Restriction of Equivalence Relation is Equivalence

Theorem
Let $S$ be a set.

Let $\mathcal R \subseteq S \times S$ be an equivalence 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 an equivalence relation on $T$.

Proof
Let $\mathcal R$ be an equivalence relation on $S$.

Then by definition:
 * $\mathcal R$ is a reflexive relation on $S$
 * $\mathcal R$ is a symmetric relation on $S$
 * $\mathcal R$ is a transitive relation on $S$.

Then:
 * from Restriction of Reflexive Relation is Reflexive, $\mathcal R \restriction_T$ is a reflexive relation on $T$
 * from Restriction of Symmetric Relation is Symmetric, $\mathcal R \restriction_T$ is a symmetric relation on $T$
 * from Restriction of Transitive Relation is Transitive, $\mathcal R \restriction_T$ is a transitive relation on $T$

and so it follows by definition that $\mathcal R \restriction_T$ is an equivalence relation on $T$.

Also see

 * Restriction of Ordering is Ordering


 * Properties of Restriction of Relation‎ for other similar properties of the restriction of a relation.