Restriction of Non-Connected Relation is Not Necessarily Non-Connected
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Theorem
Let $S$ be a set.
Let $\RR \subseteq S \times S$ be a non-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 not necessarily a non-connected relation on $T$.
Proof
Let $S = \set {a, b}$.
Let $\RR = \set {\tuple {a, a}, \tuple {b, b} }$.
$\RR$ is a non-connected relation, as can be seen by definition: neither $a \mathrel \RR b$ nor $b \mathrel \RR a$.
Now let $T = \set a$.
Then $\RR {\restriction_T} = \set {\tuple {a, a} }$.
Then $\RR {\restriction_T}$ is trivially connected on $T$.
$\blacksquare$
Also see
- Properties of Relation Not Preserved by Restriction‎ for other similar results.