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 reflexive and circular

if and only if:

$\RR$ 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.

$\blacksquare$