Definition:Identity Mapping

The identity mapping of a set $$S$$ is the mapping $$I_S: S \to S$$ defined as:

$$I_S = \left\{{\left({x, y}\right) \in S \times S: x = y}\right\}$$

That is:

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

The symbol $$1_S$$ is also seen, as is $$i_S$$ and $$id_S$$.

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