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Math Help - Help with a basic proof

  1. #1
    Junior Member
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    Help with a basic proof

    Prove that, if f:A->B and g:B->C are functions with gof:A->C bijective, then f is injective and g is surjective.

    Now, I easily saw how g is surjective:
    Let x be an element of A
    Since gof:A->C is bijective, g(f(x)) is bijective
    Implies g(f(x)) is surjective and g(f(x)) is injective
    Thus g is surjective.

    But I've been fumbling around with trying to show that f is injective and I get lost. I'm sure it's something stupid I'm missing. Can anyone start me off in the right direction?
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  2. #2
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    Proving that f is injective is the easy part.
    Suppose that f(a)= f(b) then by definition g(f(a))= g(f(b)).
    But gof is bijective so gof(a)= gof(b) implies that a=b.
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