Inclusion Mapping is Injection

From ProofWiki
Jump to navigation Jump to search

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.

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Inclusion_Mapping_is_Injection&oldid=417496"