# Restriction of Equivalence Relation is Equivalence

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## 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 $T$.

$\blacksquare$

## Also see

- Properties of Restriction of Relationâ€Ž for other similar properties of the restriction of a relation.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 14$ - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations: Exercise $3.2$