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 $T$.

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.

$\blacksquare$

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