082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

180 Chapter 5 Discrete mathematics
A
0
r : A — B
‘take plus or minus the square root of’
1
9
4
Figure5.3
Arelation between setsAandB.
2
0
1
3 –3
B
–1
–2
s : D — E
s : m — 3m + 1
D 1
0
1
2
3
4
5
4
7
10
13
16
19
22
Figure5.4
The relationsmaps elements ofDtoE.
E
andB = {−3,−2,−1,0,1,2,3}theneachelementofBisplusorminusthesquareroot
of some element ofA. We can depict thisas inFigure 5.3.
The rule, which, when given an element ofA, produces an element ofB, is called a
relation. If the ruleof the relation isgiven the symbolr wewrite
r:A→B
and say ‘the relation r maps elements of the set A to elements of the set B’. For the
example above, we can writer : 1 → ±1,r : 4 → ±2, and generallyr : x → ± √ x.
Thesetfromwhichwechooseourinputiscalledthedomain;thesettowhichwemap
is called the co-domain; the subset of the co-domain actually used is called the range.
As weshall see thisneed notbe the whole of the co-domain.
A relationrmaps elements of a setD, called the domain, to one or more elements
ofasetC, called the co-domain. We write
r:D→C
Example5.5 IfD = {0,1,2,3,4,5} andE = {1,4,7,10,13,16,19,22} and the relation with symbolsis
defined bys :D→E,s :m→3m +1, identify the domain and co-domain of
s. Draw a mapping diagramtoillustrate the relation.What isthe range ofs?
Solution Thedomainofsisthesetofvaluesfromwhichwechooseourinput,thatisD = {0,1,2,
3,4,5}.Theco-domainofsisthesettowhichwemap,thatisE = {1,4,7,10,13,16,
19,22}.Therules :m→3m+1enablesustodrawthemappingdiagram.Forexample,
s : 3 → 10 and so on. The diagram is shown in Figure 5.4. The range ofsis the subset
ofE actuallyused,inthiscase {1,4,7,10,13,16}. Wenotethatnotalltheelementsof
the co-domainareactuallyused.
The notation introduced is very similar to that for functions described in Section 2.3.
This is no accident. In fact, a function is a very special form of a relation. Let us recall
the definition ofafunction:
‘Afunctionis arulewhich whengivenaninputproducesasingleoutput’.
If we study the two relations r and s, we note that when relation r received input, it
couldproducetwooutputs.Onthemappingdiagramthisshowsupastwoarrowsleaving