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(÷èñëî åëåìåíò³â ñê³í÷åííî¿ ìíîæèíè). ª äâà ñïîñîáè äëÿ ïîð³âíÿííÿ ïîòóæíîñò³ ñê³í÷åííèõ ìíîæèí: áåçïîñåðåäíüî ïåðåðàõóâàòè åëåìåíòè ìíîæèíè àáî çàñòîñóâàòè ìåòîä â³äïîâ³äíîñò³ (ìåòîä “ãîñòèííî¿ ãîñïîäèí³”). Ìåòîä â³äïîâ³äíîñò³ ïîëÿãàº â òîìó, ùî “ãîñòèííà ãîñïîäèíÿ”, íå ðàõóþ÷è ãîñòåé, çàïðîøóº ¿õ äî ñòîëó ³ ïåðåä êîæíèì ç íèõ ñòàâèòü íàá³ð. Ïîíÿòòÿ â³äïîâ³äíîñò³ íàëåæèòü äî ïåðâèííèõ. Êàæóòü, ùî ì³æ äâîìà ìíîæèíàìè À òà Â âñòàíîâëåíà â³äïîâ³äí³ñòü, ÿêùî êîæíîìó åëåìåíòó îäí³º¿ ìíîæèíè â³äïîâ³äàº ïåâíèé åëåìåíò äðóãî¿ ìíîæèíè. Ïðè öüîìó ìîæëèâèé òàêèé âèïàäîê: êîæíîìó åëåìåíòó ³ç À â³äïîâ³äàº ëèøå îäèí åëåìåíò ³ç Â, òà, íàâïàêè, êîæíîìó åëåìåíòó ³ç Â â³äïîâ³äàº ëèøå îäèí åëåìåíò ³ç À, òîä³ ö³ ìíîæèíè áóäóòü åêâ³âàëåíòíèìè (ïîçíà÷àþòü À∼Â). Çàóâàæèìî, ùî íåñê³í÷åíí³ ìíîæèíè ìîæíà ïîð³âíþâàòè çà ïîòóæí³ñòþ, ëèøå âèêîðèñòîâóþ÷è ìåòîä â³äïîâ³äíîñò³. Äëÿ öüîãî íåñê³í÷åííó ìíîæèíó ïîð³âíþþòü ç ìíîæèíîþ íàòóðàëüíèõ ÷èñåë. Ïðè öüîìó ç’ÿñîâóºòüñÿ, ùî âñ³ íåñê³í÷åíí³ ìíîæèíè ïîä³ëÿþòüñÿ íà 2 êëàñè: êëàñ åêâ³âàëåíòíèõ ìíîæèí³ âñ³õ íàòóðàëüíèõ ÷èñåë (òàê³ ìíîæèíè íàçèâàþòüñÿ ç÷èñëåííèìè) ³ êëàñ íå åêâ³âàëåíòíèõ ìíîæèí³ âñ³õ íàòóðàëüíèõ ÷èñåë (òàê³ ìíîæèíè íàçèâàþòüñÿ íåç÷èñëåííèìè). Ââåäåìî ïîíÿòòÿ ï³äìíîæèíè. Ìíîæèíó Â íàçèâàþòü ï³äìíîæèíîþ ìíîæèíè À, ÿêùî âñ³ åëåìåíòè Â íàëåæàòü ìíîæèí³ À (ïîçíà÷àþòü Â ⊂ À). Áóäü-ÿêà ìíîæèíà À ìàº ñâî¿ ï³äìíîæèíè: ïîðîæíþ ìíîæèíó ³ ñàìó ìíîæèíó À. ßêùî äëÿ äâîõ ìíîæèí À ³ Â îäíî÷àñíî ñïðàâåäëèâ³ òâåðäæåííÿ À⊂ Â ³ Â⊂ À, òî ìíîæèíè À ³ Â ñêëàäàþòüñÿ ç îäíèõ ³ òèõ ñàìèõ åëåìåíò³â ³ íàçèâàþòüñÿ ð³âíèìè, àáî çá³æíèìè (ïîçíà÷àþòü À=Â). Ââåäåìî òàê³ îïåðàö³¿ íàä ìíîæèíàìè. 1. Îá’ºäíàííÿì (ñóìîþ) äâîõ ìíîæèí À ³ Â íàçèâàºòüñÿ ìíîæèíà Ñ, êîæíèé åëåìåíò ÿêî¿ íàëåæèòü àáî ìíîæèí³ À àáî ìíîæèí³ Â (ïîçíà÷àþòü Ñ=À∪Â). 2. Ïåðåð³çîì (äîáóòêîì) äâîõ ìíîæèí À ³ Â íàçèâàºòüñÿ ìíîæèíà Ñ, êîæíèé åëåìåíò ÿêî¿ íàëåæèòü ìíîæèí³ À ³ ìíîæèí³ Â (ïîçíà÷àþòü Ñ=À∩Â). 3. Ð³çíèöåþ äâîõ ìíîæèí À ³ Â íàçèâàºòüñÿ ìíîæèíà Ñ, êîæíèé åëåìåíò ÿêî¿ íàëåæèòü ìíîæèí³ À ³ íå íàëåæèòü ìíîæèí³ Â (ïîçíà÷àþòü Ñ=À\Â). Óâîäÿ÷è îïåðàö³¿ íàä ìíîæèíàìè, ìè íå âðàõîâóâàëè, ùî ñàì³ ìíîæèíè ìîæóòü ìàòè âíóòð³øíþ ñòðóêòóðó, òîáòî ââàæàëè, ùî âñ³ åëåìåíòè ìíîæèíè ð³âíîïðàâí³. Ïðîòå â ìàòåìàòèö³ òàê³ “÷èñò³” ìíîæèíè çóñòð³÷àþòüñÿ äóæå ð³äêî. Çíà÷íî ÷àñò³øå âèâ÷àþòü ìíîæèíè, ì³æ åëåìåíòàìè ÿêèõ ³ñíóþòü ò³ ÷è ³íø³ â³äíîøåííÿ. Îäíèì ç íàéâàæëèâ³øèõ â³äíîøåíü º â³äíîøåííÿ ïîðÿäêó. Â³äíîøåííÿ ïîðÿäêó º íå ùî ³íøå, ÿê ïðàâèëî, ùî âñòàíîâëþº ïîðÿäîê “ñë³äóâàííÿ“ åëåìåíò³â ìíîæèíè. Íåõàé À – äåÿêà ìíîæèíà. Ìíîæèíà À íàçèâàºòüñÿ óïîðÿäêîâàíîþ ìíîæèíîþ, ÿêùî äëÿ áóäü-ÿêèõ äâîõ ¿¿ åëåìåíò³â à, b âñòàíîâëåíî îäíå ç òàêèõ â³äíîøåíü ïîðÿäêó: àáî à≤b (à íå ïåðåâèùóº b), àáî b≤à (b íå ïåðåâèùóº à). Óïîðÿäêîâàí³ ìíîæèíè ìàþòü òàê³ âëàñòèâîñò³: 1) ðåôëåêñèâí³ñòü: áóäü-ÿêèé åëåìåíò íå ïåðåâèùóº ñàìîãî ñåáå; 2) àíòèñèìåòðè÷í³ñòü: ÿêùî à íå ïåðåâèùóº b, à b íå ïåðåâèùóº a, òî åëåìåíòè à ³ b çá³ãàþòüñÿ; 3) òðàíçèòèâí³ñòü: ÿêùî à íå ïåðåâèùóº b, à b íå ïåðåâèùóº ñ, òî à íå ïåðåâèùóº ñ. Ó ñôîðìóëüîâàíîìó âèùå îçíà÷åíí³ óïîðÿäêîâàíî¿ ìíîæèíè, åëåìåíòàìè ÿêî¿ ìîæóòü áóòè îá’ºêòè áóäü-ÿêî¿ ïðèðîäè, çíàê ≤ ÷èòàºòüñÿ “íå ïåðåâèùóº”. Öåé çíàê (“ìåíøå àáî äîð³âíþº”) íàáóâàº çâè÷àéíîãî çì³ñòó òîä³, êîëè åëåìåíòè ìíîæèíè À – ä³éñí³ ÷èñëà. Óïîðÿäêîâàí³ (ñê³í÷åíí³ àáî ç÷èñëåíí³) ìíîæèíè ÷àñòî çàïèñóþòü, ðîçì³ùóþ÷è ¿õí³ åëåìåíòè ó çàäàíîìó ïîðÿäêó â êðóãëèõ äóæêàõ. Íàïðèêëàä, çàïèñè (1;2;3) ³ (2;1;3) º ð³çíèìè ñê³í÷åííèìè âïîðÿäêîâàíèìè ìíîæèíàìè. 1.1.2. Ëîã³÷í³ ñèìâîëè Ó ìàòåìàòè÷íèõ òâåðäæåííÿõ ÷àñòî ïîâòîðþþòüñÿ îêðåì³ ñëîâà ³ ö³ë³ âèðàçè. Òîìó ïðè ¿õ çàïèñó êîðèñíî âæèâàòè åêîíîìíó ëîã³÷íó ñèìâîë³êó. Òóò ìè âêàæåìî ëèøå äåê³ëüêà ñàìèõ ïðîñòèõ ÷àñòî âæèâàíèõ ëîã³÷íèõ âèðàç³â. Çàì³ñòü ñë³â “³ñíóº” àáî “çíàé- 18 19
äåòüñÿ” âèêîðèñòîâóþòü ñèìâîë ∃. Öåé ñèìâîë âèíèê ç àíãë³éñüêîãî ñëîâà Existence, ùî óêðà¿íñüêîþ ìîâîþ îçíà÷àº “³ñíóº”. Äëÿ ñòâîðåííÿ ñèìâîëó ∃ âçÿëè ïåðøó áóêâó àíãë³éñüêîãî ñëîâà Existence ³ ïåðåâåðíóëè ¿¿ ñïðàâà íàë³âî íà 180 ãðàäóñ³â. Ùî ñòîñóºòüñÿ ñèìâîëó ∀, òî â³í îçíà÷àº “áóäü-ÿêèé”, “êîæíèé”, ”óñÿêèé”. Éîãî ñòâîðèëè ç ïåðøî¿ áóêâè àíãë³éñüêîãî ñëîâà Any. ×èòà÷, ìàáóòü, çäîãàäàâñÿ, ÿê âèíèê ñèìâîë ∀. 1.2. ÌÍÎÆÈÍÀ Ä²ÉÑÍÈÕ ×ÈÑÅË 1.2.1. Íàòóðàëüí³ ÷èñëà òà ä³¿ íàä íèìè ×èñëî òåæ º ïåðâèííå ïîíÿòòÿ ³ íå ìàº îçíà÷åííÿ. Íàéïðîñò³ø³ ÷èñëà º íàòóðàëüí³. Ö³ ÷èñëà âèêîðèñòîâóâàëèñÿ ëþäüìè äëÿ ë³÷áè. Âîíè âèíèêëè íà ðàíí³õ åòàïàõ ðîçâèòêó öèâ³ë³çàö³¿. Ìíîæèíà íàòóðàëüíèõ ÷èñåë º âïîðÿäêîâàíîþ ìíîæèíîþ, òîáòî äëÿ áóäü-ÿêèõ äâîõ íàòóðàëüíèõ ÷èñåë m ³ n ñïðàâäæóºòüñÿ îäíå ç òàêèõ ñï³ââ³äíîøåíü: àáî m=n (m äîð³âíþº n), àáî m
Page 1 and 2: Ì²Í²ÑÒÅÐÑÒÂÎ ÎÑÂ²Ò
Page 3 and 4: 2.4. Ðàíã ìàòðèö³ ......
Page 5 and 6: 7.9. Ïîõ³äí³ âèùèõ ïî
Page 7 and 8: 10.2. Ãðàíèöÿ ³ íåïåð
Page 9: Ïåðåäìîâà Ç óñ³õ ñî
Page 13 and 14: 1 îòðèìàºìî äð³á m = 2
Page 15 and 16: 1.2.7. Ìîäóëü ä³éñíîã
Page 17 and 18: Òåìà 2 Îñíîâè àëãåá
Page 19 and 20: Öÿ ñèñòåìà ð³âíÿíü
Page 21 and 22: 2 0 . Ó ÿêîñò³ åëåìåí
Page 23 and 24: Äîâ³ëüíèé åëåìåíò
Page 25 and 26: Ïðèêëàä 2.2.2. Çíàéòè
Page 27 and 28: a a ... a a ... a a a ... a a ... a
Page 29 and 30: Òîä³ a 12 12 ∆ 1 =∆+ m =∆
Page 31 and 32: Äâ³ ìàòðèö³ À ³ Â íà
Page 33 and 34: 3 0 . Îá÷èñëèòè àëãåá
Page 35 and 36: ÒÅÌÀ 3 ÑÈÑÒÅÌÈ Ë²Í²
Page 37 and 38: Ïðè ðîçâ’ÿçàíí³ ñè
Page 39 and 40: äå æa11 a12 ... a ö 1n a21 a22
Page 41 and 42: Ðîçâ’ÿçàííÿ. Øëÿõî
Page 43 and 44: 3.3. Êîðèñòóþ÷èñü ïð
Page 45 and 46: Äëÿ äàíî¿ òî÷êè Ì â
Page 47 and 48: Îòæå, êîîðäèíàòè òî
Page 49 and 50: ìèì îáìåæåííÿì, îáó
Page 51 and 52: Ïðèêëàä 4.2.6. Ñêëàñò
Page 53 and 54: Ïðèêëàä 4.2.14. Çíàéòè
Page 55 and 56: ð³âíÿííÿìè ïåðøîãî
Page 57 and 58: ßêùî k çàäàíå ÷èñëî,
Page 59 and 60: 3) ∆ =0 ³ ∆ x = 0, ∆ y = 0.
Page 61 and 62:
ÿêèé íàðîäèâñÿ â ëþ
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º àíàë³òè÷íå ð³âíÿ
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4.5. ÏÎÍßÒÒß ÏÐÎ Ð²ÂÍ
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Ç ðîçãëÿíóòîãî ïðè
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⎧1 ⎫ Ïðèêëàä 5.1.10. Ï
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5.1.5. Îñíîâí³ òåîðåì
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Îòðèìàëè, ùî ð³çíèö
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Â òàê³é çàãàëüí³é ï
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n n 1 ⎛⎛1+ 5⎞ ⎛1− 5⎞
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1. Àíàë³òè÷íèé ñïîñ
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òâ³, ïðè òåõí³÷íèõ
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4. Äëÿ ïîáóäîâè ãðàô
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Ðîçâ’ÿçàííÿ. Íåõàé
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Öÿ ôóíêö³ÿ ÿâëÿº ñî
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Çã³äíî ç îçíà÷åííÿ
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sin x- 0 < x- 0
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Îçíà÷åííÿ 6.3.5 Ôóíêö
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6.3.6. Ãëîáàëüí³ âëàñ
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Ñïðàâåäëèâ³ñòü ôîð
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6.23. Îá÷èñëèòè ãðàíè
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Çà ïåð³îä ÷àñó â³ä t
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Ïðèêëàä 7.3.2. Ïîêàçà
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Â ÿêîñò³ ïðèêëàäà ð
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Òåîðåìà 7.6.2 (ïðî äèô
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Òàáëèöÿ ïîõ³äíèõ ñ
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7.13. r(ϕ) =ϕ sin ϕ + cos ϕ; î
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7.9.8. Çíàõîäèìî y′: Ò
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Ôóíêö³ÿ ïðîïîçèö³¿
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Îñê³ëüêè â êðèòè÷í
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ïðè ∆x =1 i x=1000: (∆y)⏐x
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Íàñë³äîê 3. ßêùî íà
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7.14.9. = ln x 1/ x = 1/ x 1 − =
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Çàóâàæåííÿ. Ðîçêëà
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Îö³íèìî n ( ) n+ 1 θx n+ 1
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Çâ³äñè ó êð - ó äîò
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Ð î ç â ’ ÿ ç à í í ÿ.
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âè áóäåòå âèâ÷àòè ó
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8.3. ⎛ ∫ ⎜sin x+ ⎝ 3 ⎞ dx
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8.12. ∫ sin( ax + b) dx ; 8.13.
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Ïåðøèé òèï äîäàíê³
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8.5. ²ÍÒÅÃÐÓÂÀÍÍß ÄÅ
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6 4 1 1 1 5 1 3 = ∫t − dt+ ∫t
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[ñ, b] ñåãìåíòà [a, b] ¿
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9.2.1. Îçíà÷åííÿ òà óì
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4. ßêùî ôóíêö³ÿ ó = f(
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9.4. ÎÑÍÎÂÍÀ ÔÎÐÌÓËÀ
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â³äíîñò³ äî ôîðìóë
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Îçíà÷åííÿ 9.6.1. Íåâë
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³ îñê³ëüêè +∞ ∫ 1 ðî
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Ïðèêëàä 9.6.12 (òåîðåò
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Ïðèêëàä 9.7.2. Îá÷èñë
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Îñê³ëüêè ôóíêö³ÿ ó
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Ãðàíèöÿ ö³º¿ ñóìè ï
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Ð î ç â ’ ÿ ç à í í ÿ.
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Òîä³ ìîæíà ïîêàçàò
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4 3 ∆Ψ = 10 − 10 = 9000 . Îò
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ïàäàº ç íàö³îíàëüí
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Çàóâàæåííÿ 2. Àíàëî
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10.2. ÃÐÀÍÈÖß ² ÍÅÏÅÐ
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Öå ð³âíÿííÿ ïðÿìî¿,
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Öåé æàðò³âëèâèé åê
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äàòêîâ³é âàðòîñò³
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Îçíà÷åííÿ 10.4.2 Äèôå
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∂u äå ( i = 1, 2, K , n) ∂x
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Àíàëîã³÷íî ( 0, 0 ) ( 0,
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Çàóâàæèìî, ùî ïîíÿò
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Òîä³: 1) ÿêùî a a a a > 0,
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ÂÏÐÀÂÈ Äîñë³äèòè í
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Òàêèì ÷èíîì, ìè âèç
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Ðèñ. 10.21 Â³äîìèé Îìó
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ßñíî, ùî âàð³àíò ë³
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11.1.3. Ïðî â³ëüíå ïàä
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ñòå, òî ìè çìîæåìî é
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êîîðäèíàò (ðèñ. 11.4).
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ÂÏÐÀÂÈ Ðîçâ’ÿçàòè
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ÂÏÐÀÂÈ Ðîçâ’ÿçàòè
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Äàë³ âàð³þºìî ñòàë
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11.6.2. Ë³í³éí³ îäíîð³
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Òîä³ ( ) ( ) y2 x ≡λy1 x ,
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2x Çàãàëüíèé ðîçâ’ÿ
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y′ ÷.í. = e x (2Ax +Ax 2 ), y
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x = + + + . 2 2 5 y x c1 c2x c3 x
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Ô(t) =7⋅ 10 6 +3⋅ 10 6 e -0,1t
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S n 1 1 1 = + + K + 1⋅2 2⋅ 3 n
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Îñê³ëüêè ðÿä (12.2.7) ç
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Çàóâàæåííÿ. Ðÿäè ç
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íà çàâæäè íå ïîðîæí
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ÂÏÐÀÂÈ Âèçíà÷èòè ³
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12.5.6. Äîâåäåííÿ ôîðì
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êö³þ, êîòðó íàé÷àñò
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Âèðàç âèäó a+bi=z, äå à
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òî z z r (cos ϕ + isin ϕ ) r (c
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13.16. z =− 2+ 2 3i ; 13.17. z =
Page 247 and 248:
14.2.4. Êîðåí³ òà ¿õí³
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y ⎛ π ⎞ x tg α= k , k ; ctg (
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1 2 tg α+ ctg α= = sin αcos α s
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14.4. ÅËÅÌÅÍÒÈ ÂÈÙÎ¯
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14.5. ÃÐÀÔ²ÊÈ ÄÅßÊÈÕ
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14.5.11. y = arctg x (ðèñ. 14.19
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14.7. ÍÀÁËÈÆÅÍÅ ÇÍÀ×Å
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Ê36 Êåðåêåøà Ï. Â. Ëå
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