If two sets have the same cardinality, do their power sets have the same cardinality

Question

Is it true that if two sets have the same cardinality, their power sets have the same cardinality? If so, how to prove it?

Answer 1

Yes. Let $f: X \to Y$ be a bijection.

Then show that $\hat{f}: \mathscr{P}(X) \to \mathscr{P}(Y)$ given by $$\hat{f}(A) := f[A] (= \{f(x): x \in A\})$$

is a bijection between their power sets.