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Proving an equivalence relation using inverse functions

  1. Nov 27, 2011 #1
    1. The problem statement, all variables and given/known data
    Let f : A → B be a function and let Γ ⊂ B × B be an equivalence relation on B. Prove that the set (f × f)^-1 (Γ) ⊂ A × A (this can be described as {(a, a′) ∈ A × A|(f(a), f(a′)) ∈ Γ}) is an equivalence relation on A.

    2. Relevant equations

    3. The attempt at a solution
    Let (f(a),f(a’)) ⊂ Γ. Since f(a) and f(a’) hold an equivalence relation with eachother, it follows that a and a’ hold an equivalence relation with eachother. Since f(a) and f(a’) are arbitrary elements of Γ, it follows that (fxf)-1Γ ⊂ A x A is an equivalence relation on A.

    I'm not sure if thi is the right approach. In particular im not sure that i can say that f(a) and f(a') holding an equivalence relation means that a and a' hold one too.
    Last edited: Nov 27, 2011
  2. jcsd
  3. Nov 27, 2011 #2
    Does anybody know if im correct? I'm not sure if I'm skipping steps.
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