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Bijections and need of zorn's lemma

  1. May 18, 2013 #1
    Assume that there exists a bijection [itex]\phi:X\to Y[/itex]. Also assume there exists some subsets [itex]A\subset X[/itex] and [itex]B\subset Y[/itex] such that a bijection [itex]\varphi:A\to B[/itex] exists too. Now Zorn's lemma implies that there exists a bijection [itex]\psi:X\to Y[/itex] such that [itex]\psi(A)=B[/itex].

    I think I have now understood how to apply Zorn's lemma to things like this, so the above claim is clear to me (I assume).

    My question is that if [itex]A[/itex] and [itex]B[/itex] and countable, will the result also hold without Zorn's lemma?
     
  2. jcsd
  3. May 19, 2013 #2

    mfb

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    Let A={a1, a2, ...} and B={b1, b2, ...}
    This is possible as both are countable.

    Now you can just loop over the elements of A and B and "fix" ϕ to get a ϕ' with ϕ'(A)=B.
     
  4. Jun 11, 2013 #3
    Ok, I guess you are right. I was originally conserned that when I try to fix [itex]\phi[/itex] so that [itex]\tilde{\phi}(a_n)=b_n[/itex] would hold, this step could interfere with the earlier fixings at points [itex]a_1,\ldots, a_{n-1}[/itex]. But it seems now that it will not happen. But it's not trivial. Very difficult to see intuitively.
     
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