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My attempt: I know that \(\displaystyle \mathcal{P}((A\times A)\cup A)\sim\mathcal{P}(A\times A)\times\mathcal{P}(A)\sim\mathcal{P}(A) \times \mathcal{P}(A).\) How to prove that \(\displaystyle \mathcal{P}(A)\sim \mathcal{P}((A\times A)\cup A)\)? More generally, is it true that if \(\displaystyle X\) and \(\displaystyle Y\) are infinite and \(\displaystyle X\sim Y\), then \(\displaystyle X\cup Y\sim Y\)?