Restriction of Connected Relation is Connected

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 Relation‎ for other similar properties of the restriction of a relation.