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Homework Help: An intro to real analysis question. eazy?

  1. Sep 24, 2010 #1
    1. The problem statement, all variables and given/known data

    Let f : A -> B be a bijection. Show that if a function g is such that f(g(x)) = x for
    all x ϵ B and g(f(x)) = x for all x ϵ A, then g = f^-1. Use only the definition of a
    function and the definition of the inverse of a function.

    2. Relevant equations

    3. The attempt at a solution
    well since f is a bijection then there exist an f^-1 that remaps f back to A and since g does that then g is the unique inverse of f, or something like that please help im not very good that this stuff
  2. jcsd
  3. Sep 25, 2010 #2
    Yes, you only need to prove uniqueness of an inverse function. If f(g(x))=x, then g(x) must be the unique argument a for which f(a) = x. And so, this condition completely defines g for all the arguments. g is then obviously equal to f-1.
  4. Sep 25, 2010 #3
    thanks losiu for your response.
    but how do you prove that it is the unique inverse?
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