My question was inspired by this answer.
Suppose $A$ is an infinite set. Does its double, $A\times\{0,1\}$, always have the same cardinality?
In my head I quickly spotted a simple proof that the answer is yes if we assume Axion of choice / Zorn's lemma. I have stumbled upon a couple of related questions, but the question looks so simple that I would like it answered without assuming AC.