï éŽðõñŽîŽ. åñ öâéëãæòŽîàèâIJæå æé òŽóðæå,îŽïŽù ŽéIJëIJï éŽðõñŽîŽ, õãâèŽòâîæ ðõñæèæŽ, éŽöæê (1){(6) øŽîåãâIJæöâàãæúèæŽ öâãùãŽèëå âçãæãŽèâêðñîæ ïæéIJëèëâIJæå.éŽàŽèæåŽá, åñ IJŽîIJŽîŽ ᎠçèŽîŽ éŽîåèæŽ éåóéâèâIJæŽ, éŽöæê îŽáàŽêŽèæïŽ ŽéIJëIJï, îëé âï ŽïâŽ, Žéæðëé ŽèæïŽù éŽîåèæïéëóéâáæŽ, â. æ.A = B ∩ B,C = (E ∩ F ) ∪ (E ′ ∩ F ′ ),D = A ∪ B,B = (E ∪ F ′ ) ∪ (E ′ ∩ F ),E = A ∩ B,F = (B ∩ C) ′ .(1a)(2a)(3a)(4a)(5a)(6a)îëàëîù äâéëå, A = ∅, E = ∅ ᎠŽéæðëé ŽèæïŽ áŽ âèäŽ ŽîŽïëáâïŽéIJëIJâê ïæéŽîåèâï. (2a), (4a) Ꭰ(6a)-áŽê àŽéëéáæêŽîâëIJï, îëé F = E, â. æ.(3a), (4a)-áŽê éæãæôâIJå, îëé D = B = F = E. ŽóâáŽê (2a)-ï áŽýéŽîâIJæåéæãæôâIJå, îëé C = ∅, ŽóâáŽê àŽéëéáæêŽîâëIJï, îëé öâïŽúèâIJâèæŽ éýëèëáâîåæ éðçæùâIJŽ: x ∈ A ′ ∩ B ∩ C ′ ∩ D ∩ E ′ ∩ F . âï êæöêŽãï, îëé ŽèæïŽ, çèŽîŽáŽ âèäŽ ðõñæŽê, ýëèë IJŽîIJŽîŽ, áâIJæ ᎠòèëîŽ ïæéîŽåèâï ŽéIJëIJâê.ï Ž ã Ž î þ æ ö ë 1.1.1. Žøãâêâå, îëé (B, ∗, +, ′ ) IJñèæï ŽèàâIJîŽöæ õëãâèæ x, y, z ∈ B-åãæïïŽéŽîåèæŽê掎) (x + y) ∗ (x ′ + y) = y;IJ) (z + x) ∗ (z ′ + y) = (z ∗ y) + (z ′ ∗ x);à) ((x ∗ z) + (y ∗ z ′ )) ′ = (x ′ ∗ z) + (y ′ ∗ z ′ ).2. ᎎéðçæùâå, îëé åñ (B, ∗, +, ′ )-öæ ïîñèáâIJŽ åŽêŽòŽîáëIJŽ x ∗ y =x ∗ z, éŽöæê x + y = x + z.3. 1.12 Ꭰ1.13 ïóâéâIJæáŽê éææôâå éŽåæ âçãæãŽèâêðñîæ ñòîë \éŽîðæãæ"ïóâéâIJæ.

218êŽý. 1.12êŽý. 1.13êŽý. 1.144. 1.14 ïóâéæïŽåãæï ŽŽàâå ùýîæèæ a Ꭰb öâïŽïãèâèæïŽåãæï x Ꭰy àŽéëïŽåãèâèæïöâïŽIJŽéæïæ òñêóùææïŽåãæï. ŽŽàâå ëîæ ïóâéŽ 1.8 ùýîæèæï àŽéëõâêâIJæå.5. ŽŽàâå \ŽîŽ Ꭰ" ëìâîŽùææï öâïŽIJŽéæïæ ïóâéŽ, îëéâèïŽù Žóãï ëåýæöâïŽïãèâèæ Žê àŽéëïŽïãèâèäâ âîåæŽêæŽ éŽöæê Ꭰéýëèëá éŽöæê, îëùŽ õãâèŽöâïŽïãèâèæ êñèæŽ.

2196. ãåóãŽå, ïîñèáâIJŽ:Ž) õãâèŽ éîIJëèâèæ æéìñèïñîæŽ;IJ) õãâèŽ çŽîàæ ìîëàîŽéæïðæ éâèŽêóëèæñîæŽ.à) ŽîŽãæꎎ æéìñèïñîæ ᎠéâèŽêóëèæñîæ.á) éçæåýãâèæ éâèŽêóëèæñîæŽ.Žîæï åñ ŽîŽ éçæåýãâèæ çŽîàæ ìîëàîŽéæïðæ ᎠŽê éîIJëèâèæ?7. ïŽâîåŽöëîæïë çëêòâîâêùææï ëîàŽêæäŽðëîâIJéŽ çëéâîùæñèæ æêðâîâïâIJæïåŽãæáŽê ŽùæèâIJæï éæäêæå, àŽáŽûõãæðâï, îëé çëéìæñðâîâIJæï õãâèŽéûŽîéëâIJâèæ æïŽîàâIJèâIJᎠéýëèëá âîåæ éŽôŽäææå. åãæåëê éûŽîéëâIJèâ-IJæ æóêâIJëáêâê ìŽïñýæïéàâIJèâIJæ æéæï åŽëIJŽäâ, åñ ãæê éææôâIJᎠéëêŽûæèâëIJŽïûŽîéëéŽáàâêèëIJŽöæ. ýñåéŽ âãîëìñèéŽ áŽ ýñåéŽ ŽéâîæçñèéŽ çëéìŽêæŽé åŽêýéëIJŽàŽêŽùýŽáŽ éëêŽûæèâëIJŽäâ (Žôãêæöêëå æïæêæ A, B, C, D, E ᎠF , G,H, I, J, öâïŽIJŽéæïŽá). éŽàîŽé ïŽãŽüîë ìëèæðæçæïŽ áŽ çëêðîŽóðæï ãŽèáâ-IJñèâIJâIJæï àŽéë, ŽñùæèâIJâèæŽ àŽîçãâñèæŽ öâäôñáãâIJæ. âãîëìâèâIJæ æåýëãâê,îëé G ᎠI-ï Žî öâñúèæŽå âîåáîëñèŽá éëêŽûæèâëIJæï éæôâIJŽ. ŽêŽèëàæñîŽá,âîåáîëñèŽá Žî öâñúèæŽå F , G ᎠJ-ï. ŽéâîæçâèâIJæ æåýëãâê, îëéA ᎠD ãâî éææôâIJâê éëêŽûæèâëIJŽï, ãæáîâ G Žî éææôâIJï éëêŽûæèâëIJŽï.ŽêŽèëàæñîŽá àŽéëæîæùýâIJŽ C ᎠD-ï éëêŽûæèâëIJŽ G-éáâ. åñ E éëêŽûæèâëIJï,éŽöæê ñêᎠéëêŽûæèâëIJáâï J-ù. éŽàîŽé, åñ ëîæãâ éëêŽûæèâëIJï, éŽöæêB ãâî éææôâIJï éëêŽûæèâëIJŽï, ᎠIJëèëï, B ᎠC-ï âîåŽá Žî öâñúèæŽåéëêŽûæèëâIJæï éæôâIJŽ. öâïŽúèâIJâèæŽ åñ ŽîŽ åæåëâñèæ þàñòæáŽê ïŽé-ïŽéæéëêŽûæèæï öâîøâãŽ, æïâ, îëé Žî áŽæôãâï ìæîëIJâIJæ Ꭰîëàëî?§ 2. øŽçâðæèæ êŽýâãŽîîàëèâIJæà Ž ê é Ž î ð â IJ Ž . S ïæéîŽãèâï ⊛ Ꭰ⊕ IJæêŽîñèæ ëìâîŽùæâIJæå,âûëáâIJŽ øŽçâðæèæ êŽýâãŽîîàëèæ, åñ ïîñèáâIJŽ:Ž) ⊕ ëìâîŽùæŽ ŽïëùæŽùæñîæŽ;IJ) ⊕ ëìâîŽùææï éæéŽîå ŽîïâIJëIJï âîåâñèëãŽêæ 0 âèâéâêðæ;à) ⊛ ëìâîŽùæŽ ŽïëùæŽùæñîæŽ;á) ⊛ ëìâîŽùææï éæéŽîå ŽîïâIJëIJï âîåâñèëãŽêæ 1 âèâéâêðæ;â) ∀ x ∈ S-åãæï x ⊛ 0 = 0 = 0 ⊛ x;ã) ⊕ ëìâîŽùæŽ çëéñðŽùæñîæŽ x ⊕ y = y ⊕ x ∀ x, y ∈ S.ä) ⊕ ëìâîŽùæŽ æáâéìëðâêðñîæŽ x ⊕ x = x ∀ x ∈ S.å) ⊛ Ꭰ⊕ ëìâîŽùæâIJæ áæïðîæIJñùæñèæŽ,x ⊛ (y ⊕ z) = (x ⊛ y) ⊕ (x ⊛ z) ∀ x, y, z ∈ S,(x ⊕ y) ⊛ z = (x ⊛ z) ⊕ (y ⊛ z) ∀ x, y, z ∈ S.

220æ) S-æï ïŽïîñèæ îŽëáâêëIJŽ âèâéâêðâIJæï þŽéæ ŽîïâIJëIJï Ꭰæï âîåŽáâîåæŽ(â. æ. ŽîŽŽ áŽéëçæáâIJñèæŽ öâþŽéâIJæï îæàäâ).ç) ⊛ ëìâîŽùæŽ áæïðîæIJñùæñèæŽ ñïŽïîñèë þŽéæï éæéŽîå:( ∑ ) ( ∑ )a i ⊛ b j = ∑ (a i ⊛ b j ),iji,jïŽáŽù∑∑a i = a 1 ⊕ a 2 ⊕ + · · · , b j = b 1 ⊕ b 2 ⊕ + · · · , .iâï ïŽçéŽëá ñùêŽñîæ ïðîñóðñîŽ ýöæîŽá àŽéëæõâêâIJŽ àîŽòåŽ øŽçâðãæï ŽèàëîæåéâIJöæ.øŽçâðæè êŽýâãŽîîàëèâIJöæ öâáâàâIJæï ïŽæèñïðîŽùæëá öâãŽáŽîëå(Z 2 , ·, +) Ꭰ(Z 2 , ∧, ∨), ïŽáŽù ëìâîŽùæâIJæ àŽêéŽîðâIJñèæŽ 2.1 ùýîæèæï ïŽöñ-ŽèâIJæå:∗ 0 10 0 01 0 1+ 0 10 0 11 1 0j∧ 0 10 0 01 0 1ù ý î æ è æ 2.1.∨ 0 10 0 11 1 1àŽêãæýæèëå øŽçâðãæï ëìâîŽùææï àŽêéŽîðâIJŽ. Žé ëìâîŽùææï öâáâàæ Žôãêæöêëåa ∗ -æå:∞∑a ∗ = a i ,i=0a 0 = 1, a 1 = a, a 2 = a ∗ a Ꭰa i = a i−1 ∗ a((Z 2 , ∧, ∨)-öæ ∨ Žîæï \+" Ꭰ∧ Žîæï \·"). øŽçâðæè êŽýâãŽîîàëèöæ Žé àŽêéŽîðâIJŽïŽäîæ Žóãï, 1 ∗ = 1 0 ∨ 1 1 ∨ 1 2 ∨ ∨ · · · = 1 ∨ 1 ∨ 1 ∨ · · · = 1.(ä áŽ å ŽóïæëéâIJæï úŽèæå) éŽàîŽé åñ öâãùáâIJæå, æàæãâ àŽêãéŽîðëå(Z 2 , ·, +)-öæ éæãæôâIJå, îëé1 ∗ = 1 0 + 1 1 + 1 2 + · · · = 1 + 1 + 1 + · · · .âï þŽéæ öâàãæúèæŽ áŽåãèæèæ öâéáâàêŽæîŽá:1 + 1} {{ } + } 1 {{ + 1 } + · · · = 0 + 0 + · · · = 0Žê1 + 1 + 1 + 1 + 1 + 1 + 1 + · · · = 1 + 0 + 0 + · · · = 1.â. æ. þŽéæ Žî Žîæï àŽêéŽîðâIJñèæ, éŽöŽïŽáŽéâ, øŽçâðãæï ëìâîŽùæŽï Žäîæ ŽîŽŽóãï.ï Ž ã Ž î þ æ ö ë 2.1. öâŽéëûéâå, îëé ({∅, E}, ∩, ∪) Žîæï øŽçâðæèæêŽýâãŽîîàëèæ.

gamoyenebani:kodireba, kriptografia, kompiuterulitomografia, globaluri pozicionirebissistemakodireba1. SesavalivixilavT informaciis ,,xmauriani” arxiT gadacemis problemas,kerZod orobiT kodirebasa da dekodirebas.tipiuri situacia aseTia: vTqvaT gvinda gadvceT Setyobinebaromelic aris romeliRac alfabetis ramodenime simbolosaganSemdgari mwkrivi. es alfabeti SeZleba iyos Semgdariasoebisgan, aTobiTi an orobiTi ricxvebisgan, an sxva raimesimboloebisgan. aseTi gadacema SeiZleba xorcieldebodeserTi kompiuteridan meoreSi, an iyos, vTqvaT, satelefonisaubari, an telegadacema da sxva.praqtikulad yvela arxi araidealuria im gagebiT, rom gadacemuligadacemuli simbolo SeiZleba aranulovani albaTobiTarasworad iyos miRebuli arxSi arsebuli “xmauriT” damaxinjebisgamo.magaliTad, vTqvaT gadasacemia ricxvi 6. gadacemisas es ricxviSeiZleba damaxinjdes da abonentma miiRos ricxvi 7. ararsebobs meTodi, romliTac SeiZleba imis amocnoba, rom 7-isnacvlad 6 unda yofiliyo. am problemis movla ase SeiZleba.gadmcemi da mimRebi unda SeTanxmdnen kodis gasaRebze –vTqvaT ricxvze 234. gadamcemi axdens gadasacemi ricxviskodirebas: gadasacem ricxvs a mimdevrobiT amravlebs 2-ze, 3-ze,4-ze da a-is nacvlad agzavnis sam ricxvs – 2a, 3a, 4a. vTqvaT Tugadasacemia ricxvi 6, maSin gadacems sameuls12, 18, 24.abonenti axdens miRebul ricxvebis dekodirebas: hyofs maTmimdevrobiT 2-ze, 3-ze, 4-ze. Tu gzavnili uSecdomod Cavida,samive SemTxvevaSi gayofisas miiReba 6, e.i gamogzavnili ricxviyofila 6. magram Tu gzavnili gzaSi damaxinjda da vTqvaTabonentma miiRo 12, 19, 24, maSin dekodirebisas abonenti miiRebssameuls6, 6,33..., 6.221

advili misaxvedria, rom Sua ricxvi 6,33... Secdomis Sedegia, e.i.gamogzavnili yofila 6. cxadia, Tu damaxinjda erTi ki ara,ori an samive ricxvi, maSin gamogzavnili ricxvis amocnobaSeuZlebeli iqneba. magram ori an sami Secdomis albaTobagacilebiT naklebia, videre erTisa, erT Secdomas ki kodirebises meTodi advilad amoicnobs da gaasworebs. kodireba miTufro kargad imuSavebs, rac ufro grZeli iqneba koduri sityva.magaliTad xuTcifriani koduri sityva amoicnobs dagaasworebs 2 Secdomas, Svidcifriani – sams.kodirebis es nimuSi mxolod sailustraciod mogvyavs.sinamdvileSi is ar gamodgeba orobiTi, e.i 0 da 1-iT Sedgeniligzavnilebis gadasacemad, kompiuterebi ki swored am orisimbolosgan Semdgar alfabets iyeneben. sinamdvileSigamoiyeneba gacilebiT ufro faqizi kodebi, romlebsac qvemoTaRvwerT.amgvarad, informaciis gadacemis zogadi sqema aseTia:gasagzavni informacia kodirebuli informacia miRebuliinformacia dekodirebuli informacia.kodebi or klasad iyofa: Secdomebis aRmomCeni da Secdomebisgamsworebeli kodebi.magaliTad, zemoT aRwerili kodirebis nimuSSi koduri sityvaorniSna rom yofiliyo, maSin es kodi mxolod SecdomebisaRmomCeni iqneboda. marTlac, vTqvaT koduri sityvaa 23, e.i. a-snacvlad igzavneba ricxvTa wyvili 2a, 3a. igive 6-iani kodirdebawyvilad 12, 18. vTqvaT pirveli 1-iani damaxinjda da abonentmamiiRo wyvili 22, 18. maSin dekodirebisas miviRebT 11, 6 dasavsebiT gaugebaria, ra ricxvi iyo gadmocemuli. e.i. am kodsSecdomis mxolod aRmoCena SeuZlia, xolo gasworeba ki – ara.zemoT aRweril samniSna kods ki SeuZlia Secdomis ara martoaRmoCena, aramed erTi Secdomis gasworebac.moviyvanoT kodirebis ramdenime martivi meTodi.vTqvaT, gasagzavnia nulebisa da erTebisgan Semdgari mwkrivi(e.i. Cveni alfabeti Sedgeba ori asosgan - 0 da 1)0010111010110100110011000110101000101011davyoT es mwkrivi xuTeulebad (5 asoian sityvebad)00101 11010 11010 01100 11000 11010 10001 01011.kodirebis meTodi aseTi iyos: daviTvaloT TiTeuli sityviscifrTa jami, da Tu es jami luwia, sityvas meeqvse asodmivuweroT 0, xolo Tu kentia – mivuweroT 1. maSin kodirebuligzavnili ase gamouyureba001010 110101 110101 011000 110000 110101 100010 010111.222

aseTi kodirebis Semdeg yoveli sityvis cifrTa jami aucilebladluwia! da Tu gzavnilSi romelime sityvis (eqvseulis)cifrTa jami kenti iqneba, maSin utyuarad davaskvniT, rom essityva Secdomas Seicavs. dekodirebisTvis sakmarisia CamovacaloTeqvseuls bolo cifri. rogorc vxedavT kodirebis am me-Tods Secdomebis mxolod aRmoCena SeuZlia, Tanac mxolodzogierTis. Tu magaliTad romelime sityvaSi ori Secdomamoxda, maSin am Secdomas ver aRmovaCenT: Tu 110000-is nacvladmovida 110110 (e.i. damaxinjda meoTxe da mexuTe asoebi) cifrTajami isev luwi gamodis.2. veli Z 2rogorc simravle Z 2 Sedgeba ori elementisagan {0,1}. Sekrebisoperacia aseTia:0 + 0 = 0, 0 + 1 = 1 + 0 = 1, 1 + 1 = 0.xolo gamravleba ki aseTi:0 0 = 0, 0 1 = 1 0 = 0, 1 1 = 1.es operaciebi aqceven Z 2 -s velad. Z 2 n -iT aRvniSnavT n-ganzomilebian veqtorul sivrces Z 2 -ze:Z 2 n = Z 2 . . . Z 2 .yoveli n-niSna orobiTi ricxvi sinamdvileSi aris (Z 2 ) nveqtori(e 1 ,…,e n ), e k = 0,1.Cven dagvWirdeba Semdegi metrika (Z 2 ) n -Si: manZili a = (a 1 ,…,a n ) dab = (b 1 ,…,b n ) veqtorebs Soris d(a,b) aris im poziciebisraodenoba, sadac a k b k .magaliTad manZili d((01011),(10110)) = 4, radgan es veqtorebigansxvavdebian oTx poziciaSi – pirvelSi, meoreSi, mesameSi damexuTeSi. manZilis es cneba akmayofilebs saWiro aqsiomebs:1. d(a,b) = 0 maSin da mxolod maSin, rodesac a = b;2. d(a,b) = d(b,a);3. d(a,b) d(a,c) + d(c,b).SemovitanoT kidev aseTi cneba. a=(a 1 ,…,a n ) (Z 2 ) nw(a) iyos im poziciaTa raodenoba, sadac a k =1.-isveqtoris wonaTeorema. manZili or veqtors Soris udris maTi jamis wonas.223

3. kodirebisa da dekodirebis zogadi sqemaorobiTi n-niSna blokebis kodirebuli gadacemis SedgebaSemdegi etapebisagan:1. kodireba – funqcia f: (Z 2 ) n (Z 2 ) m , amasTan es funqcia cxadiaunda iyos ineqcia da xSirad – wrfivi, amitom n < m..2. gadacema - funqcia g: (Z 2 ) m (Z 2 ) m , wesiT igivuri unda iyos,magram xdeba Secdomebi, da kodireba-dekodirebis arsiswored am Secdomebis gasworebaSia. es asaxva sazogadodwrfivi ar aris.dekodireba - funqcia h: (Z 2 ) m (Z 2 ) n , romlis arsi imaSia romkompozicia h g f : (Z 2 ) n (Z 2 ) n iyos maqsimalurod axlosigivurTan. es asaxvac ar aris wrfivi sazogadod.aseT kods aRvniSnavT (n,m)-iT.Teorema. (n,m)-kodis SeuZlia Secdomis aRmoCena yvelapoziciaSi maSin da mxolod maSin, rodesac manZili kodirebulsityvebs Soris iyos n-ze meti, anu d(f(x),f(y))>n.Teorema. (n,m)-kodis SeuZlia Secdomis gasworeba yvelapoziciaSi maSin da mxolod maSin, rodesac manZili kodirebulsityvebs Soris iyos 2n-ze meti anu d(f(x),f(y))>2n.siluwis testikodireba5. kodirebis ZiriTadi sqemebiyovel gadasacem sityvas a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 emateba me-8 biti(testbiti) t ise, rom miRebuli 8 bitiani sityvis a 1 , a 2 , a 3 , a 4 , a 5 , a 6 ,a 7 ,t cifrTa jami luwi gamovides, anua 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 → a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , t = (a 1 +a 2 +a 3 +a 4 +a 5 +a 6 +a 7 ) mod 2.magaliTebi1,1,0,1,0,0,1 → 1,1,0,1,0,0,1, 0; 0,1,0,1,0,0,1 → 1,1,0,1,0,0,1, 1,dekodirebaTu miRebuli sityvis a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 ,t cifrTa jami luwia,maSin sityva sworadaa gadmocemuli, Tu kentia, maSin mcdarad,anuif (a 1 +a 2 +a 3 +a 4 +a 5 +a 6 +a 7 +t) mod 2 = 0 then true else falsemagaliTebi1,1,0,1,0,0,1, 0 true; 1,1,0,0,0,0,1, 0 false.siluwis testiT SeiZleba mxolod Secdomis aRmoCena, magaramgasworeba – ara.siluwis testis reitingia 8:7 (8 bitian gzavnilSi informatiulia7 biti, me-8 ki sasinji bitia).224

hemingis 8:4 kodikodirebagadasacemi 4 bitiani sityva a,b,c,d CavweroT matricis saxiT damivuweroT oTxi testbiti x = (a + c)mod 2, y = (b + d)mod 2,z = (a + b)mod 2, t = (c + d)mod 2x ya b zc d tgadasacemi koduri sityva ase Caiwereba: x,y,a,z,b,c,d,t (adgilebze,romelTa nomrebia 2-is xarisxebi – 1,2,4,8 iwereba testbitebi,danarCen adgilebze ki gadasacebi bitebi).magaliTebi1 10 10 1 10,1,1,0 → → → 1,1,0,1,1,1,0,11 01 0 1dekodirebamiRebuli 8 bitiani sityva x,y,a,z,b,c,d,t Caiwereba matricad damowmdeba siluweze misi striqonebi da svetebi. Tu aRmoCnddeba,rom Secdomaa i-ur striqonsa da j-ur svetSi, maSin mcdaria amstriqonisa da svetis gadakveTaze mdgari biti. Tu Secdomaamxolod erT striqonSi an mxolod erT svetSi, maSin gzavnilisoTxive biti sworia.magaliTebi1 11 11,1,0,1,0,1,0,1 → 0 0 1 → 0 0 1 →1 0 1 1 0 11 10→1 1→ 0,1,1,01 0 1225

hemingis 7:4 kodikodirebagadasacemi 4 bitiani sityva a,b,c,d CavweroT matricis saxiT damivuweroT sami testbiti x = (c + a + b)mod 2, y = (a + b + d)mod 2,z = (c + d + b)mod 2 (c + d)mod 2xacbdyzgadasacemi koduri sityva ase Caiwereba: x,y,a,z,b,c,d (adgilebze,romelTa nomrebia 2-is xarisxebi – 1,2,4 iwereba testbitebi,danarCen adgilebze ki gadasacebi bitebi).magaliTi0 10,1,1,0 → 0 1 → 0 1→ 0,1,0,0,1,1,01 01 00dekodirebamiRebuli 7 bitiani sityva x,y,a,z,b,c,d nawildeba matricad damowmdeba siluweze Semdegi blokebi:x-is bloki x+c+a+b; y-is bloki y+a+b+d; z-is bloki z+c+d+b.Tu mcdaria x da y blokebi, maSin unda Sewordes biti a;Tu mcdaria z da y blokebi, maSin unda Sewordes biti d;Tu mcdaria x da z blokebi, maSin unda Sewordes biti c;Tu mcdaria x,y da z blokebi, maSin unda Sewordes biti b;yvela sxva SemTxvevaSi gzavnili a,b,c,d sworia.magaliTi0 10,1,0,0,1,0,0 →0 1→0 000 10 10 000 1→ → 0,1,1,01 0savarjiSoebi daSifreT siluwis kodiT gzavnili1011001 0011010 1010101 0101110 0010011 1011100 0111001 1010101226

aRmoaCineT mcdari baitebi siluwis kodiT daSifrulgzavnilSi10110010 00110101 10101011 01011100 00100110 10111000 01110011 10101010daSifreT hemingis 8:4 kodiT gzavnili0001 1010 0010 1101aRmoaCineT da gaasworeT Secdomebi hemingis 8:4 kodiTdaSifrul gzavnilSi01000011 00110101 10000101 10101011aRmoaCineT da gaasworeT Secdomebi hemingis 8:4 kodiTdaSifrul gzavnilSi01000011 00010101 10001101 10101010daSifreT hemingis 7:4 kodiT gzavnili0001 1010 0010 1101aRmoaCineT da gaasworeT Secdomebi hemingis 7:4 kodiTdaSifrul gzavnilSi0101001 0111010 1001010 0110101aRmoaCineT da gaasworeT Secdomebi hemingis 7:4 kodiTdaSifrul gzavnilSi0101001 0101010 1001110 01100016. kriptografiakriptografiis mizani diametrulad gansxvavdeba kodirebismiznisagan. Tu kodirebis mizania gazavnili maqsimaluradgasagebi gaxados (damaxinjebebis SemTxvevaSic ki),kriptografiaSi – piriqiT, mizania gzavnili gasagebi iyosmxolod misTvis, vinc icis „gasaRebi“, sxvebisTvis ki gaugebaridarces.kriptografiuli paradoqsi: me da Tqven vsubrobT telefoniTan internetiT. viRaca (mteri) ismens yvela Cvens saubars.miuxedavad amisa Cven SegviZlia SevTanxmdeT misTvismiuwvdomel saidumlo ricxvze – kodze, romlis meSveobiTacukve nebismier informacias gavcvliT.me Cavifiqre ricxvi a (personal key). misgan gavakeTe gzavnili x =f(a) = 2 a mod 5 (public key) da gavagzavne.man Caifiqra ricxvi b (personal key). misgan gaakeTa gzavnili y =f(b) = 2 b mod 5 (public key) da gamomigzavna.es x da y mterma icis, magram ar icis a da b.axla, Cemi personal key a da misi public kay y-iT me vakeTeb ricxvs c= F(y,a) = y a mod 5.227

xolo is Tavisi personal key b da Cemi public kay x-iT akeTebs c’=F(x,b) = x b mod 5. .Teorema. F(f(b),a) = F(f(a),b).damtkiceba.F(f(b),a) = f(b) a = (2 b mod 5) a = 2 ba mod 5 = 2 ab mod 5 = (2 a mod 5) b = f(a) b = F(f(a),b),aq gamoyenebulia aseTi Tviseba: (m mod k) n = m n mod kniutonis binomidan gamodis.romelicamrigad, es ricxvebi c da c’ erTmaneTs emTxveva. da es aris issaidumlo ricxvi, romelic Cven orma viciT mxolod.magaliTi.me Cavifiqre a = 3 da gavgzavne x = 2 3 mod 5 = 3. mterma ar icis raaris a. es SeiZleba iyos 3 (aq 2 3 mod 5 = 8 mod 5 = 3), an 7 (aq 2 7 mod 5= 128 mod 5 = 3). an sxva ricxvi.man Caifiqra b = 2 da gamomigzavna y = 2 2 mod 5 = 4. aqac mterma aricis ra aris b. es SeiZleba iyos 2 (aq 2 2 mod 5 = 4 mod 5 = 4), an 6(aq 2 6 mod 5 = 64 mod 5 = 4). an sxva ricxvi.me gavakeTe c = y a = 4 3 mod 5 = 64 mod 5 = 4. is gaakeTebs c’ = x b = 3 2 mod 5= 9 mod 5 = 4. es 4 aris Cveni koduri ricxvi.ra SeuZlia mters? mas SeiZleba egonos (orive sxva unda)a = 7, b= 3, maSinac, cxadia isev x = 3 da y = 4, da misi varaudiTkoduri ricxvi iqneba c = y a = 4 7 mod 5 = 64 mod 5 = 4. c’ = x b = 3 6 mod 5=anu, Cven jer virCevT funqcias f : R → R (araSeqcevads) da kideverT funqcias F : R x: R → R; romlebic akmayofilebs pirobasF(f(a),b) = F(f(b),a).mere me vigzavni f(a)-s, is migzavnis f(b)-s da orive vakeTeds kodsF(f(a),b) = c =F(f(b),a). Cvens SemTxvevaSi f(a) = 2 a mod 5 da F(a,b) = a b mod 5.228

kompiuteruli tomografiis maTematikakompiuteruli tomografi aris samedicino danadgari, romlissaSualebaT xdeba daxSul areebSi damaluli obieqtebis (mag.tvini Tavis qalaSi, RviZli muclis RruSi) damzera,introskopia.safuZveli aris klasikuri rentgenoskopia an misi ZalianTanamedrove variantebi, mag. magnituri rezonansi.klasikuri rentgenis suraTi aCvenebs damaluli obieqtisproeqcias erT sibrtyeSi, mis mxolod brtyel suraTs, Crdils.tomografia damyarebulia am meTodis mravaljerad gamoyenebaze– iReben obieqtis rentgenis suraTs ramdenime mimarTulebidanda miRebuli ramdenime proeqciis meSveobiT maTematikurimeTodebis gamoyenebiT xdeba obieqtis moculobiTi (3D)suraTis Seqmna. cxadia es idea – mravalproeqciianirentgenoskopia – Tavidanve gaCnda, magram imis gamo, rommiRebuli proeqciebis maTematikuri damuSaveba uzarmazargamoTvlebs moiTxovs, kompiuterebis epoqamde am meTodispraqtikuli gamoyeneba ver xerxdeboda.qvemoT gaCvenebT kompiuteruli tomografiis maTematikismeTodebis azrs Zalze gamartivebul – diskretul viTarebaSi.erTi wertilis lokalizebavTqvaT erTi wertili A ucnobi koordinatebiT moTavsebuliaxiluli sinaTlisTvis gaumWirvale, magram rentgenissxivebisaTvis gamWvirvale yuTSiA229

zemdan gadaRebuli (vertikaluri, 90 ) rentgenis suraTimogvcems mxolod 1x1xkoordinats, anu ucnobi wertili imyofebawertilze gamaval vertikalur wrfeze, magram misi zustimdebareoba ucnobi rCebaAx 1wertilis saboloo lokalizaciisTvis saWiroa kidev erTirentgenis suraTis gadaReba raime kuTxiT. es suraTimogvcems 2mdebareobs x 2x wertils x RerZze da viRebT, rom A wertiliwertilze gamaval kuTxiT daxril wrfezeAαx 2 x 1sabolood, es ori rentgenis suraTi gvaZlevs saSualebasdavadginoT sad mdebareobs wertili – am ori wrfisgadakveTaze230

Aαx 2 x 1aqedan dadgindeba A wertilis meore koordinatic:y1 ( x1 x2)tg .ori wertilis lokalizebaaxla vTqvaT ori wertili mdebareobs gaumWvirvale yuTSi. orirentgenis suraTiT ver xerxdeba maTi lokalizebaB’ABA’αx 4 x 3 x 2 x 30am naxazze, sadac mocemulia ori rentgenoskopirebis ( 90 da kuTxeebiT) Sedegi, gaurkvevelia, romeli wyvilia realuri,A,B Tu A', B ' . amis garkveva mxolod mesame rentgenis suraTiTmoxerxdeba.gasagebia, rom ufro meti wertilis SemTxvevaSi bevrad metirengenoskopirebi iqneba saWiro, Sesabamisi gamoTvlebicarsebiTad garTuldeba da moxerxdeba mxolod kompiuterebisgamoyenebiT.231

GPS - Global Positioning Systemadgilmdebareobis garkvevis (pozicionirebis)globaluri sistemamartivi SemTxveva _ pozicionireba sibrtyezeBM ‘AMA Tanamgzavris koordinatebi (a 1 ,a 2 ),B Tanamgzavris koordinatebi (b 1 ,b 2 ),M wertilis ucnobi koordinatebi (x, y),t A - A Tanamgzavridan M wertilamde signalis mosvlis dro,r A = c · t A - A Tanamgzavridan M wertilamde manZili,t B - B Tanamgzavridan M wertilamde signalis mosvlis dro,r B = c · t B - B Tanamgzavridan M wertilamde manZili.( x a ) ( y a ) r( x b ) ( y b ) r2 2 21 2 A2 2 21 2 Bmxolod am ori gantolebisgan Semdgari sistemis amonaxsnimogvcems or wertils _ M-s da M ’–s.232

Tu gveqneba t c monacemi mesame C Tanamgzavridan, maSin SevZlebTdavadginoT, romelia Cveni namdvili mdebareoba, M Tu M ‘:( x a ) ( y a ) ( z a ) r ( x b ) ( y b ) ( z b ) r ( x c ) ( y c ) ( z c ) r2 2 2 21 2 3 A2 2 2 21 2 3 B2 2 2 21 2 3 Camocana. vTqvaT cnobilia, rom A Tanamgzavris koordinatebia(0,0), B Tanamgzavrisa (0, 6). xolo C Tanamgzavrisa (3, 6).manZili M wertilidan A Tanamgzavramde aris 5, manZili Mwertilidan B Tanamgzavramde aris 5i da manZili Mwertilidan C Tanamgzavramde aris 2. ipoveT M wertiliskoordinatebi.amoxsna. 2 Tanamgzavris monacemebiT:2 2 x y 25 2 2( x 6) y 25ori amonaxsni: (x = 3, y = -4) da (x = 3, y = 4) .3 Tanamgzavris monacemebiT: xerTi amonaxsni: (x = 3, y = 4) .pozicionireba sivrceSi2 2 xy252 2( 6) 25x y 2 2( 3) ( 6) 4samganzomilebian SemTxvevaSi wertils aqvs sami koordinati,amitom saWiro iqneba monacemebi 4 Tanamgzavridan.pozicionireba saaTis koreqciiTsinamdvileSi amocana ufro rTulia: Tanamgzavrebze ZviradRirebulizusti saaTebia, kargad sinqronizebuli, xolo TqveniGPS-is saaTi ki iafia, arazusti, romlis Cvenebac gansxvavdebaTanamgzavris saaTis Cvenebisgan Δt droiT. amgvarad Semodis kideverTi, meoTxe ucnobi Δt, amitom saWiro xdeba monacemi me-5Tanamgzavridanac: 2 2 2( x a1 ) ( y a2 ) ( z a3) c t rA2 2 2( x b1 ) ( y b2 ) ( z b3) c t rB2 2 2 ( x c1 ) ( y c2 ) ( z c3) c t r .C2 2 2 ( x d1) ( y d2) ( z d3) c t rD2 2 2 ( x e1 ) ( y e2 ) ( z e3) c t rEy233

masala gameorebisTvissimravleTa Teoria1. simravlis cnebasimravle – sawyisi cnebaa (ar ganimarteba).garkveul elementTa erToblioba.magaliTebi1. am auditoriaSi myof studentTa simravle;2. kursis studentTa simravle;3. fakultetis studentTa simravle;4. naturalur ricxvTa simravle N = {1,2,3,4,…};5. mTel ricxvTa simravle Z = {…,-2,-1,0,1,2,…};6. racionalur ricxvTa (wiladTa) simravle Q;7. namdvil ricxvTa simravle Q;8. luw ricxvTa simravle {…,-4,-2,0,2,4,…};9. kent ricxvTa simravle {…,-5,-3,-1,1,3,5,…};tavtologiurad _10. 6-ze nakleb naturalur ricxvTa simravle { 1,2,3,4,5};11. orniSna ricxvTa simravle {10,11,12,…,98,99};12. orniSna kent simravle {11,13,…,97,99};aRniSvna:xX “elementi x ekuTvnis X simravles”.xX “elementi x ar ekuTvnis X simravles”.magaliTebi5N, 3 N, -3 N, -3Z, 0.33N, 0.33Z, 0.33Q, 4{1,4,9,25},7{1,4,9,25}.aRniSvna:X Y “X simravle Sedis Y simravleSi” = “X simravle aris Ysimravlis qvesimravle”.XY “X simravle ar Sedis Y simravleSi” = “X simravle araris Y simravlis qvesimravle”.magaliTebiN Z Q R, {1,3,5} {1,2,3,4,5,6,7}, {1,3,9}{1,2,3,4,5,6,7}.235

simravleTa TanakveTaA B = {x, x A da x B}2. moqmedebani simravleebzeABsimravleTa gaerTianebaA B = {x, x A an x B}ABmagaliTebi{1,2,3,4} {2,3,5}= {2,3}, {1,2,3,4}{3,4,5} = {1,2,3,4,5}simravleTa sxvaobaA \ B = {x, x A , x B}ABukiduresi SemTxvevebi: carieli simravle da universi U(damokidebulia konteqstze).simravleTa toloba: A = B Tu A B da B A.A simravlis damateba:A’ = U \ A = {xU, xA}.A’ AUoperaciaTa TvisebebiricxvebSisimravleebSia bA Ba + bA B0 a 0 = 0A = a + 0 = aA = Aa b = b aA B = B Aa + b = b + aA B = BA236

a (b c) = (a b) cA (B C) = (A B) Ca (b + c) = a b + a cA (B C) = A B A C1 Ua 1 = aA U = AA U = UA A’ = A A’ = U - A A = AA A = A3. asaxvebiasaxva (funqcia) X simravlidan Y simravleSi f : X Y ariswesi, romliTac X simravlis yovel elements Seesabameba Ysimravlis erTi garkveuli elementi. simravles X ewodeba fasaxvis gansazRvris are, xolo simravles Y misi mniSvnelobaTaare.magaliTebi1. X = {1,2,3,4}, Y = {7,8,9}, xolo wesi f aseTia: f(1) = 7, f(2) = 8, f(3) =8, f(4) = 9,1237894SeTanadebis es wesi asaxvaa.2. X = {1,2,3,4}, Y = {7,8,9}, xolo wesi f aseTia: f(1) = 7, f(1) = 8, f(2) =8, f(3) = 9, f(4) = 9, grafikulad1237894237

SeTanadebis es wesi ar aris asaxva, radgan x = 1Seesabameba ori sxvadasxva elementi - y = 7 da y = 8.elements3. X iyos adamianebis simravle, Y-ic aseve adamianebissimravle, xolo SeTanadebis wesi iyos aseTi: yovel x Xelements (adamians) Seesabamebodes misi Zma. es ar arisasaxva: (a) arsebobs erTi mainc adamiani, visac Zma ar hyavs,anu arsebobs iseTi x X, romelsac araferi ar Seesabameba,(b) arsebobs erTi mainc adamiani, romelsac or Zma hyavs, anuarsebobs iseTi x X, romelsac ori sxvadasxva y YSeesabameba, rac agreTve akrZalulia asaxvis ganmartebiT.4. es magaliTi gaswordeba, Tu Sesabamisobis wess aseSevcvliT: yovel adamians Seesabamebodes misi deda. maSinyovel x X elements Seesabameba Tavisi erTaderTi y Y.5. X = R, Y = R, xolo SeTanadebis wesi f : X Y iyos mocemuliformuliT f(x) = x 2 , kerZod f(1) = 1, f(2) = 4, f(5) = 25, … .ganmarteba. asaxvas f : X Y ewodeba siureqcia, Tu Y-is yovelelementSi gadmodis romelime x, anu Tu yY xX, f(x) = y.magaliTebiasaxva f:RR, f(x) = x 2 ar aris siureqcia: elementSi y = -4 araferiar gadmodis.xolo asaxva f:RR, f(x) = 3x siureqciaa: elementSi y gadmodis x= y / 3.ganmarteba. asaxvas f:XY ewodeba ineqcia, Tu X-isgansxvavebuli elementebi Y-is gansxvavebul elementebSigadadian, anu Tu x 1 x 2 f(x 1 ) f(x 2 ) .magaliTebiasaxva f(x) = x 2 ar aris ineqcia: gansxvavebuli elementebi x = -2da x = 2 erT elementSi gadadian – f(-2) = (-2) 2 = 4 = 2 2 = f(2).xolo asaxva f(x) = 3x ineqciaa: vTqvaT x 1 x 2 anu x 1 - x 2 0, maSinf(x 1 )- f(x 2 ) = 3x 1 - 3x 2 = 3(x 1 - x 2 ) 0 e.i. f(x 1 ) f(x 2 ) .ganmarteba. asaxvas f:XY ewodeba bieqcia, Tu is erTdrouladsiureqciacaa da ineqciac.aseT asaxvas urTierTcalsaxa asaxvasac uwodeben.magaliTebiasaxva f(x) = x 2 ar aris bieqcia: is arc siureqciaa da arcineqcia.xolo asaxva f(x) = 3x bieqciaa: is siureqciac iyo da ineqciac.238

4. asaxvaTa kompoziciaasaxvebi f : X Y da g : Y Z gansazRvraven asaxvas (g f ): X Z ,romelsac maTi kompozicia ewodeba da is moicema tolobiT (g f )(x) = g(f(x)).magaliTivTqvaT f : R R da g : R R mocemulia tolobebiT f (x) = x + 2,g(x) = x 2 , maSin maTi kompozicia (g f ): R R moicema tolobiT (g f )(x) = (x + 2) 2 .xolo kompozicia (f g): R R ki tolobiT (f g)(x) = x 2 +2..asaxvas id X : X X romelic mocemulia tolobiT id X (x) = xigivuri asaxva ewodeba.Teorema. nebismieri asaxvebisaTvis f : Y X da g : X ZsamarTliania tolobebi id X f = f, g id X = g.f : X Y asaxvis Seqceuli ewodeba iseT asaxvas f -1 : Y X, romsruldeba pirobebi f –1 f = id X da f f -1 = id Y . yvela asaxvas argaaCnia Seqceuli. asaxvas ewodeba Seqcevadi, Tu mas aqvsSeqceuli asaxva.Teorema. asaxva f : X Y siureqciaa maSin da mxolod maSin,rodesac arsebobs asaxva g: Y X iseTi, rom sruldeba piroba f g = id Y .Teorema. asaxva f : X Y ineqciaa maSin da mxolod maSin,rodesac arsebobs asaxva g: Y X iseTi, rom sruldeba pirobag f = id x .Teorema. asaxva f : X Y bieqciaa maSin da mxolod maSin,rodesac arsebobs asaxva g: Y X iseTi, rom sruldebapirobebi g f = id x , f g = id Y.es niSnavs, rom g = f –1 . amrigad miviReT, rom asaxva aris bieqciamaSin da mxolod maSin, rodesac is Seqcevadia.5. simravlis simZlavreganmarteba. X da Y simravleebs uwodeben toli simZlavrissimravleebs, Tu arsebobs bieqcia f : X Y.Teorema. intervali (0,1) da intervali (0,2) toli simZlavrissimravleebia.damtkiceba.0123902

Teorema. intervali (0,1) da mTeli RerZi (namdvil ricxvTasimravle) toli simZlavris simravleebia.damtkiceba.Teorema. naturalur ricxvTa simravle N = {1,2,3,…} da luwnaturalur ricxvTa simravle {2,4,6,…} toli simZlavrissimravleebia.damtkiceba. asaxva f : {1,2,3,…} {2,4,6,…} moccemuli formuliTf(n) = 2n amyarebs saWiro bieqcias.Teorema. naturalur ricxvTa simravle N = {1,2,3,…} da kentnaturalur ricxvTa simravle {1,3,5,…} toli simZlavrissimravleebiadamtkiceba. asaxva f : {1,2,3,…} {3,5,7,…} moccemuli formuliTf(n) = 2n + 1 amyarebs saWiro bieqcias.simravles ewodeba Tvladi, Tu is tolZalovania naturalurricxvTa simravlisa. wina ori Teorema niSnavs, rom luwricxvTa simravle da kent ricxvTa simravle orive Tvladia.Teorema. mTel ricxvTa simravle Tvladia, anu simravleebi Nda Z toli simZlavrisani arian.Teorema. racionalur ricxvTa simravle Tvladia, anusimravleebi N da Q toli simZlavrisani arian.Teorema. namdvil ricxvTa simravle Tvladi ar aris, anusimravleebi N da R ar arian toli simZlavris.amrigad, erTmaneTSi Calagebuli simravleebidan N Z Q Rpirveli sami Tvladia, anu isini toli simZlavrisani arian,xolo ukanaskneli, namdvil ricxvTa simravle, anu kontinuumi,R ki araTvladia, is arsebiTad ufro mZlavria, vidre N.savarjiSoebi1. vTqvaT U = {1,2,3,4,5}, X = {1,5}, Y = {1,2,4}, Z = {2,5}. ipoveT(a) X Y’; (b) (X Z) Y’; (c) X (Y Z); (d) (X Y) (X Z);240

(e) (X Y)’; (f) X’ Y’; (g) (X Y)’; (h) (X Y) Z; (i) X (Y Z);(j) X \ Z; (k) (X \ Z) (Y \ Z).2. vTqvaT A B = , ipoveT A \ B da B \ A.3. ipoveT X X’, X X’, X \ X’.4. mocemulia simravleebi A, B da C, amasTan C B. daamtkiceT,rom(a) A C A B; (b) A C A B; (c) A \ B A \ C; (d) C \ A B\ A;(e) B’\ A C’\ A.5. daamtkiceT, rom A (B C) = (A B) (A C).6. daamtkiceT, rom (A B)’ = A’ B’.7. daamtkiceT, rom A B maSin da mxolod maSin, rodesac A B= B.8. daamtkiceT, rom A B maSin da mxolod maSin, rodesac A B= A.9. vTqvaT asaxvebi f : R R da g : R R mocemulia tolobebiTf (x) = 2x + 3, g(x) = x 3 , ipoveT kompozicia (g f ): R R.10. vTqvaT asaxvebi f : R R da g : R R mocemulia tolobebiTf (x) = 2x + 3, g(x) = x 3 , ipoveT kompozicia (f g): R R.11. vTqvaT asaxvebi f : R R da g : R R mocemulia tolobebiTf (x) = x 2 + 3, g(x) = x 3 , ipoveT kompozicia (f g ): R R.12. vTqvaT asaxvebi f : R R da g : R R mocemulia tolobebiTf (x) = x 2 + 3, g(x) = x 3 , ipoveT kompozicia (g f ): R R.13. vTqvaT asaxvebi f : R R da g : R R mocemulia tolobebiTf (x) = 2x, g(x) = 0,5x, ipoveT kompozicia (g f ): R R.14. vTqvaT asaxva f : R R mocemulia tolobiT f(x) = 2x, arisTu ara es asaxva (a) siureqcia, (b) ineqcia, (g) bieqcia?15. vTqvaT asaxva f : R R mocemulia tolobiT f(x) = x 2 , arisTu ara es asaxva (a) siureqcia, (b) ineqcia, (g) bieqcia?16. vTqvaT asaxva f : R R mocemulia tolobiT f(x) = x 3 , arisTu ara es asaxva (a) siureqcia, (b) ineqcia, (g) bieqcia?17. aCveneT, rom igivuri asaxva id X : X X bieqciaa.18. aCveneT, rom ori siureqciis kompozicia siureqciaa.19. aCveneT, rom ori ineqciis kompozicia ineqciaa.20. aCveneT, rom ori bieqciis kompozicia bieqciaa.6. simravleTa namravliA da B simravleTa namravli ewodeba simravles, romliselementebia wyvilebi (a,b), sadac a aris A simravlis elementi,xolo b ki B simravlisa, anu241

A B = {(a,b): aA, bB}.magaliTebi1. A iyos simravle, Semdgare 8 laTinuri asosagan {a,b,c,d,e,f,g,h}xolo simravle B iyos {1,2,3,4,5,6,7,8}, maSin maTi namravliaris 64 elementisgan Semdgari simravlea8,b8,c8,d8,e8,f8,g8,h8a7,b7,c7,d7,e7,f7,g7,h7a6,b6,c6,d6,e6,f6,g6,h6a5,b5,c5,d5,e5,f5,g5,h5a4,b4,c4,d4,e4,f4,g4,h4a3,b3,c3,d3,e3,f3,g3,h3a2,b2,c2,d2,e2,f2,g2,h2a1,b1,c1,d1,e1,f1,g1,h1rac aris Wadrakis dafis standartili notacia.2. namravli R R aris sibrtye.7. mimarTebebimimarTeba A da B simravleebs Soris ewodeba A B namravlisqvesimravles R A B. Tu (a,b) R A B maSin amboben, rom aaris R-mimarTebaSia b-sTan da es ase aRiniSneba aRb.mimarTebaTa SesaZlo Tvisebebi:1. mimarTebas ewodeba refleqsuri Tu x -Tvis xRx;2. mimarTebas ewodeba simetriuli Tu x,y -Tvis xRy yRx;3. mimarTebas ewodeba tranzituli Tu x,y,z -Tvis xRy, yRz xRz;4. mimarTebas ewodeba antisimetriuli Tu x,y -Tvis xRy, yRx x= y.ganvixiloT aseTi mimarTebebi:(1) R ={(x,y): x,y N, x | y (x yofs y-s)};(2) R ={(x,y): x,y N, x < y};(3) R ={(x,y): x,y N, x y};(4) R ={(x,y): x,y N, x – y luwia}.amocanagaarkvieT TiTeuli am mimarTebisaTvis aqvT Tu ara maT zemoTCamoTvlili Tvisiebebi.8. mimarTebis matricamimarTeba sasrul simravleze SeiZleba matricis saxiTCaiweros. vTqvaT A = {a 1 , a 2 , …, a n }, B = {b 1 , b 2 , …, b m } da242

mocemulia raime mimarTeba R A B . am mimarTebas Seesabamebam n matrici ||α ij ||, sadac α ij = 1 Tu x i R y j da α ij = 0 winaaRmdegSemTxvevaSi.magaliTivTqvaT A = {a 1 , a 2 }, B = {b 1 , b 2 ,b 3 } da mimarTeba R A B aseTia:a 1 Rb 1 , a 1 Rb 3 , a 2 Rb 2 , maSin am mimarTebis matriciaamocanebi dawereT matrici {1,2,3,4} simravleze gansazRvruli aseTimimarTebisa: iRj Tu i – j luwia. dawereT matrici {1,2,3,4} simravleze gansazRvruli aseTimimarTebisa: iRj Tu i – j kentia. dawereT matrici {1,2,3,4} simravleze gansazRvruli aseTimimarTebisa: iRj Tu i < j . dawereT matrici {1,2,3,4} simravleze gansazRvruli aseTimimarTebisa: iRj Tu i ≤ j . dawereT matrici {1,2,3,4} simravleze gansazRvruli aseTimimarTebisa: iRj Tu i > j . dawereT matrici {1,2,3,4} simravleze gansazRvruli aseTimimarTebisa: iRj Tu i ≥ j .2a 2 0 1 0b 1 b b 3a 1 1 0 1gaarkvieT TiTeuli am mimarTebisaTvis aqvT Tu ara maTzemoT CamoTvlili Tvisiebebi.9. mimarTebaTa ZiriTadi saxeebiaseTi zogadobiT mimarTebis cneba iSviaTad gamoiyeneba. CvendagvWirdeba mimarTebaTa sami kerZo saxe.1. asaxva aris mimarTebis kerZo saxe: yoveli asaxva f : A BaCens aseT mimarTebas R = {(x,f(x))} X Y, am simravles fasaxvis grafiki ewodeba. sinamdvileSi asaxva aris mimarTeba(anu qvesimravle) R X Y iseTi, rom sruldeba Semdegi 2piroba:(1) x X y Y x R y;(2) xRy, xRy’ y = y’.2. mimarTebas ewodeba eqvivalentoba, Tu is refleqsuria, simetriulida tranzituli, anu sruldeba Semdegi aqsiomebi(1) xRx;(2) xRy yRx;243

(3) xRy, yRz xRz.Tu R mimarTeba am aqsiomebs akmayofilebs, maSin xRy simbolosnacvlad xmaroben aRniSvnas: x ~ y, rac ase ikiTxeba “xeqvivalenturia y-is”. maSin es aqsiomebi ufro nacnob saxesiReben:(1) x ~ x;(2) x ~ y x ~ y;(3) x ~ y, y ~ z x ~ z.magaliTebiSemovitanoT mTel ricxvTa simravleSi aseTi mimarTeba: x ~ y Tux – y iyofa 5-ze. aCveneT, rom es eqvivalentobis mimarTebaa.vTqvaT X simravleze mocemulia eqvivalentobis mimarTeba x ~ y,x elementis eqvivalentobis klasi [x] ewodeba simravles [x] = {y X, x ~ y }.Teorema. yoveli eqvivalentobis mimarTeba hyofs X simravleserTmaneTis araTanamkveT eqvivalentobis klasebad.damtkiceba. unda vaCvenoT ori ram: (1) rom eqvivalentobisklasebis simravle faravs mTel X-s da (2) rom ori klasi an arTanaikveTeba, an mTlianad emTxveva erTmaneTs.pirveli winadadeba cxadia – X-is yoveli elementi x SedisTundac, Tavis klasSi [x]: x ~ x refleqsurobis gamo.axla davamtkicoT meore winadadeba. vTqvaT [x][y] , esniSnavs, rom z i.r. z [x][y], anu z ~ x da z ~ y. simetriulobiTes igivea rac x ~ z da z ~ y, tranzitulobiT ki es gvaZlevs x ~ y,amitom [x]= [y].rogorc vxedavT damtkicebaSi gamoyenebulia eqvivalentobissamive aqsioma.magaliTizemoT naxsenebi eqvivalentobis mimarTeba mTel ricxvTa ZsimravleSi “x ~ y Tu x – y iyofa 5-ze” hyofs Z-s Semdeg 5eqvivaletobis klasad[0] = {…, 0,5,10,15,…}, [1] = {…,1,6,11,…}, [2] = {…,2,7,12,…}, [3] ={…,3,8,13,…}, [4] = {4,9,14,…}. am klasTa simravles qvia 5-zegayofis naSTTa klasebi da ase aRiniSneba: Z 5 . analogiuradimarteba n-ze gayofis naSTTa klasebi Z n .amocanebi adamianTa simravleze ganvixiloT aseTi mimarTeba “x ~ y Tux aris y-is winapari”. aris Tu ara es eqvivalentobismimarTeba?adamianTa simravleze ganvixiloT aseTi mimarTeba “x ~ y TumaT saerTi winapari hyavT”. aris Tu ara es eqvivalentobismimarTeba?244

ganvixiloT {0,1,2,3,4,5,6} simravleze aseTi mimarTeba “x ~ yTu x – y iyofa 3-ze”. aCveneT, rom es eqvivalentobismimarTebaa, CamowereT ewvivalentobis klasebi, dawereT ammimarTebis matrici.3. mimarTebas ewodeba nawilobrivi dalageba Tu isrefleqsuria, antisimetriuli da tranzituli, anu Tusruldeba Semdegi aqsiomebi:(1) xRx;(2) xRy, yRx x =y;(3) xRy, yRz xRz.Tu R mimarTeba am aqsiomebs akmayofilebs, maSin xRy simbolosnacvlad xmaroben ufro nacnob aRniSvnas: x y. maSin esaqsiomebi ufro nacnob saxes iReben:( 1 ) x x;(2) x y, y x x =y;(3) x y, y z x z.mimarTebas ewodeba sruli dalageba, Tu is nawilobrividalagebaa da damatebiT sruldeba aseTi aqsiomac: nebismieriori elementi sadaria, anu (4) x,y an x y, an y x.magaliTebiU iyos raime simravle, xolo X iyos am simravlis yvelaqvesimravleTa simravle, anu x X Tu x U. SemovitanoT X-zeaseTi dalageba: x y Tu x y. es nawilobrivi dalagebaa.mimarTeba naturalur ricxvTa simravleze “ x y Tu x yofs y-s”aseve nawilobrivi dalagebaa.X iyos adamianebis simravle, xolo dalageba SemovitanoT ase:x y Tu x aris y-is winapari. esec nawilobrivi dalagebaa.namdvl ricxvTa simravleze SemovitanoT mimarTeba, romelicmocemulia sibrtyis R 2 = R R aseTi qvesimravliT: es iyos I daIII sakoordinato kuTxeebis biseqtrisis qveS moTavsebulinaxevarsibrtye.aCveneT, rom es dalageba sinamdvileSi sruli dalagebaa,romelic emTxveva RerZis Cveulebriv dalagebas.245

10. nawilobrivi dalageba kubis wveroTa simravleSi(hemingis dalageba)erTi wvero metia meoreze, Tu mis koordinatebSi meti 1-biaz(0,0,1) (1,0,1)(0,0,0) < (0,0,1) < (0,1,1) < (1,1,1)(0,0,0) < (0,0,1) < (1,0,1) < (1,1,1)(0,1,1) (1,1,1)(0,0,0) (1,0,0)(0,1,0) (1,1,0)yx(0,0,0) < (0,1,0) < (0,1,1) < (1,1,1)(0,0,0) < (0,1,0) < (1,1,0) < (1,1,1)(0,0,0) < (1,0,0) < (1,1,0) < (1,1,1)(0,0,0) < (1,0,0) < (1,0,1) < (1,1,1)udidesi da umciresi, minimaluri da maqsimalurivTqvaT (X, ≤ ) nawilobrivad dalagebuli simravlea. elements mX ewodeba umciresi Tu x - Tvis m ≤ x. analogiurad, elementsM X ewodeba udidesi Tu x - Tvis x ≤ M.Teorema. nebismier nawilobrivad dalagebul simravleSiSeiZleba arsebobdes araumetes erTi umciresi (udidesi)elementisa.damtkiceba. vTqvaT m da m’ ori umciresi elementia. m-isumciresobis gamo m ≤ m’, xolo m’-is umciresobis gamo m’ ≤ m.antisimetriulobiT viRebT m = m’. analogiurad damtkicdebaudidesi elemetis erTaderTobac.dalagebul simravleTa namravlivTqvaT (X, ≤ ) da (Y, ≤ ) nawilobriv dalagebuli simravleebia.ganvmartoT maT namravlze X Y aseTi dalageba: (x 1 ,y 1 ) ≤ (x 2 ,y 2 )Tu x 1 ≤ x 2 da y 1 ≤ y 2 . am dalagebas vuwodoT namravlis dalageba.leqsikografiuli dalagebaimave namravlze X Y imarteba sxvanairi dalagebac (msgavsiimisa, Tu rogoraa dalagebuli sityvebi leqsikonSi, amitom amdalagebas leqsikografiuli dalageba ewodeba): (x 1 ,y 1 ) ≤ (x 2 ,y 2 )Tu x 1 < x 2 , xolo Tu x 1 = x 2 , maSin y 1 ≤ y 2 .amocanebi (nawilobriv) dalagebul simravleTa yvela zemoT moyvanilmagaliTSi aRmoaCineT umciresi da udidesi elementebi, TukiaseTebi arseboben.246

adamianTa simravleze ganvixiloT aseTi mimarTeba “x R y Tux aris y-is winapari”. aris Tu ara es dalageba?eqvivalentoba? asaxva?adamianTa simravleze ganvixiloT aseTi mimarTeba “x R y TumaT saerTi winapari hyavT”. aris Tu ara es dalageba?eqvivalentoba? asaxva?ganvixiloT {1,2,3,4} simravleze aseTi mimarTeba “x R y Tu x≤ y”. aCveneT, rom es dalagebaa, dawereT am mimarTebismatrici.ganvixiloT {1,2,3,4} simravleze aseTi mimarTeba “x R y Tu xyofs y-s”. aCveneT, rom es dalagebaa, dawereT am mimarTebismatrici. ganvixiloT {1,2,3,4} simravleze aseTi mimarTeba “x R y Tu x- y iyofa 3-ze”. aCveneT, rom es eqvivalentobis mimarTebaa,CamowereT ewvivalentobis klasebi, dawereT am mimarTebismatrici. daamtkiceT, rom nawilobriv dalagebuli vTqvaT (X, ≤ ) da(Y, ≤ ) imravleebis zemoT aRwerili X Y namravlis dalagebanawilobrivi dalagebaa. vTqvaT (X, ≤ ) da (Y, ≤ ) dalagebebi orive srulia. sworia Tuara, rom X Y namravlis dalagebac srulia? daamtkiceT, rom nawilobriv dalagebuli vTqvaT (X, ≤ ) da(Y, ≤ ) simravleebis X Y namravlis zemoT aRwerilileqsikografiuli dalageba nawilobrivi dalagebaa. vTqvaT (X, ≤ ) da (Y, ≤ ) orive srulia. sworia Tu ara, rom X Y–is leqsikografiuli dalagebac srulia? daasaxeleT N N simravlis is elementebi, romelTaTvisacsruldeba (x,y) ≤ (5,4) namravlis dalagebiT. daasaxeleT N N simravlis is elementebi, romelTaTvisacsruldeba (x,y) ≤ (5,4) leqsikografiuli dalagebiT. emTxveva Tu ara erTmaneTs namravlis da leqsikografiulidalagebebi?247

algebra1. jgufebiganmarteba. jgufi ewodeba simravles G operaciiT : GG G, (a,b) = a * b,romelic akmayofilebs Semdeg aqsiomebs:1. asociaturoba: a * (b * c) = (a * b) * c;2. erTeuli: e R i.r. yoveli elementisaTvis a Rsruldeba piroba a * e = e * a = a3. mopirdapire: a R â i.r. a * â = â * a = e;jgufi komutaturia (abelisaa) Tu damatebiT sruldeba aqsioma4. a * b = b * a .abelis jgufebisaTvis gamoiyeneba aditiuri Cawera: a * b=a+b, e = 0,â = -a, xolo araabelurebisTvis – multiplikaturi: a * b=ab, e = 1,â = a -1 .magaliTebi1. luwi ricxvebi Sekrebis mimarT abelis jgufia, kentebi kiara.2. (Z, +) jgufia;3. racionaluri ricxvebi gamravlebis mimarT ar aris jgufi.4. (Q \ 0, . ) jgufia.5. Z 4 = {0,1,2,3} jgufia Semdegi operaciis mimarT:+ 0 1 2 30 0 1 2 31 1 2 3 02 2 3 0 13 3 0 1 26. araabeluri jgufis magaliTia aragadagvarebul matricTajgufi matricTa gamravlebis mimarT:xolo( 1 2 ) . ( 2 3 ) ( 10 13 )=3 4 4 5 28 29( 2 3 ) . ( 1 2 ) ( 11 16 )=4 5 3 4 19 28Teorema. jgufSi neitraluri elementi erTaderTia.248

Teorema. jgufSi yovel elemnts gaaCnia mxolod erTimopirdapire.damtkiceba.2. qvejgufiganmarteba. jgufis qvesimravles H G ewodeba qvejgufi, Tu HTviTon aris jgufi igive operaciis mimarT, anu sruldebapirobebi1. Tu a,b H, maSin a * b H;2. e H;3. Tu a H, maSin â HmagaliTebi.1. N Z ar aris qvejgufi.2. kenti ricxvebis simravle ar aris Z-is qvejgufi.3. luwi ricxvebis simravle aris Z-is qvejgufi:4. Z 6 = {0,1,2,3,4,5) jgufis qvesimravleTagan qvejgufebia mxolod{0}, {0,2,4}, {0,3}.Teorema. nZ ZnZ saxisaa.qvejgufia, piriqiTac, Z-is nebismieri qvejgufi3. homomorfizmebiganmarteba. jgufebis asaxvasf : G G’ewodeba homomorfizmi, Tu sruldeba Semdegi pirobebi1. f(e) = e’;2. f(a * b) = f(a ) * f(b).magaliTebi1. asaxva f : Z Z mocemuli tolobiT f(k) = 3k+1 ar arishomomorpizmi.2. aseve ar aris homomorpizmi asaxva f(k) = k 2 :3. asaxva f : Z Z mocemuli tolobiT f(x) = 3x homomorfizmia4. asaxva f : Z Z mocemuli tolobiT f(x) = nx homomorfizmia,piriqiTac, nebismieri homomorfizmi f : Z Z aucilebladf(x) = nx tipisaa:5. asaxva f : Z Z 2 mocemuli tolobebiT f(2n) = 0, f(2n+1) = 1homomorfizmia.249

4. anasaxi da birTviganmarteba. f : G G’ homomorfizmis anasaxi ewodebaqvesimravlesIm f = {g G’, g = f(h) }.Im f yovelTvis aracarielia: e’ = f(e) Im f.ganmarteba. f : G G’ homomorfizmis birTvi ewodebaqvesimravlesKer f = {g G, f(g) = e’ }.Ker f yovelTvis aracarielia: e Ker f.magaliTebi.1. f : Z Z, f(x) = 3x homomorfizmisTvisIm f = { …, -6,-3,0,3,6,…}, Ker f = {0}.2. f : Z Z 2 , f(2x) = 0, f(2x+1) = 1 homomorfizmisTvisIm f = Z 2 , Ker f = { …,-4,-2,0,2,4,… }.Teorema. Im f qvejgufia.Teorema. Ker f qvejgufia.Teorema. homomorfizmi f : H G ineqciaa maSin da mxolod maSin,rodesac Ker f = e.5. rgolebi da velebiganmarteba. rgoli ewodeba simravles R aRWurvils orioperaciiT, SekrebiTa da gamravlebiT a + b, a b, romlebicakmayofileben Semdeg aqsiomebs1. (R, + ) komutaturi jgufia;2. Sekreba da gamravleba dakavSirebulni ariandistribuciulobis kanonebiT: a (b + c) = a b + a c, (a + b) c = a c + b c;3. gamravleba asociaturia: a (b c) = (a b) c;rgols hqvia erTeuliani, Tu damatebiT sruldeba aqsioma4. arsebobs elementi e R, romelic gamravlebis mimarTneitralur elemnts warmoadgens: a e = e a = a;rgols hqvia komutaturi, Tu damatebiT sruldeba piroba5. a b = b a.ganmarteba. rgols (R, +, ) ewodeba veli, Tu is erTeuliania,komutaturia da yovel aranulovan elements gaaCniaSebrunebuli, anu a 0 R â R i.r. a â = e.magaliTebi.1. (Z, +, ) rgolia, magram ar aris veli.250

2. (Q, +, ) velia.3. Z 4 rgolia Semdegi operaciebis mimarT:+ 0 1 2 3 0 1 2 30 0 1 1 3 0 0 0 0 01 1 2 3 0 1 0 1 2 32 2 3 0 1 2 0 2 0 23 3 0 1 2 3 0 3 2 1magram ar aris veli:4. Z 3 velia.5. kompleqsur ricxvTa veli. R 2 = {(a,b), a,b R } SemdegioperaciebiT (a,b) + (c,d) = (a+c, b+d), (a,b) (c,d) = (a c - b d, a d + b c)velia.ganmarteba. R rgolis aranulovan elements a hqvia 0-isgamyofi, Tu arsebobs aranulovani bR iseTi, rom a b = 0.rgols hqvia unulgayofo, Tu mas nulis gamyofebi ar aqvs.magaliTebi.1. Z da Q unulgamyofo rgolebia.2. Z 4 - s aqvs nulis gamyofi: 2 2 = 0.Teorema. vels ar SeiZleba hqondes nulis gamyofebi.Teorema. Z nmartivia.Teorema. Z nunulgamyofoa maSin da mxolod maSin, rodesac nvelia maSin da mxolod maSin, rodesac n martivia.amocanebi1. 2+4 Z 5 -Si aris(a) 6 (b) 0 (g) 1 (d) 32. Z 6 -Si 4-is mopirdapire (Sekrebis mimarT) aris(a) 6 (b) 0 (g) 1 (d) 23. am qvesimravleTagan Z-is qvejgufia(a) naturaluri ricxvebi (b) kenti ricxvebi(g) 3-is jeradi ricxvebi (d) sruli kvadratebi4. am qvesimravleTagan Z-is qvejgufia(b) dadebiTi ricxvebi (b) uaryofiTi ricxvebi(g) 0 (d) 100-ze naklebi ricxvebi5. am qvesimravleTagan Z 4 -is qvejgufia(a) {1,2,3} (b) {0,1,2} (g) {2,4} (d) {0,2}6. am asaxvaTagan f : R R romelia homomorfizmi(a) f(x) = x 2 (b) f(x) = sin x (g) f(x) = 2 x (d) f(x) = 5x7. (2,4) da (1,3) kompleqsur ricxvTa namravlia(a) (2,12) (b) (-10,10) (g) (10,-10) (d) (14,10)251

8. (0,1) kompleqsuri ricxvis kvadratia(a) (0,-1) (b) (1,0) (g) (1,1) (d) (-1,0)9. (0,1) kompleqsuri ricxvis Sebrunebulia(a) (1,0) (b) (0,0) (g) (1,1) (d) (0,-1)10. am rgolTagan romeli ar aris veli(a) racionaluri ricxvebi Q (b) mTeli ricxvebi Z(g) namdvili ricxveebi R (d) kompleqsuri ricxvebi C11. romelia am rgolTagan veli(a) Z 4 (b) Z 3 (g) Z 6 (d) Z12. romelia am rgolTagan unulgamyofo(a) Z 4 (b) Z 8 (g) Z 6 (d) Z13. am rgolTagan romels aqvs 0-is gamyofebi(a) Z 2 (b) Z 3 (g) Z 6 (d) Z14. 3 4 Z 5 -Si aris(b) 12 (b) 2 (g) 0 (d) 315. Z 5 -Si 4-is Sebrunebuli (gamravlebis mimarT) aris(a) 0,25 (b) 4 (g) 1 (d) 316. Z 6 -Si 0-is gamyofia(a) 3 (b) 4 (g) 1 (d) 517. Z 7 –Si 2-is mopirdapire Sekrebis mimarT aris(a) -2 (b) 4 (g) 1 (d) 518. Z 7 –Si 2-is mopirdapire gamravlebis mimarT aris(a) 0.5 (b) 4 (g) 1 (d) 519. Z 7 –Si 3 – 5 aris(a) 0 (b) 4 (g) 1 (d) 520. Z 7 –Si 3:2 aris(b) 0 (b) 4 (g) 1 (d) 5252

propoziciuri logikapropozicia _ “gamonaTqvami”, “winadadeba”, an WeSmaritia,magaliTad,“am winadadebaSi oTxi sityvaa”,an mcdari“studentebi gamocdaze ar iweren”,zogis WeSmariteba ki gaurkvevelia, mag.“xval, albaT, iwvimebs”, “romeli saaTia?”.propoziciuri logika operirebs mxolod im gamonaTqvamebiT,romlebic an WeSmaritia, an mcdari.propoziciuri logika, anu propoziciaTa aRricxva aris aRweraimisa, rogor miviRoT WeSmariti winadadebebidan (vTqvaTaqsiomebidan) sxva WeSmariti winadadebebi (vTqvaT Teoremebi).propoziciuri logika eyrdnoba or ZiriTad princips:winaaRmdegobis principi. ar arsebobs winadadeba, romelicerdroulad WeSmariticaa da mcdaric.mesame gamoricxulis principi. yoveli winadadeba an WeSmaritia,an mcdari.sxva sityvebiT propoziciur logikaSi mxolod aseT – anWeSmarit, an mcdar – winadadebebs ixilaven. amitomac hqvia amlogikas sxvanairad absoluturi logika.operaciebi winadadebebzearis ramdenime operacia, romlebic erTi an ramdenimewinadadebisgan sxva winadadebebs aCenen.uaryofis operacia - ara. yovel winadadebas “A” SeeTanadebamisi uaryofa, winadadeba “NOT A”, sxva aRniSvniT ¬ A. Tu “A”WeSmaritia, maSin “NOT A” mcdara; Tu “A” mcdaria, maSin “NOTA” WeSmaritia.koniunqciis operacia - da. yovel or winadadebas “A”, “B”SeeTanadeba winadadeba “A AND B” , sxva aRniSvniT A ^ B. . eswinadadeba WeSmaritia maSin da mxolod maSin, rodesac orive -“A”-c da “B”-c WeSmaritia. yvela sxva SemTxvevaSi “A AND B”mcdaria.diziunqciis operacia - an. yovel or winadadebas “A”, “B”SeeTanadeba winadadeba “A OR B” , sxva aRniSvniT A v B. eswinadadeba WeSmaritia maSin da mxolod maSin, rodesac erTi253

mainc, an “A” an “B” WeSmaritia. Tu orive mcdaria, maSinmcdaria “A OR B”-c.implikaciis operacia - gamomdinareobs. yovel or winadadebas“A” da “B” SeeTanadeba winadadeba “A IMPLIES B”, sxva aRniSvniTA→B. amis eqvivalenturi formebia1. A → B;2. Tu A maSin B;3. A aris sakmarisi B-sTvis;4. B aris aucilebeli A-sTvis.WeSmaritebis cxrilebiaqaa “gamravlebis tabulebi” am operaciebisaTvisaq T (TRUE) niSnavs WeSmarits, xolo F (FALSE) _ mcdars.operaciaTa aRmniSvneli inglisuri sityvebis AND, OR, IMPLIESnacvlad iyeneben aRniSvnebsNOT = ¬ ,OR = v , AND = ^ , IMPLIES = →ixmareba agreTve aRniSvna A B rac niSnavs, rom A → B da B →A. es niSnavs, rom A da B eqvivalenturni arian.magaliTi. ganvixiloT sami sawyisi winadadebaT = Tovs, y = yinavs, g = gareT gavdivar.CavweroT winadadebebi operaciebis terminebSi1. “Tu Tovs an yinavs, gareT ar gavdivar”anu(T OR y) IMPLIES NOT g(T v y) → ¬ g.2. „Tu arc Tovs da arc yinavs, gareT gavdivar“(¬ T ^ ¬ y) → g.3. „roca gareT gavdivar, maSin Tovs“4. “an Tovs, an yinavs”g → T.T v y.254

amrigad, mocemuli ramdenime winadadebidan logikurioperaciebis meSveobiT SeiZleba SevqmnaT axali winadadebebi dada Semdeg vcadoT gavarkvioT, rodisaa miRebuli winadadebaWeSmariti da rodis mcdari.tavtologia, absurdi, saTuoTu Sedgenili winadadeba WeSmaritia Semadgeneli winadadebebisnebismieri mniSvnelobisaTvis, maSin mas tavtologiaewodeba.Tu yovelTvis mcadria – absurdi, , Tu xan WeSmaritia da xanmcdari – saTuo.magaliTad, P v (¬P) tavtologiaa, is kovelTvis WeSmaritia:P ^ (¬P) absurdia, is yovelTvis mcdaria:P →(¬P) saTuoa:P ¬P P v (¬P)0 1 11 0 1P ¬P P ^ (¬P)0 1 01 0 0P ¬P P → (¬P)0 1 11 0 0is mcdaric SeiZleba iyos da WeSmaritic.ZiriTadi igiveobebi (tavtologiebi)de morganis kanonebi1);2);kontrpoziciis kanoniSTanTqmis kanonebi;1);2);distribuciulobis kanonebi1)2);.255

TiTeuli am kanonis Semowmeba SesaZlebelia cxrilebiT,SevamowmoT magaliTad de morganis kanoni:PQ P v Q ¬(P v Q) ¬P ¬Q ¬Q ^ ¬P0 0 0 1 1 1 10 1 1 0 1 0 01 0 1 0 0 1 01 1 1 0 0 0 0aq me-3 da me-7 svetebi erTnairia, e.i. ¬(P v Q) da ¬Q ^ ¬PerTdroulad arian mcdari an WeSmarito (maT erTnairiWeSmaritebis funqciebi aqvT), amitom isini eqvivalenturniarian.qvemoT mogvyavs ramdenime sxva tavtologiaP (P P)P (P P)(P Q) (Q P)(P Q) (Q P)[(P Q) R] [P (Q R)][(P Q) R] [P (Q R)](P Q) ( P Q)(P Q) ( P Q)[P (Q R] [(P Q) (P R)][P (Q R] [(P Q) (P R)](P True) True(P False) False(P False) P(P True) P(P P) True(P P) FalseP ( P)(P Q) ( P Q)(P Q) [(P Q) (Q P)][(P Q) R] [P (Q R)][(P Q) (P Q)] P(P Q) ( Q P)256

davamtkicoT (PdavamtkicoT (PQ) ( P Q)P Q P→Q ¬P Q ¬Q v P0 0 1 1 0 10 1 1 1 1 11 0 0 0 0 01 1 1 0 1 1Q) ( Q P)hilbertis aqsiomebiP Q P→Q ¬P ¬Q ¬Q→¬P0 0 1 1 1 10 1 1 1 0 11 0 0 0 1 01 1 1 0 0 1;;;;;;;;;;.cnobili antikuri tavtologiebi(( A B ) ( B C )) ( A C) - silogizmi (sokrate adamiania,yvela adamiani mikvdavia, e.i. sokratec mokvdavia)( A ( A B )) B - Modus Ponens - gamoyvanis wesi (Tu A-dangamodis B da A WeSmaritia, maSin WeSmaritia B-c)(( B ) ( A B )) ( A) - Modus Tollens - winaaRmdegobisdaSveba (Tu A-dan gamodis B, magram Bmcdaria, maSin mcdariyofila A-c).257

xidi lagikidan simravleebiskenvTqvaT X raime simravlea, xolo P raime gamonaTqvami X-iselementebis Sesaxeb. Tu x X elements aqvs es P Tviseba,maSin vwerT P(x).magaliTad, X = N iyos naturalur ricxvTa simravle, xolo Piyos Tviseba „aris martivi ricxvi“. maSin P(2), P(3), P(5), P(7), . . . ,magram ara P(4), P(6), P(8), . . . .simravle P T ganimarteba, rogorc erToblioba yvela im x-sa,romelTac aqvT es P Tviseba, anuP { x X , P( x)}.TCvens magaliTSi P = „aris martivi ricxvi“ P T = { 2,3,5,7,9,11,13, ... }.leqsikoni logikuri operaciebidan simravlur operaciebSisimravluri da logikuri de morganis kanonebikvantorebix X P( x)niSnavs „arsebobs x X-dan, romlisTvisac P WeSma-x X P( x)ritia“.niSnavs „yoveli x -Tvis X-dan P WeSmaritia“.kvantorebi da uaryofasavarjiSoebidaamtkiceT zemoT moyvanili tavtologiebi.258

grafTa Teoria(mokle kursi)1. grafis ganmarteba da magaliTebigrafi Sedgeba wveroebisagan da wiboebisagan. wveroebiwarmoadgenen wertilebs, xolo zogirTi wyvili wveroebisaSeerTebulia wiboTi. wveroTa simravle aRiniSneba V-Ti (vertices), xolo wiboTa simravle E-Ti (edges). |V| aRniSnavs grafiswveroTa raodenobas, xolo |E| ki wiboTa raodenobas.mogvyavs ramdenime mniSvnelovani grafis magaliTi1. tetraedris grafi 2. kubis grafi|V| = 4, |E| = 4 |V| = 8, |E| = 123. n-wveroiani sruli grafi – am grafs aqvs n wvero da wveroTayvela wyvili SeerTebulia wiboTi. aseTi grafisaTvis |V| = n, |E| =n (n-1) / 2.kerZod,3. 4-wveroiani 4. Ria konverti 5. 5-wveroianisruli grafi |V| = 5, |E| = 8 sruli grafi(daxuruli konverti) |V| = 5, |E| = 10|V| = 4, |E| = 62. grafTa mocemis xerxebigrafi SeiZleba aRiweros misi wveroebisa da wiboebis ubraloCamoTvliT:259

ABdcDaebCam grafSi 4 wveroa A,B,C,D da 5 wibo c = (A,B), b = (A,C), a = (C,B), d= (D,B), e = (D,C). es grafi ase SeiZleba aRiwerosV ={A,B,C,D}, E = {(A,B),(A,C),(B,C),(B,D),(D,C)}.grafis Cawera SeiZleba e.w. SeerTebaTa matriciTac. es aris |V| ×|V| zomis kvadratuli matrici, sadac (i,j) adgilze aris 1 Tu i-uri da j-uri wveroebi SeerTebulia wibiTi, da aris 0winaamRdeg SemTxvevaSi. Cveni grafisTvis es matriciaA B C DA 0 1 1 0B 1 0 1 1C 1 1 0 1D 0 1 1 0arsebobs gansxvavebuli gza grafis matriciT aRwerisa -incidentobis matrici. es aris |E| × |V| zomis matrici, sadac (i,j)adgilze aris 1 Tu i-ur adgilze mdgomi wvero ekuTvnis j-uradgilze mdgom wibos, da aris 0 winaamRdeg SemTxvevaSi. CvenigrafisTvis es matriciaa b c d eA 0 1 1 0 0B 1 0 1 1 0C 1 1 0 0 1D 0 0 0 1 13. grafTa komponentebigza grafis v da w wveroebs Soris ewodeba wveroTamimdevrobas (v 0 ,v 1 ,…,v n ), sadac v = v 0 , w = v n da yoveli wyvili[v i ,v i+1 ] qmnis wibos.gzas ewodeba cikli, Tu is Sekrulia, anu v 0 = v n .grafs ewodeba bmuli, Tu SesaZlebelia misi yoveli ori wverosgziT SeerTeba.260

grafs ewodeba xe, Tu is bmulia da acikluri (e.i. ar aqvsciklebi).grafis wveros xarisxi (valentoba) ewodeba im wiboTaraodenobas, romlebic Seicaven am wveros.Teorema. grafis yvela wveros xarisxTa jami luwi ricxvia.Teorema. grafis kent xarisxiani wveroebis ricxvi luwia.4. planaruli grafebigrafs ewodeba planaruli (brtyeli) Tu SesaZlebelia misisibrtyeze ise daxazva, rom wiboebi erTmaneTs ar kveTdnen.sibrtyeze daxazul grafi warmoqmnis regionebs – areebs,romlebic SemosazRvrulia garkveuli ciklebiT. aseT regionTasimravle aRvniSnoT R-iT (aRvniSnoT, rom regonebSi iTvlebaagreTve erTi usasrulo regioni ganlagebuli grafis garekonturis gareT).eileris formula. |V| - |E| + |R| = 2.sami Wis amocana. SevaerToT sami saxli sam WasTan ise, rombilikebi erTmaneTs ar kveTdnen.am naxazze bilikebi erTmaneTs kveTen, magram SeiZleba Tu araam bilikebis ise gatareba, rom isini ar ikveTebodnen? eilerisformulidan SeiZleba imis gamoyvana, rom es SeuZlebelia, anues grafi planaruli (brtyeli) ar aris. am grafs ewodeba K 3,3 .araplanaruli garfis meore magaliTia 5-wveroiani sruligrafi K 5 .:Teorema (grafis planarulobis kriteriumi). grafi planaruliamaSin da mxolod maSin, rodesac is ar Seicavs K 3,3 an K 3,3qvegrafebs.261

wiboTa Semovlis amocana5. Semovlis amocanebi1. SeiZleba Tu ara mocemuli grafis yvela wibos ise Semovla,rom TiTeul wiboze gaviaroT mxolod erTxel? es formulirebaeqvivalenturia Semdegis: arsebobs Tu ara grafSi iseTi gza,romelic grafis yovel wibos TiTojer Seicavs? aseT gzaseileris gza ewodeba.2. SeiZleba Tu ara grafis Semovla ase: daviwyoT romelime wverodan,SemoviaroT grafi ise, rom yvela wibo mxolod erTxelgaviaroT da davbrundeT sawyis wveroSi. es formulireba eqvivalenturiaSemdegis: arsebobs Tu ara grafSi iseTi cikli, romelicgrafis yovel wibos TiTojer Seicavs? aseT cikls eileriscikli ewodeba.leonard eileris Teorema.1. eileris cikli arsebobs maSin da mxolod maSin, rocam amgrafs aqvs mxolod luwxarisxiani wveroebi.2. eileris gza arsebobs maSin da mxolod maSin, roca amgrafis kentxarisxiani wveroebis ricxvi ors ar aRemateba.kenigsbergis xidebiSeiZleba Tu ara kenigsbergis Svidive xidis Semovla ise, romar gaviaroT orjer erT xidze?es amocana eqvivalenturia Semovlis amocanisa am grafisaTvis:napiri 1kunZuli 1 kunZuli 2napiri 2eileris TeoremiT es SeuZlebelia, radgan am garfs aqvs 4kentxarisxiani wvero.wveroTa Semovlis amocana1. SeiZleba Tu ara mocemuli grafis yvela wveros ise Semovla,rom TiTeul wveroze gaviaroT mxolod erTxel? esformulireba eqvivalenturia Semdegis: arsebobs Tu ara grafSi262

iseTi gza, romelic grafis yovel wveros TiTojer Seicavs?aseT gzas hamiltonis gza ewodeba.2. SeiZleba Tu ara grafis Semovla ase: daviwyoT romelimewverodan, SemoviaroT grafi ise, rom yvela wvero mxoloderTxel gaviaroT da davbrundeT sawyis wveroSi. esformulireba eqvivalenturia Semdegis: arsebobs Tu ara grafSiiseTi cikli, romelic grafis yovel wveros TiTojer Seicavs?aseT cikls hamiltonis cikli ewodeba.komivoiaJoris amocanavTqvaT grafSi dafiqsirebulia yvela wiboTa sigrZeebi. vipoviTumciresi sigrZis cikli, romelic Semoivlis grafis yvelawveros.Tu grafs aqvs hamiltonis cikli, maSin is aris am amocanisamoxsna.5. oTxi feris problemaminimum ramdeni feria saWiro nebismieri rukis ise SesaRebad,rom mezobel qveynebs sxvadasxva feri hqondeT?sami feri amisTvis ar kmara:SedarebiT advili saCvenebelia, rom 5 feri kmara nebismierorukisaTvis. didi xnis ganmavlobaSi problemad rCeboda,sakamarisia Tu ara 4 feri. es problema gadaiWra mxolodkompiuterebis daxmarebiT Catarebuli vrceli gamoTvlebiT.es problema SeiZleba grafebis enaze iTargmnos. amisaTvisnebismier rukas SevuTanadoT aseTi grafi: wveroebi iyosqveynebis dedaqalaqebi, ori wvero (dedaqalaqi) SevaerToTwiboTi (rkinigziT), Tu am qveynebs saerTo sazRvris monakveTiaqvT. maSin rukis SeRebvis problema dadis Semdeg amocanaze:mocemuli grafis wveroebs mivaniWoT nomrebi (leiblebi) ise,rom mezobel wveroebs sxvadasxva nomeri hqondeT. minimumramdeni nomeri aris amisaTvis sakmarisi? zemoT mocemuli rukisgrafia4132mis gadanomrvas Wirdeba 4 nomeri (4 feri).263

6. xeebiyvelaze martivi, magram yvelaze gamoyenebadi grafebia xeebi –bmuli acikluri grafebi.yoveli xe pirvel rigSi brtyeli grafia: aciklurobis gamo isar SeiZleba Seicavdes K 3,3 da K 5 grafebs (romlebsac aqvTciklebi). amitom xisaTvis samarTliania eileris formula |V| -|E| + |R| = 2. kvlav aciklurobis gamo ar arsebobs Siga regionebi,aris marTo gare, e.i. |R| = 1, amitom |V| - |E| = 1, anu xisaTviswveroTa ricxvi erTiT metia wiboTa ricxvze.xis struqtura aqvT veb gverdebs, direqtoriebs, internetismisamarTTa domenur sistemas.Tavad xis struqtura aseTia. xis erTerT wveros uwodebenfesvs (root). amis Semdeg Semdeg avtomaturad Cndeba ierarqia:nebismieri ori wverodan, romlebis wiboTia SeerTebuli,erTerTi aris mSobeli (romelic ufro axloa fesvTan), meoreki Svili. fesvs hyavs mxolod Svilebi, Sinagan wveroebs hyavTrogorc mSoblebi, aseve Svilebi, ylorti ewodeba wveros, Tumas mxolod mSobeli hyavs.a b c d e fxyrzam xeSi r fesvia, x, y, z Siga wveroebi, a, b, c, d, e, f ylortebi.yoveli ylortis valentoba 1-is tolia. binaruli ewodeba xes,Tu misi yoveli wveros (ylortebis) garda, 2-is tolia.amocanebiamoxseniT Semdegi amocanebi (a) tetraedris grafisaTvis; (b)kubis grafisaTvis; (g) 4-wveroiani sruli grafisaTvis; (d) RiakonvertisaTvis:1. dawereT am grafis SeerTebaTa matrici;2. dawereT am grafis incidenciis matrici;3. daxazeT am grafis brtyeli naxazi;4. gamoTvaleT kombinacia |V| - |E| + |R|;5. aqvs Tu ara am grafs eileris gza?6. aqvs Tu ara am grafs eileris cikli?7. amoxseniT komovoiaJeris amocana am grafisaTvis.8. minimum ramdeni feriT SeiZleba am grafis SeRebva?264

kombinatorikan-elementis gadanacvlebaTa raodenobaP(n) = n ! = 1· 2 · 3 · … · n.n-elementidan k-elementis amorCevis kombinaciaTa raodenoba(binomialuri koeficienti)k n!Cn .k ! ( n k)!Tvisebebi01k n kCn1;Cn n;CnC 1 k k 1n; Cn Cn 1C n1.niutonis binomi1 1 2 2 2 2 2 2 1 1( ) n n n n ... k n k k ... n a b a C a b C a b C a b C a b n C n ab n bn .n n n n nnn k nk k( a b) Cna b .k 0Teorema.nC C ... C C 2 .0 1 n1 0n n n ndamtkiceba.nnn n k nk k k Cn Cnk 0 k 02 (1 1) 1 1 .paskalis samkuTxedi1 11 2 11 3 3 11 4 6 4 1CC0 11 1C C C0 1 22 2 2C C C C0 1 2 33 3 3 3C C C C C0 1 2 3 44 4 4 4 4amocanebi1. daamtkiceT C 0 n1;C1nk n k n;C C ;2. gamoTvaleT (a + b) 5 , (a + b) 6 , (a + b) 7 .nn1 k k 1Cn Cn 1C n1.3. kalaTaSi 20 burTulaa, maT Soris 7 wiTeli, danaCeni Savi.rogoria albaToba imisa, rom SemTxvevitad amoRebul 5burTulaSi iqneba 2 wiTeli?amoxsna. sul SesaZlo kombinaciebis (xuTeulebis amorCevisocidan) aris C , xolo kombinaciebi, romlebic Seicaven 2wiTels arisC520 C2 37 13. amitom saZiebeli albaTobaaCP C2 37 135C20.265

4. ipoveT albaToba imisa, rom SemTxveviT arCeuli 5-niSnaricxvi ar Seicavdes cifrs 5.4. dawesebulebaSi 5 vakansiaa kacebisaTvis, 5 vakansiaqalebisaTvis da 4 vakansia, romelTaTvisac sqess mniSvnelobaar aqvs. konkurSi monawileobs 12 mamakaci da 8 qali. ramdeniavakansiaTa Sevsebis variantebis raodenoba?5. klasSi 13 biWia da 12 gogona. (a) ramdennairad SeiZleba amklasis gamwkriveba ise, rom biWebi simaRlis mixedviT iyvnendalagebulni? (b) igive amocana gogonebis simaRlis mixedviTdalagebaze.266

gamoyenebuli literatura1. Mazzola, G., Milmeister, G., Weissmann, J. Comprehensivemathematics for computer scientists. 1. Sets and numbers, graphsand algebra, logic and machines, linear geometry. Universitext.Springer-Verlag, Berlin, 2004.2. Graham, R. L., Knuth, D. E., Patashnik, O. Concrete mathematics.A foundation for computer science. Second edition. Addison-Wesley Publishing Company, Reading, MA, 1994.3. Cooke, D. J., Bez, H. E. Computer Mathematics. CambridgeUniversity Press, 1984; Russian transl.: Nauka, Moscow, 1990.4. Birkhoff, G., Bartee, T. Modern Applied Algebra, McGraw Hill,1970; Russian transl.: Nauka, Moscow, 1989.5. Lang, S. Algebra. Revised third edition. Graduate Texts inMathematics, 211. Springer-Verlag, New York, 2002.6. Seymour Lipschutz, Essential Computer Mathematics. Schaum'sOutline, 2003.7. Birkgof, G. Lattice theory. (Russian) Translated from the Englishand with an appendix by V. N. Saliĭ. Translation edited by L. A.Skornyakov. Nauka, Moscow, 1984.267

s a r C e v iwinasityvaoba ……………………………………………………… 3simravleTa Teoria ………………………………………………….. 5$ 1. simravleebi da maTi specifikacia ……………………… 5$ 2. umartivesi operaciebi simravleebze …………………... 10$ 3. venis diagramebi …………………………………………… 14$ 4. qvesimravleebi …………………………………………….. 17$ 5. simravleTa namravli …………………………………….. 24mimarTebebi ………………………………………………………… 27$ 1. ZiriTadi cnebebi ………………………………………… 27$ 2. grafikuli warmodgenebi ……………………………….. 30$ 3. mimarTebebis Tvisebebi …………………………………… 33$ 4. dayofa da ekvivalentobis mimarTeba ………………… 36$ 5. dalagebis mimarTeba …………………………………….. 39$ 6. Sedgenili mimarTebebi …………………………………… 42$ 7. mimarTebebis Caketva ……………………………………… 44ariTmetikis ZiriTadi cnebebi …………………………………… 49$ 1. `mcire~ sasruli ariTmetika …………………………… 49$ 2. `didi~ sasruli ariTmetika …………………………… 53$ 3. orobiTi ariTmetika ……………………………………… 57$ 4. logikuri ariTmetika …………………………………… 61grafTa Teoria (sawyisi cnebebi) ……………………………… 65$ 1. sawyisi cnebebi …………………………………………… 65$ 2. marSutebi, ciklebi da bmuloba ……………………… 72planaruli grafebi ……………………………………………… 77$ 1. planaruli grafebi ……………………………………… 77$ 2. monacemTa struqturuli grafis warmosadgenad …… 83orientirebuli grafebi ………………………………………… 89$ 1. orientirebuli grafebi ………………………………… 891.1. Sesavali ………………………………………………… 891.2. marSutebi da bmuloba orientirebul grafebSi .. 901.3. dalagebuli orientirebuli grafebi da Semovlebi 95268

algebruli struqturebi ………………………………………… 103binaruli operaciebi ………………………………………….. 103komutaciuri da asociuri binaruli operaciebi …………. 105neitraluri elementi ………………………………………… 107naxevarjgufebi …………………………………………………. 108monoidebi ……………………………………………………….. 109jgufebi …………………………………………………………. 110jgufebis elementaruli Tvisebebi ………………………….. 113qvejgufebi ……………………………………………………… 115lagranJis Teorema …………………………………………….. 120normaluri jgufebi …………………………………………… 123faqtor-jgufi ………………………………………………….. 125jgufTa homomorfizmi ………………………………………… 125izomorfuli jgufebi ………………………………………… 129homomorfizmis birTvi da anasaxi …………………………… 131rgolebi ………………………………………………………… 135idealebi ………………………………………………………… 139maxasiaTebeli …………………………………………………... 141rgolTa homomorfizmi ………………………………………... 142rgolTa lokalizacia ………………………………………… 146sasruli rgolebi da velebi ………………………………… 149bulis algebrebi ……………………………………………… 155savarjiSoebi …………………………………………………… 164jgufi, rgoli, veli ……………………………………………… 169$ 1. algebruli struqturebi da qvestruqturebi ………. 169$ 2. umartivesi operaciuli struqturebi ……………….. 171$ 3. rgolebi da velebi ……………………………………… 1743.1. rgolebi ……………………………………………….. 1743.2. velebi …………………………………………………. 1763.3. sasruli velebi ……………………………………… 1783.4. dalagebuli velebi ………………………………….. 180wrfivi algebra ……………………………………………………. 185$ 1. wrfivi algebra …………………………………………... 185269

1.1. veqtoruli sivrceebi da wrfivi gardaqmnebi …….. 1851.2. struquturli gamosaxulebebi n R -Si …………….. 192meserebi da bulis algebrebi …………………………………… 201$ 1. meserebi da bulis algebrebi …………………………. 2011.1. meserebi ……………………………………………….. 2011.2. bulis algebrebi ……………………………………. 2011.3. bulis algebris zogierTi gamoyeneba ……………. 209$ 2. Caketili naxevarrgolebi ……………………………… 219gamoyenebani: kodireba, kriptografia, kompiuterulitomografia, globaluri pozicionirebis sistema …………… 221kodireba ………………………………………………………….. 2211. Sesavali ……………………………………………………… 2212. veli Z 2…………………………………………………….. 2233. kodirebisa da dekodirebis zogadi sqema ………………. 2245. kodirebis ZiriTadi sqemebi ………………………………. 2246. kriptografia ………………………………………………. 227kompiuteruli tomografiis maTematika ……………………… 229GPS - Global Positioning Systemadgilmdebareobis garkvevis (pozicionirebis)globaluri sistema ……………………………………………… . 232masala gameorebisTvis …………………………………………….. 235simravleTa Teoria ………………………………………………… 2351. simravlis cneba …………………………………………….. 2352. moqmedebani simravleebze ………………………………… 2363. asaxvebi ……………………………………………………… 2374. asaxvaTa kompozicia ………………………………………. 2395. simravlis simZlavre ……………………………………… 2396. simravleTa namravli ……………………………………… 2417. mimarTebebi …………………………………………………. 2428. mimarTebis matrica ………………………………………… 2429. mimarTebaTa ZiriTadi saxeebi …………………………….. 24310. nawilobrivi dalageba kubis wveroTa simravleSi(hemingis dalageba) ……………………………………… 246270

algebra ……………………………………………………………… 2481. jgufebi ……………………………………………………… 2482. qvejgufi …………………………………………………….. 2493. homomorfizmebi ……………………………………………... 2494. anasaxi da birTvi …………………………………………... 2505. rgolebi da velebi ………………………………………… 250propoziciuri logika …………………………………………….. 253grafTa Teoria (mokle kursi) …………………………………... 2591. grafis ganmarteba da magaliTebi ………………………… 2592. grafTa mocemis xerxebi …………………………………… 2593. grafTa komponentebi ……………………………………….. 2604. planaruli grafebi ………………………………………… 2615. Semovlis amocanebi ………………………………………… 2625. oTxi feris problema ……………………………………… 2636. xeebi ………………………………………………………….. 264kombinatorika ………………………………………………………. 265271