Identity Function

Definition

The identity function is a function \(\mathrm{id} : X \to X\) defined by:

\[ x \mapsto x.\]

It is important to note that when there is an obvious canonical embedding of one set into another, the function that does so is sometimes called the identity function. For example \(\mathrm{id} : \mathbb{N} \to \mathbb{Q}\) defined by

\[ x \mapsto x.\]

Although this is often better called the inclusion map.