TITLE: Does every set have a derangement?
QUESTION [5 upvotes]: A derangement of a set $A$ is a bijection from $A$ to itself with no fixed points. Is it the case that every infinite set has a derangement?

REPLY [2 votes]: If $A$ is infinite, then it is in bijection with $\lbrace 0,1\rbrace\times A$. Swapping $0$ with $1$ is a derangement of $\lbrace 0,1\rbrace\times A$, and hence, using the mentioned bijection, a derangement of $A$.