TITLE: How to prove using Zorn's lemma that if $X$ has at least two elements, then is a 1-1 function $H\colon X\to X$ such that $H(x)\neq x$ for all $x$?
QUESTION [5 upvotes]: This is the problem.
Let $X$ be an arbitrary set having at least two elements. Show that there is 1-1 function $H: X \to X$ such that for all $x \in X$, $H(x) \neq x$, by using Zorn's Lemma.
To prove by using Zorn's Lemma, I have to construct an ordered set.. but I have no idea. :(

REPLY [4 votes]: Use Zorn's lemma to get a maximal family of pairwise disjoint two-element subsets of $X$. If you're lucky, these pairs will cover all of $X$ and you can define $H$ to be the function that maps each $x\in X$ to the other element of its pair. If you're unlucky and there's some $a\in X$ not covered by your maximal family of pairs, then there's only one such $a$ by maximality. Adjoin $a$ to one of your pairs, say $\{p,q\}$, define $H$ on the $3$-element set $\{a,p,q\}$ by $a\mapsto p\mapsto q\mapsto a$, and define $H$ on all your other pairs as in the "lucky" case.