# Inverse of Identity Mapping

## Theorem

Let $S$ be a set.

Let $I_S: S \to S$ be the identity mapping on $S$.

Then the inverse of $I_S$ is itself:

$\paren {I_S}^{-1} = I_S$

## Proof

From the nature of the identity mapping, we have:

$I_S \circ I_S = I_S$

from which it follows by definition that $I_S$ is the inverse of itself.

$\blacksquare$