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

5.2 Set theory 181
someelementsinA.Whenrelationsreceivedaninput,itproducedasingleoutput.This
shows up as a single arrow leaving each element in D. Hence the relation r is not a
function, whereas the relationsis. This leads to the following more rigorous definition
of a function.
A function f is a relation which maps each element of a set D, called the
domain, toasingleelement of a setC, called the co-domain. We write
f:D→C
Example5.6 IfM = {off, on},N = {0,1} and wedefine a relationr by
r:M→N
r:off→0
r:on→1
then the relationris a function since each element inM is mapped to a single element
inN.
Example5.7 IfP = {0,1} andQ = {high} and wedefine a relationr by
r:P→Q
r:1→high
thenrisnotafunction since each element inPisnot mapped toan element inQ.
All of the functions described in Chapter 2 have domains which are subsets of the real
numbers R.Theinputtoeachfunctionistheparticularvalueoftheindependentvariable
chosen from the domain and the output is the value of the dependent variable. When
dealing with continuous domains the graphs we have already considered replace the
mapping diagrams.
Example5.8 Find the domain,D, of the rational function f :D→Rgiven by
f:x→
3x
x −2
Solution Since no domain is given, we choose it to be the largest set possible. This is the set
of all real numbers except the value x = 2 at which point f is not defined. We have
D={x:x∈R,x≠2}.
EXERCISES5.2
1 Useset notationto describethe intervals onthexaxis
given by
(a) (−3,2) (b) [0,2] (c) [−2, −1)
(d) (3,6] (e) |x| < 1
2 Sketch the followingsets. [Hint:seeSection2.2 on
open andclosedintervals.]
(a) {x:x∈Rand2<x4}
(b) {x:x∈Rand−1x0}