Possible Duplicate:
power set cardinal equality
Let $X$ and $Y$ be sets, and suppose that $|\mathscr{P}(X)| = |\mathscr{P}(Y)|$ (where $\mathscr{P}$ denotes the power set).
Does it follow that $|X|=|Y|$?
Remark: It's obviously true for finite sets, as $2^m=2^n$ implies $m=n$.