Inverse Relations and Inverse Functions

The identity functionDefinition: Let A be a set. The identity function iA : A æ A is thefunction defined by iA(x) = x for all x œ A.As a set of ordered pairs, iA µ A ◊ A consists of all pairs (a, a) fora œ A.

0

iA ga IAEA ip fan _x

a'go Kasa lR

ga A

r Ir s

A

The identity functionProposition: The identity function is bijective.

Proof ia is injective

We thief ga c A and in a ia ca then a a

Must Since each a and iaca _ashow clearly a a

EA is surjective

we must show that forany AoAthe tsana'eA

so that iata a Set a a then iala iAlaA

The inverse of a relationDefinition: Let A and B be sets and let R be a relation onR µ A ◊ B. The inverse relation R≠1 to R is the relation on B ◊ Adefined by

R≠1 = {(b, a) œ B ◊ A : (a, b) œ R}.

s

B r Aq

I

iit B

yr o i

ki tei i

Rmarkedpb

Examples of inverse relationsExample: Let A = R and let R be the relation <. Then R consistsof all pairs (a, b) œ R ◊ R with a < b. The inverse relation R≠1

consists of all pairs (b, a) œ R ◊ R with (a, b) œ R. Thus R≠1 isthe relation >.

R Cab f a ab a be R

pi Cb as I as b a beIR a b a b

R y J x y r

i cops Y ri

r rR i

r r

R pi y

Another exampleExample: Let A = Z and let R be the relation “divides”, so that Rconsists of pairs (a, b) œ Z ◊ Z where a|b. The inverse relation R≠1

consists of pairs (a, b) where b|a, or, in other words, where a is amultiple of b.So the inverse relation to a|b, meaning a is a divisor of b, is therelation aRb when a is a multiple of b.

R a lb f 7C c 2 so that beac RE 2 2

R Cb a db a be 2 ca b bla

api'b means that a is a m pkof b

iii