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

  1. #1
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    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.
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  2. #2
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    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 I_A with an injection and 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 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?
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