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Math Help - Proof regarding inverse functions

  1. #1
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    Proof regarding inverse functions

    Hello,
    I've got to solve a following problem.

    f is a function like this f: X \rightarrow Y. And two sets A,B  \subseteq X. I've got to figure out how are the following two sets related:
    f^{-1}(f(A)) and A

    I got to here:
    x \in  f^{-1}(f(A)) \Leftrightarrow f(x)  \in f(A)

    Can I assume through this that  x \in A? If so then how to correctly justify it?
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  2. #2
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    Re: Proof regarding inverse functions

    Quote Originally Posted by MachinePL1993 View Post
    Hello,
    I've got to solve a following problem.

    f is a function like this f: X \rightarrow Y. And two sets A,B  \subseteq X. I've got to figure out how are the following two sets related:
    f^{-1}(f(A)) and A

    I got to here:
    x \in  f^{-1}(f(A)) \Leftrightarrow f(x)  \in f(A)

    Can I assume through this that  x \in A? If so then how to correctly justify it?
    Let f=\{(1,a),~(2,a),~(3,b),~(4,b)\} and A=\{1,3\}

    Then f^{-1}(f(A))=~?
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  3. #3
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    Re: Proof regarding inverse functions

    Then f^{-1}(f(A))=\{1,2,3,4\}

    So is the following proof correct?

    So let's assume that set  C is defined like this C=\{f(x)|x \in A\}, this means that  C \subseteq Y and  C=f(A) .

    So let's take any x such that x \in A, then  f(x) \in C

    Since  C=f(A) , this means that f^{-1}(f(A))=f^{-1}(C)

    So I have proven that  x \in A \rightarrow x \in f^{-1}(f(A)) , in other words A \subseteq f^{-1}(f(A))

    But  f^{-1}(f(A)) is not a subset of  A which can be proven by a counterexample like the one above.

    Is this right?
    Last edited by MachinePL1993; December 3rd 2012 at 03:03 PM.
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  4. #4
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    Re: Proof regarding inverse functions

    Quote Originally Posted by MachinePL1993 View Post
    So is the following proof correct?
    Since  C=f(A) , this means that f^{-1}(f(A))=f^{-1}(C)
    So I have proven that  x \in A \rightarrow x \in f^{-1}(f(A)) , in other words  f^{-1}(f(A)) \subseteq A
    But  f^{-1}(f(A)) is not a subset of  A which can be proven by a counterexample like the one above.

    That is too much. Try to prove
    A\subseteq f^{-1}(f(A))
    Thanks from MachinePL1993
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