Restriction of Equivalence Relation is Equivalence

Theorem

Let $S$ be a set.

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

Let $\RR$ be an equivalence relation on $S$.

Then by definition:

$\RR$ is a reflexive relation on $S$

$\RR$ is a symmetric relation on $S$

$\RR$ is a transitive relation on $S$.

Then:

from Restriction of Reflexive Relation is Reflexive, $\RR {\restriction_T}$ is a reflexive relation on $T$

from Restriction of Symmetric Relation is Symmetric, $\RR {\restriction_T}$ is a symmetric relation on $T$

from Restriction of Transitive Relation is Transitive, $\RR {\restriction_T}$ is a transitive relation on $T$

and so it follows by definition that $\RR {\restriction_T}$ is an equivalence relation on $T$.

$\blacksquare$

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

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings
2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations: Exercise $3.2$