# 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.

$\blacksquare$

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 8$: Functions - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections: Example $5.4$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.3$: Mappings