Prove Theorem (b): If f: X ? Y is a one-to-one and onto

Chapter 7, Problem 14E

(choose chapter or problem)

Problem 14E

Prove Theorem (b): If f: X → Y is a one-to-one and onto function with inverse function f−−1: Y → X, then f o f−1 = IY, where IY is the identity function on Y.

Theorem

Composition of a Function with Its Inverse

If f : X → Y is a one-to-one and onto function with inverse function f−1: Y → X, then

(a) f −1 ◦ f = IX and (b) f ◦ f−1 = IY.

Proof:

Part (a):Suppose f : X → Y is a one-to-one and onto function with inverse function f−1: Y → X. [To show that f−1 ◦ f = IX , we must show that for all x∈X, (f−1 ◦ f)(x)= x.] Let x be any element in X. Then

(f−1 ◦ f)(x)= f−1(f(x))

by definition of composition of functions. Now the inverse function f−1 satisfies the condition

Property 1

f−1(b)= a ⇔ f(a)= b for all a ∈ X and b ∈ Y.

Let

Equation 2

x' = f−1(f(x)).

Apply property 1 with x' playing the role of a and f(x)playing the role of b. Then

f(x')= f(x).

But since f is one-to-one, this implies that x' = x. Substituting x for x' in equation 2 gives

x = f−1(f(x)).

Then by definition of composition of functions,

(f−1 ◦ f)(x)= x,

as was to be shown.

Part (b):This is exercise 14 at the end of this section.