# Restriction of Connected Relation is Connected

## 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 $T$.

$\blacksquare$