# Inclusion Mapping is Injection

## Theorem

Let $S, T$ be sets such that $S$ is a subset of $T$.

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

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

is an injection.

For this reason the inclusion mapping can be known as the canonical injection of $S$ to $T$.

## Proof

Suppose $\map {i_S} {s_1} = \map {i_S} {s_2}$.

 $\ds \map {i_S} {s_1}$ $=$ $\ds s_1$ Definition of Inclusion Mapping $\ds \map {i_S} {s_2}$ $=$ $\ds s_2$ Definition of Inclusion Mapping $\ds \map {i_S} {s_1}$ $=$ $\ds \map {i_S} {s_2}$ by definition $\ds \leadsto \ \$ $\ds s_1$ $=$ $\ds s_2$ from above

Thus $i_S$ is an injection by definition.

$\blacksquare$