If two sets $A$ and $B$ have equal power sets, can we conclude that $A=B$
I feel like they are equal, by intuition and by analyzing particular cases, but I don't know how to write a formal proof of this. Thanks in advance.
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Note that for any set $X$ we have $\bigcup \mathcal P(X) = X$ where $\mathcal P(X)$ is a power set of $X$. Thus $$A=\bigcup \mathcal P(A) = \bigcup \mathcal P(B) = B$$.
Yes. Since A is in the power set of B, A is a subset of B. Likewise, B is a subset of A. Therefore, A = B.
Yes. If $\mathcal P(A)=\mathcal P(B)$, as $A\in\mathcal P(A)$, $A\in\mathcal P(B)$, which means $A\subset B$. Similarly, $B\subset A$, so $A=B$.