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Math Help - Equality of functions

  1. #1
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    Equality of functions

    Hello,

    I have a following problem to solve:

    For some function  f:X \rightarrow Y and C \subseteq Y I have to find something, that would equal f(f^{-1}(C)) = ?.

    I managed to show that f(f^{-1}(C)) \subseteq C and I have a counterexample that disproves  C \subseteq f(f^{-1}(C)) . Yet I have no idea what f(f^{-1}(C)) could equal to.
    Last edited by MachinePL1993; December 4th 2012 at 10:49 AM.
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  2. #2
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    Re: Equality of functions

    it's always included in C, that's all we can say. Exemple f R-R x|-> x, and C is [-4,4], we have f(f^-1(C))=[0,4] =/= C
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  3. #3
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    Re: Equality of functions

    Could I just write f(f^{-1}(C))=\{y| y \in C \wedge \exists_{x \in X}(f(x)=y)\}? Because  f(f^{-1}(C)) equals the elements of C that can be reached from X.
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  4. #4
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    Re: Equality of functions

    Quote Originally Posted by MachinePL1993 View Post
    For some function  f:X \rightarrow Y and C \subseteq Y I have to find something, that would equal f(f^{-1}(C)) = ?.

    This is much like your other post.

    f(f^{-1}(C)) = C if and only if f is surjective.
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  5. #5
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    Re: Equality of functions

    Yes you can write that.
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