Bijections and need of zorn's lemma

[jostpuur]

Assume that there exists a bijection $\phi:X\to Y$. Also assume there exists some subsets $A\subset X$ and $B\subset Y$ such that a bijection $\varphi:A\to B$ exists too. Now Zorn's lemma implies that there exists a bijection $\psi:X\to Y$ such that $\psi(A)=B$.

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 $A$ and $B$ and countable, will the result also hold without Zorn's lemma?

 

[mfb]

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.

 

[jostpuur]

Ok, I guess you are right. I was originally conserned that when I try to fix $\phi$ so that $\tilde{\phi}(a_n)=b_n$ would hold, this step could interfere with the earlier fixings at points $a_1,\ldots, a_{n-1}$. But it seems now that it will not happen. But it's not trivial. Very difficult to see intuitively.