TITLE: Proving: If $|A\times B| = |A\times C|$, then $|B|=|C|$.
QUESTION [0 upvotes]: Prove or disprove:
If $A\times B\sim A\times C$, then $B\sim C$.
("$\sim$": "numerically equivalent" / "has the same cardinality as")
What bijection/counterexample should I use to prove/disprove it?

REPLY [0 votes]: Consider $A=\mathbb{N}$. Now take $B$ and $C$ any different, finite, non-empty sets. Actually it seems that you can take $A$ any infinite set, but for this one I'm not sure.
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