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Spivak chapter 3 problem 24 - proof of a composition

  1. Jul 18, 2017 #1
    1. The problem statement, all variables and given/known data
    Suppose g is a function with the property that g(x) =/= g(y) if x=/=y.
    Prove that there is a function f such that f( g(x) ) = x. (The composition)

    2. Relevant equations
    Definition of a function, collection of ordered pairs;
    g(x) =/= g(y) if x=/=y;
    x → g(x) → x (The composition that has to be proven).

    3. The attempt at a solution
    Since all g(x) are actually unique, that means there is a function f whose domain is a collection of all g(x) and that assigns the value x to all g(x), so that f is a collection of ordered pairs of the form (g(x), x). In other words, there is no contradiction from definition of a function, thus such a function does exist.

    My problem is that I am extremely careful with proofs and to be honest this "proof" of mine seems lazy and wrong and full of holes. The part where I assume that f can assign x to g(x) seems very sketchy. I also found out that this is somehow a proof that there is an inverse function f for function g, if g(x) are all unique.

    So I wonder if this is actually the right way of doing things, or did I miss sth crucial and my "proof" has bunch of holes in it or is it even plain miss from the start.
     
  2. jcsd
  3. Jul 18, 2017 #2
    I would say it is correct.
     
  4. Jul 18, 2017 #3
    Thank you :)
     
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