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Set theoretic problems

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

    if A is not empty and s:A->B is injective, then there is a surjective function f:B->A such that f(s(a))=a for all a in A. Do not use the Axiom of Choice

    3. The attempt at a solution

    for all b', b'' in B s(b')=s(b'') means b'=b''. So s^-1 (c) is unique in A. Because any point in A is sent to at most one point in B, we can just let f send every point in B of the form s(c) to c.

    now we only need to send all the other points somewhere. Here I need to 'pick' some point once again. Why do I not need the AC here?
  2. jcsd
  3. Oct 23, 2008 #2


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    Hi jacobrhcp! :smile:

    Because the axiom of choice is only for choosing the whole set … here, you only need to choose one element. :wink:

    For revision, see http://en.wikipedia.org/wiki/Axiom_of_choice
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