Relation is Equivalence iff Reflexive and Circular
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Theorem
Let $\RR \subseteq S \times S$ be a relation in $S$.
Then:
- $\RR$ is an equivalence relation.
Proof
Sufficient Condition
Let $\RR$ be reflexive and circular.
Then from Reflexive Circular Relation is Equivalence:
- $\RR$ is an equivalence relation.
$\Box$
Necessary Condition
Let $\RR$ be an equivalence relation.
We have a fortiori:
- $\RR$ is reflexive.
Then from Equivalence Relation is Circular:
- $\RR$ is circular.
$\blacksquare$