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 $T$.

$\blacksquare$


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