# 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.

## Properties

• Function $f:X \rightarrow X$ is an involution $\iff$ $\forall x \in X$ $f(x) = y \land f(y) = x$. This induces that both $(x,y)$ and $(y,x)$ are in f. By the definition of the inverse of a function, $\{ (y,x) | (x,y) \in f \}$ is the inverse of the function f. Therefore, the function f must contain $f^{-1}$. From this, it is obtained that $f^{-1} \subseteq f$. Simmilalry, we can show that $f \subseteq f^{-1}$. Thus, $f = f^{-1}$.

Rewriting the first line we have: function $f:X \rightarrow X$ is an involution $\iff$ $f = f^{-1}$.

• A function is an involution iff it is symmetric about the line $f(x)=x$ in the coordinate plane.