Restriction of Non-Connected Relation is Not Necessarily Non-Connected

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Theorem

Let $S$ be a set.

Let $\mathcal R \subseteq S \times S$ be a non-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 not necessarily a non-connected relation on $T$.


Proof

Proof by Counterexample:

Let $S = \set {a, b}$.

Let $\mathcal R = \set {\tuple {a, a}, \tuple {b, b} }$.

$\mathcal R$ is a non-connected relation, as can be seen by definition: neither $a \mathrel {\mathcal R} b$ nor $b \mathrel {\mathcal R} a$.

Now let $T = \set a$.

Then $\mathcal R {\restriction_T} = \set {\tuple {a, a} }$.

Then $\mathcal R {\restriction_T}$ is trivially connected on $T$.

$\blacksquare$


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