Results 1 to 4 of 4
Like Tree1Thanks
  • 1 Post By Plato

Thread: Proof regarding inverse functions

  1. #1
    Junior Member
    Joined
    Nov 2012
    From
    Poland
    Posts
    40

    Proof regarding inverse functions

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

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

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

    Can I assume through this that $\displaystyle x \in A$? If so then how to correctly justify it?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,782
    Thanks
    2824
    Awards
    1

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

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

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

    Then $\displaystyle f^{-1}(f(A))=~?$
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Nov 2012
    From
    Poland
    Posts
    40

    Re: Proof regarding inverse functions

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

    So is the following proof correct?

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

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

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

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

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

    Is this right?
    Last edited by MachinePL1993; Dec 3rd 2012 at 02:03 PM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,782
    Thanks
    2824
    Awards
    1

    Re: Proof regarding inverse functions

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

    That is too much. Try to prove
    $\displaystyle A\subseteq f^{-1}(f(A))$
    Thanks from MachinePL1993
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proof: Inverse Functions
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Sep 2nd 2012, 02:46 AM
  2. Replies: 3
    Last Post: Feb 20th 2011, 08:24 AM
  3. Replies: 2
    Last Post: Dec 9th 2009, 05:09 AM
  4. Replies: 2
    Last Post: Oct 19th 2009, 02:47 AM
  5. Inverse Composition of Functions Proof
    Posted in the Discrete Math Forum
    Replies: 9
    Last Post: Sep 17th 2007, 05:26 AM

Search Tags


/mathhelpforum @mathhelpforum