inverse relations and inverse functions · 2020. 12. 17. · inverse relations and inverse...
TRANSCRIPT
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