Background
We are proving that two free groups on sets $S_1,S_2$ are isomorphic iff the sets have the same cardinality. If they are finite, it's easy to show, by the freeness of the groups, that $2^{|S_1|}=2^{|S_2|}$, and thus the cardinalities are the same. Otherwise the power sets are in bijection, which is shown in the same way.
The problem is that in the infinite case, I'm not sure how to prove that bijection of power sets implies bijection of sets. I tried by contradiction, but only got stuck with one p.s. being in bijection with a proper subset of the other and not being able to rule out that this proper subset be in bijection with the whole p.s.
Question
Is it true that if the power sets of two sets are in bijection, then the two sets also are? If so, how do I prove it? Otherwise, can you provide a counterexample? And if this does not hold how do I complete the proof in the background section?
