Hey guys, quick question, and I apologize if it's already been asked recently. But is it possible to prove two sets are equal if they have the same power sets? That is, can you say that A=B if A and B have the same power set?
I have a feeling the answer's no, but I can't think of an example where it wouldn't be true.
If two sets, X and Y, have the same power sets (the power set of set A is the collection of all subsets of A) then the singleton sets in the power sets must be the same. But X is just equal to the union of all singleton sets in the power set of X and Y is the union of all singleton sets in the power set of Y.