Inclusion Mapping is Injection

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

\(\displaystyle \map {i_S} {s_1}\) \(=\) \(\displaystyle s_1\) Definition of Inclusion Mapping
\(\displaystyle \map {i_S} {s_2}\) \(=\) \(\displaystyle s_2\) Definition of Inclusion Mapping
\(\displaystyle \map {i_S} {s_1}\) \(=\) \(\displaystyle \map {i_S} {s_2}\) by definition
\(\displaystyle \leadsto \ \ \) \(\displaystyle s_1\) \(=\) \(\displaystyle s_2\) from above


Thus $i_S$ is an injection by definition.

$\blacksquare$


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