# Restriction of Connected Relation is Connected

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## Theorem

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 $T$.

## Proof

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

That is:

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

So:

\(\ds a, b\) | \(\in\) | \(\ds T\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds \tuple {a, b}\) | \(\in\) | \(\ds T \times T\) | |||||||||||

\(\, \ds \land \, \) | \(\ds \tuple {b, a}\) | \(\in\) | \(\ds T \times T\) | 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 $T$.

$\blacksquare$

## Also see

- Properties of Restriction of Relationâ€Ž for other similar properties of the restriction of a relation.