# Thread: Let f:A->B and g:B->A. Let IA and IB be the identify functions on the sets A and B...

1. ## Let f:A->B and g:B->A. Let IA and IB be the identify functions on the sets A and B...

I also got stuck on this one, while I was studying for my final, any help appreciated!

Let f:A->B and g:B->A. Let IA and IB be the identity functions on the sets A and B, respectively. Prove each of the following:

a) If g of f = IA, then f is an injection.

b) If f of g = IB, then f is a surjection.

c) If g of f = IA and f of g = IB, then f and g are bijections and g = f^-1

**f^-1 means f inverse.

2. ## Re: Let f:A->B and g:B->A. Let IA and IB be the identify functions on the sets A and

We can generalize a) and b) by replacing $\displaystyle I_A$ with an injection and $\displaystyle I_B$ with a surjection. Clearly, an identity function is both an in injection and a surjection.

For a), assume that f is not an injection. Then it maps two distinct elements of A into one element of B. Can $\displaystyle g\circ f$ be an injection in this case?

Similarly, for b), if the second function is not a surjection, i.e., it does not cover all elements of B, then how a composition can be a surjection? Would you agree to live in the neighborhood where, even though the post office sorts the mail correctly, the letter carrier does not serve all houses?