Definition:Inclusion Mapping

Definition
The inclusion mapping $i_S: S \to T$ is a mapping on a set $S$ defined when $S \subseteq T$:
 * $i_S: S \to T: \forall x \in S: i_S \left({x}\right) = x$

It can be seen that the inclusion mapping $i_S$ is the restriction to $S$ of the identity mapping $I_T: T \to T$.

Alternative Names
This is also known as:
 * the canonical inclusion of $S$ in $T$
 * the (canonical) injection of $S$ into $T$
 * the embedding of $S$ into $T$.

However, beware of confusing this with the use of the term canonical injection in the field of abstract algebra.

Notation
Beware the notation. Always be sure you understand what is being used.

Some authors use $i_S$ (or similar) for the identity mapping, and so use something else, probably $\iota_S$ (Greek "iota"), for the inclusion mapping.

Another notation is:
 * $f: S \subseteq T$

or
 * $f: S \stackrel f {\subseteq} T$

The symbol $\iota$ is also used in the context of analytic number theory to denote the Identity Arithmetic Function:
 * $\iota \left({n}\right) = \begin{cases}

1 & : n = 1 \\ 0 & : n \ne 1 \end{cases}$