Relation is Equivalence iff Reflexive and Circular

Theorem
Let $\RR \subseteq S \times S$ be a relation in $S$.

Then:
 * $\RR$ is reflexive and circular


 * $\RR$ is an equivalence relation.
 * $\RR$ is an equivalence relation.

Sufficient Condition
Let $\RR$ be reflexive and circular.

Then from Reflexive Circular Relation is Equivalence:
 * $\RR$ is an equivalence relation.

Necessary Condition
Let $\RR$ be an equivalence relation.

We have :
 * $\RR$ is reflexive.

Then from Equivalence Relation is Circular:
 * $\RR$ is circular.