Identity function

The function that maps every element of a set to itself

Identity function

An identity function on a set $A$ is the function $\mathrm{id}_A:A\to A$ defined by

\mathrm{id}_A(a)=a\quad\text{for all }a\in A.

Identity functions are neutral elements for composition : if $f:A\to B$ is any function, then $f\circ \mathrm{id}_A=f$ and $\mathrm{id}_B\circ f=f$ . The identity function is always bijective , and its inverse function is itself.

Examples:

On the real numbers , $\mathrm{id}_{\mathbb{R}}(x)=x$ for all $x\in\mathbb{R}$ .
On the power set $\mathcal{P}(A)$ of a set $A$ , the identity function sends each subset $S\subseteq A$ to itself.