Restriction of Connected Relation is Connected
Jump to navigation
Jump to search
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.