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Math Help - Proving a function to be injective

  1. #1
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    Proving a function to be injective

    Hi all,

    I'll get right to the question:

    Suppose you are given functions f:A->B and g:B->C such that the composite function g(f(x)) is injective, prove that f is injective.

    I'm pretty sure I have it, but the last step seems a bit of a jump, and although i can't seem to find counter example to refute my proof, i'd like to make sure!

    here's what i think:

    Suppose x1 not= x2 for x1 and x2 in a set A

    =>g(f(x1)) not=g(f(x2))

    Therefore f(x1) not= f(x2) because if they WERE equal to each other, it would mean the composite function g(f(x1)) would =g(f(x2)) and we know this isn't true because it was explicitly stated that x1 not= x2 and that for the composite function to remain injective (another thing that is true at start of question) g(f(x1)) must not=g(f(x2)) with x1 not= x2.

    So by contradiction f(x1) not= f(x2) and so f is injective.

    Erm, it all just melts my brain reading over it, it's like it loops itself so many times...but here's hoping someone can help if i am wrong!

    Edit: How do I use proper math notation here, i've seen it on a few posts, but haven't figured it out yet ;/
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  2. #2
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    Quote Originally Posted by scorpio1 View Post
    Suppose you are given functions f:A->B and g:B->C such that the composite function g(f(x)) is injective, prove that f is injective.

    Suppose x1 not= x2 for x1 and x2 in a set A =>g(f(x1)) not=g(f(x2))
    herefore f(x1) not= f(x2) because if they WERE equal to each other, it would mean the composite function g(f(x1)) would =g(f(x2))
    Yes that is correct if I have read your notation correctly.

    Quote Originally Posted by scorpio1 View Post
    Edit: How do I use proper math notation here, i've seen it on a few posts, but haven't figured it out yet ;/
    You need to learn to use LaTeX. It is a type setting language.
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by scorpio1 View Post

    Edit: How do I use proper math notation here, i've seen it on a few posts, but haven't figured it out yet ;/
    see our LaTex forum. it is all you need
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