\begin{definition}[Definition:Identity Mapping]
The '''identity mapping''' of a set $S$ is the mapping $I_S: S \to S$ 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:
:$I_S: S \to S: \forall x \in S: \map {I_S} x = x$
Informally, it is a mapping in which every element is a fixed element.
\end{definition}