Restriction of Connected Relation is Connected

From ProofWiki
Jump to navigation Jump to search


Let $S$ be a set.

Let $\RR \subseteq S \times S$ be a connected relation on $S$.

Let $T \subseteq S$ be a subset of $S$.

Let $\RR {\restriction_T} \subseteq T \times T$ be the restriction of $\RR$ to $T$.

Then $\RR {\restriction_T}$ is a connected relation on $T$.


Suppose $\RR$ is connected on $S$.

That is:

$\forall a, b \in S: a \ne b \implies \tuple {a, b} \in \RR \lor \tuple {b, a} \in \RR$


\(\ds a, b\) \(\in\) \(\ds T\)
\(\ds \leadsto \ \ \) \(\ds \tuple {a, b}\) \(\in\) \(\ds T \times T\)
\(\, \ds \land \, \) \(\ds \tuple {b, a}\) \(\in\) \(\ds T \times T\) Definition of Ordered Pair and Definition of Cartesian Product
\(\ds \leadsto \ \ \) \(\ds \tuple {a, b}\) \(\in\) \(\ds \paren {T \times T} \cap \RR\)
\(\, \ds \lor \, \) \(\ds \tuple {b, a}\) \(\in\) \(\ds \paren {T \times T} \cap \RR\) as $\RR$ is connected on $S$
\(\ds \leadsto \ \ \) \(\ds \tuple {a, b}\) \(\in\) \(\ds R \restriction_T\)
\(\, \ds \lor \, \) \(\ds \tuple {b, a}\) \(\in\) \(\ds R {\restriction_T}\) Definition of Restriction of Relation

and so $\RR {\restriction_T}$ is connected on $T$.


Also see