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Math Help - onto and composition proof

  1. #1
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    onto and composition proof

    I've got this review problem:

    Let f: X -> Y and g: Y -> Z be functions such that gf: X -> Z is onto. Prove that g must be onto.
    When I draw it out on paper, it seems quite intuitive: if g is not onto, then there is a "connection" that can't be made between X and some member of Z. But I'm having trouble thinking of how to formally express that...
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  2. #2
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    Re: onto and composition proof

    Quote Originally Posted by infraRed View Post
    I've got this review problem:

    Let f: X -> Y and g: Y -> Z be functions such that gf: X -> Z is onto. Prove that g must be onto.

    When I draw it out on paper, it seems quite intuitive: if g is not onto, then there is a "connection" that can't be made between X and some member of Z. But I'm having trouble thinking of how to formally express that...
    try contradiction.

    Suppose that g is not onto. Then $\exists z_0 \in Z \ni g(y) \neq z_0~\forall y \in Y$

    Does $gf(x) = z_0$ for some $x \in X$ ? Can $gf(\cdot)$ thus be onto?
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