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Homework Help: Finite Axiom of Choice

  1. Oct 23, 2008 #1
    1. The problem statement, all variables and given/known data

    Prove, without using the Axiom of Choice:

    if f: X->Y is surjective and Y is finite, there exists a 'section', a function s:Y->X such that f(s(y))=y for all y in Y

    Hint: perform induction over the cardinality of Y

    3. The attempt at a solution

    Induction over the cardinality of Y? I don't even know what they mean. I'd say this would require the AC because we need to pick one of the elements of f^-1 (y) for all y in Y. since X is infinite, one of these f^-1 (y) has to be infinite as well, and I don't know how to pick a specific element of this if I can't just say 'I pick some'.

    And I also don't understand the hint. Some help would be appreciated.
     
  2. jcsd
  3. Oct 23, 2008 #2

    HallsofIvy

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    Science Advisor

    You don't know "proof by induction"? Surely you are just misunderstanding. Since Y is finite, its cardinality is some positive integer, say n. Prove this statement by induction on n.

    If n= 1, that means that Y contains exactly one member, call it "y". There exist some function from X such that f(x)= y for all x in X. Can you find a function from Y to X, s(y) such that f(s(y))= y? That should be pretty easy.

    Now, suppose that statement is true when y contains n members and prove that it is true and prove it is true when Y contains n+1 members. (Hint: remove a specific one of the members of Y.)
     
    Last edited by a moderator: Oct 24, 2008
  4. Oct 24, 2008 #3
    Ah, thank you, yes I was misunderstanding. I do know proof by induction yes, but 'over the cardinality of Y' was a weird statement to me. This clears things up, I should be fine now.
     
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