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Math Help - Surjective functions

  1. #1
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    Surjective functions

    let f: From A to B. let g: From B to A.

    How do I prove f(g(x)) = x (indentity function) implies f is onto (surjective).
    Last edited by mr fantastic; November 3rd 2009 at 10:59 PM. Reason: Removed shouting
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  2. #2
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    Quote Originally Posted by l888l888l888 View Post
    let f: From A to B. let g: From B to A.

    How do I prove f(g(x)) = x (indentity function) implies f is onto (surjective).
    Suppose f is NOT surjective. Then there exist y in B such that, for all x in A, f(x)\ne y. Now, what is f(g(y))?
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    f(g(y))= y because f(g(x))=x
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    MHF Contributor Drexel28's Avatar
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    More explicitly (like HallsOfIvy said)

    Problem: Suppose that f:X\mapsto Y and g:Y\mapsto X are functions such that f\circ g=\iota_Y (identity mapping on Y). Prove that f:X\mapsto Y is surjective.

    Proof: Let y\in Y. Since g:Y\mapsto X we know that g(y)\in X. Therefore f(g(y))=y since f\circ g=\iota_Y. And since y was arbitrary this proves surjectivity.
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