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One example I can think of is $f: \mathbb{Z_2} \to \mathbb{Z_2}$ given by $f(1) = 0$ and $f(0) = 1$.

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Such a function is called an involution. The Wikipedia article has a few examples, as does the previous question, "What's the name for the property of a function $f$ that means $f(f(x))=x$?". –  Rahul Jul 7 '12 at 2:36

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up vote 5 down vote accepted

In a sense, every such bijection is going to look the same. If $a,b \in X$, then we will either have things that look like $f(a) = a$ or $f(a) = b, f(b) = a$ (which I'm going to refer to as a single 'transposition.'

But this gives us infinitely many examples to choose from, even just in $\mathbb{Z}$. You might let $f$ be the identity on every element except, say, $1$ and $5$, such that $f(1) = 5, f(5) = 1$. Or you can have as many transpositions as you'd like.

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This is marginally related: godplaysdice.blogspot.com/2008/02/… –  user12014 Jul 7 '12 at 2:02
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It should be noted that we can only have as many disjoint transpositions as we like. I wonder if this question becomes deeper if we consider it in a category besides Set. –  Don L. Jul 7 '12 at 2:08

Interesting example. $f(x)=-x, x\in \mathbb{R}$, or $f(x)=\frac{1}{x}, x\in \mathbb{R}\backslash \{0\}$ would come to mind first for me. In general such a function can clearly be defined on any group-like structure with inverses, just by defining a function that takes every element to its inverse, and the identity (if it exists) to itself.

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