Question

2. Let ????:???? →???? be a bijective function and ????-1:????→???? be its inverse function. Prove that ????-1o????= ????A and ????o????-1= ????b where ????A is the identity function on the set ???? and ????B is the identity function on the set ????

Answer 1

`f^{-1` o f = A and f o
`f^{-1` = B, where f is a bijective function and f^-1 is its inverse function.

Let f: A → B be a bijective function, and let
`f^{-1`: B → A be its inverse function.

We need to show that
`f^{-1` o f = A and f o
`f^{-1` = B.

To prove
`f^{-1` o f = A, we need to show that (
`f^{-1` o f)(x) = x for all x in A.

Let x be an arbitrary element in A. Since f is a bijective function, there exists a unique y in B such that f(x) = y. Therefore, we can write:

(
`f^{-1` o f)(x) =
`f^{-1`(f(x))

=
`f^{-1`(y)

Since
`f^{-1` is the inverse function of f, we know that
`f^{-1`(y) = x. Therefore, we can write:

(
`f^{-1` o f)(x) = x

This holds for all x in A, so we have shown that
`f^{-1` o f = A.

To prove f o
`f^{-1` = B, we need to show that (f o f^-1)(y) = y for all y in B.

Let y be an arbitrary element in B. Since f is a bijective function, there exists a unique x in A such that f(x) = y. Therefore, we can write:

(f o
`f^{-1`)(y) = f(
`f^{-1`(y))

= x

Since
`f^{-1` is the inverse function of f, we know that
`f^{-1`(y) = x. Therefore, we can write:

(f o
`f^{-1`)(y) = y

This holds for all y in B, so we have shown that f o
`f^{-1`= B.

Therefore, we have proven that
`f^{-1`o f = A and f o
`f^{-1` = B, where f is a bijective function and
`f^{-1` is its inverse function.

Complete question: Let