Definition:Identity Mapping

Definition
The identity mapping of a set $S$ is the self-map $I_S: S \to S$ defined as:
 * $I_S = \set {\tuple {x, y} \in S \times S: x = y}$

or alternatively:
 * $I_S = \set {\tuple {x, x}: x \in S}$

That is:
 * $I_S: S \to S: \forall x \in S: \map {I_S} x = x$

That is, it is a mapping in which every element is a fixed element.

Also see

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


 * Definition:Diagonal Relation $\Delta_S$ on $S$: the same as the identity mapping on $S$