Definition:Identity Mapping

Definition
The identity mapping, or identity transformation, of a set $S$ is the mapping $I_S: S \to S$ defined as:
 * $I_S = \left\{{\left({x, y}\right) \in S \times S: x = y}\right\}$

or alternatively:
 * $I_S = \left\{{\left({x, x}\right): x \in S}\right\}$

That is:
 * $I_S: S \to S: \forall x \in S: I_S \left({x}\right) = x$

Informally, it is a transformation in which every element is a fixed element.

The symbol $1_S$ is also seen, as are $i_S$, $id_S$, $\operatorname {id}_S$, $I$, $\iota_S$ and $\iota$.

Beware of the possibility of confusing with the inclusion mapping.

Also see

 * Identity Mapping is a Bijection
 * Inverse of Identity Mapping
 * Identity Mapping is Left Identity
 * Identity Mapping is Right Identity

Note that the identity mapping on $S$ is the same as the diagonal relation $\Delta_S$ on $S$.