TITLE: Trying to prove that cardinality of power sets are equal
QUESTION [5 upvotes]: So if we have two sets $X$ and $Y$, we know that if $|X| = |Y|$, then $|P(X)| = |P(Y)|$.  This means that there is a bijection $f: X → Y$.
What would a function be that maps elements of $P(X)$ to elements of $P(Y)$?
I can try to use the existing function $f$ to prove this new function.

REPLY [10 votes]: Big HINT: There’s really only one reasonable thing to try: for $A\subseteq X$ let $F(A)=\{f(x):x\in A\}$.