Does every set have a derangement?

Question

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?

Answer 1

Assuming all the elements are different, the bijections of a set of size $n$ are equivalent to permutations: label the elements $1,2,\ldots,n$. So there are as many derangements for a set of size $n$ as there are for the numbers $1,2\ldots,n$

Answer 2

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 $A$.