Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    2017 Award

    Staff: Mentor

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Similar Threads for Bijections need zorn's
I Differentiation on R^n ...need/ use of norms ...