Subset of an infinite set with same cardinality

Question

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.

Answer 1

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).