Properties of Restriction of Relation

Theorem
Let $S$ be a set.

Let $\mathcal R \subseteq S \times S$ be a 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 $T$.

If $\mathcal R$ on $S$ has any of the properties:


 * Reflexive
 * Antireflexive
 * Symmetric
 * Antisymmetric
 * Asymmetric
 * Transitive
 * Antitransitive
 * Connected

... then $\mathcal R \restriction_T$ on $T$ has the same properties.