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Math Help - Properties of injective functions

  1. #16
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    Part 1 should start as: let y\in f(f^{-1}(H)). Then y=f(x) for some x\in f^{-1}(H) ...

    Part 2 should start as: let y\in H. Now, by surjectivity of f, there exists an x\in A such that f(x)=y ...
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  2. #17
    Member Mollier's Avatar
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    1)

    Let y\in f(f^{-1}(H)). Then y=f(x) for some x\in f^{-1}(H).
    f^{-1}(H)=\{x\in A: f(x)\in H\},
    hence if x\in f^{-1}(H) then y=f(x)\in H.

    2)

    Let y\in H. Now, by surjectivity of f, there exists an x\in A such that f(x)=y.
    Since f(x)=y and y\in H we have that x \in f^{-1}(H) and so f(x)=y \in f(f^{-1}(H)).

    Not too pleased with the last one..

    Thanks!
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  3. #18
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by ojones View Post
    Plato:

    "Proofs by contradiction should not be used if a direct proof is available" - Paul Halmos on mathematical writing.

    "All students are enjoined in the strongest possible terms to eschew proofs by contradiction!" - Halsey Royden in Real Analysis.
    By the mathematicians we picked I guess it's Topology vs. Measure Theory :P
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  4. #19
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    Not really Drexel28. I was just trying to find comments by known mathematicians on the subject. RL Moore is more famous for teaching than research and his comments should be taken in that context.
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