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).
Asaf Karagila
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Would you mind showing the proof in details? – math112358 Sep 19 '19 at 13:25
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You can find more details elsewhere on the site, e.g. https://math.stackexchange.com/questions/1041731/prove-that-if-a-is-an-infinite-set-then-a-times-2-is-equipotent-to-a – Asaf Karagila Sep 19 '19 at 16:46