Showing that the cardinality of two power sets are equal

Question

For two sets have A and B (which aren't finite), where |A| =|B| is it true that the cardinality of power set of A is equal to the cardinality of the power set of B?

I know its true I just want to know a formal proof.

Thanks for the help :).

Answer 1

Hint: The assertion $\lvert A\rvert=\lvert B\rvert$ means that there is a bijection $f\colon A\longrightarrow B$. Use it to define a bijection from $\mathcal{P}(A)$ onto $\mathcal{P}(B)$.

Answer 2

The power set $\mathcal{P}(X)$ of a set $X$ can be seen as the set of all functions from $X$ to $\{0,1\}$. Clearly, the cardinality of this set depends only on the cardinality of $X$.

Therefore, if $\lvert A\rvert = \lvert B \rvert$, then $\lvert \mathcal{P}(A) \rvert = \lvert \mathcal{P}(B) \rvert$.