Definition:Identity Mapping

Definition
The identity mapping, also known as identity operator 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 $i_S$, $id_S$, $\operatorname {id}_S$ and $\iota_S$.

The subscript is frequently removed if there is no danger of confusion as to which set is under discussion.

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 $S$.