Involution

An involution is a function whose inverse is itself. From the perspective of set theory and functions, if a relation is a function and is symmetric, then it is an involution.

Examples

The function $y(x)=x$ has the inverse $x(y)=y$ , which is the same function, and thus $f(x)=x$ is an involution.
The logical NOT is an involution because $\neg { \neg p} \equiv p$ .
The additive negation is an involution because $--x=x$ .
The identity function $I_x$ is an involution because $I_x:X \rightarrow X = \{(x,x) | x \in X\}$ therefore, $\forall (x,x) \in I_x$ $f(x) = x$ and $f(f(x)) = x$ . Hence, it is an involution.
The multiplicative inverse is an involution because $\frac{1}{\frac{1}{x}}=x$ . In fact, for any $n \neq 0, f(x)=\frac{n}{x}$ is an involution.