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Let $A$ be an infinite set. Show that there is a subset $B\subseteq A$ such that $|B|=|A-B|=|A|$. I've tried using Zorn's lemma with $P(A)$ and $\subset$ and got nowhere. I would like a hint.

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HINT: Turn the question around. Use Zorn's lemma to prove there is a bijection between $B$ and $B\times\Bbb \{0,1\}$, or $B\times\Bbb N$ or $B\times B$ (either one works out).

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