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Math Help - Prove a composite function is one-to-one...

  1. #1
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    Prove a composite function is one-to-one...

    Hi, can someone check a proof I did? It's a very rough draft and I'm probably over-thinking it...

    Let f:X->Y and g:Y->Z be functions.

    Prove that if g o f is one-to-one, then f is one-to-one.


    If f is NOT one to one, then there exists several y belonging Y that correspond to each x of X.

    Since g is one-to-one, then there corresponds one z of Z for each y of Y.

    So if f is not one to one then f(x) has multiple values of y and g(f(x)) has multiple values of z corresponding to each y, and hence, g(f(x)) is not one-to-one.

    But since g(f(x)) IS one-to-one, then f(x) must be as well.


    Does this make sense to anyone?
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  2. #2
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    Suppose that f(a)=f(b) does that mean g(f(a))=g(f(b))? WHY?
    What can you then conclude?
    How does that prove it?
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