# Restriction of Connected Relation is Connected

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## Theorem

Let $S$ be a set.

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

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

Let $\mathcal R \restriction_T \ \subseteq T \times T$ be the restriction of $\mathcal R$ to $T$.

Then $\mathcal R \restriction_T$ is a connected relation on $T$.

## Proof

Suppose $\mathcal R$ is connected on $S$.

That is:

- $\forall a, b \in S: a \ne b \implies \left({a, b}\right) \in \mathcal R \lor \left({b, a}\right) \in \mathcal R$

So:

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

\(\displaystyle \implies \ \ \) | \(\displaystyle \left({a, b}\right)\) | \(\in\) | \(\displaystyle T \times T\) | ||||||||||

\(\, \displaystyle \land \, \) | \(\displaystyle \left({b, a}\right)\) | \(\in\) | \(\displaystyle T \times T\) | by definition of ordered pair and cartesian product | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle \left({a, b}\right)\) | \(\in\) | \(\displaystyle \left({T \times T}\right) \cap \mathcal R\) | ||||||||||

\(\, \displaystyle \lor \, \) | \(\displaystyle \left({b, a}\right)\) | \(\in\) | \(\displaystyle \left({T \times T}\right) \cap \mathcal R\) | as $\mathcal R$ is connected on $S$ | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle \left({a, b}\right)\) | \(\in\) | \(\displaystyle R \restriction_T\) | ||||||||||

\(\, \displaystyle \lor \, \) | \(\displaystyle \left({b, a}\right)\) | \(\in\) | \(\displaystyle R \restriction_T\) | by definition of restriction of relation |

and so $\mathcal R \restriction_T$ is connected on $T$.

$\blacksquare$

## Also see

- Properties of Restriction of Relation for other similar properties of the restriction of a relation.