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

 * Inclusion Mapping is Restriction of Identity