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.

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

Also see

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