# Definition:Inclusion Mapping

## Definition

Let $T$ be a set.

Let $S\subseteq T$ be a subset.

The inclusion mapping $i_S: S \to T$ is the mapping defined as:

$i_S: S \to T: \forall x \in S: \map {i_S} x = x$

## Also known as

This is also known as:

the canonical inclusion of $S$ in $T$
the (canonical) injection of $S$ into $T$
the embedding of $S$ into $T$
the insertion of $S$ into $T$.

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

## Notation

Some sources use merely the symbol $i$ to denote the inclusion mapping.

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:

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

Some sources use the same symbol for the identity mapping as for the inclusion mapping without confusion, on the grounds that the domain and codomain of the latter are different.

## Also see

• Results about inclusion mappings can be found here.