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Math Help - Proof of being onto given a composition is onto?

  1. #1
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    Proof of being onto given a composition is onto?

    Suppose that g: A -> B and f: B-> C and that f o g is onto. Show that f is onto.

    Here's what I have written on my pad:

    Proof: Suppose g: A -> B and f: B-> C and that f o g is onto. Observe that f o g: A -> C. Since we have that f o g is onto, then C = (f o g)(A) = f(g(A)).
    The next step isn't quite coming to me so easily. I want to say this:

    Proof: Suppose g: A -> B and f: B-> C and that f o g is onto. Observe that f o g: A -> C. Since we have that f o g is onto, then C = (f o g)(A) = f(g(A)). Since the composition of g with f maps to C, then f must be onto.
    But I'm not too confident in that.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mjlaz View Post
    Suppose that g: A -> B and f: B-> C and that f o g is onto. Show that f is onto.

    Here's what I have written on my pad:



    The next step isn't quite coming to me so easily. I want to say this:



    But I'm not too confident in that.
    Suppose that f was not onto, then since f(A)\subseteq B we see that g(f(A))\subseteq g(B)\subset C
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