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Functions of (AUB)

  1. Oct 18, 2013 #1
    1. The problem statement, all variables and given/known data
    Let g be a function from ℝ to ℝ and for all subsets A and B of R.

    2. Relevant equations
    Prove that:
    [tex]g^{-1}(A\cup B)=g^{-1}(A)\cap g^{-1}(B)[/tex]
    and
    [tex]g^{-1}(A\cap B)=g^{-1}(A)\cup g^{-1}(B)[/tex]


    3. The attempt at a solution
    Our teacher gave us this problem but I think it's wrong because I was easily able to prove that:
    [tex]g^{-1}(A\cup B)=g^{-1}(A)\cup g^{-1}(B)[/tex]
    and
    [tex]g^{-1}(A\cap B)=g^{-1}(A)\cap g^{-1}(B)[/tex]
    but when i try to solve what he gave us this is all I can do:

    [tex]x\in g^{-1}(A\cup B)\Longleftrightarrow g(x)\in A\cup B\Longleftrightarrow g(x)\in Aorg(x)\in B\Longleftrightarrow x\in g^{-1}(A)\cup g^{-1}(B)[/tex]

    and the same goes for the second one. Can anyone tell me where I'm going wrong?
     
  2. jcsd
  3. Oct 18, 2013 #2

    pasmith

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    Homework Helper

    These are wrong. A counterexample to both is the zero function [itex]g(x) \equiv 0[/itex] with [itex]A = \{0\}[/itex] and [itex]B = \{1\}[/itex].

    These are correct.
     
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