Reflexive and Transitive Relation is Idempotent

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

Let $\RR$ be both reflexive and transitive.

Then $\RR$ is idempotent, in the sense that:
 * $\RR \circ \RR = \RR$

where $\circ$ denotes composition of relations.

Proof
Let $\RR$ be both reflexive and transitive.

By definition of transitive relation:


 * $\RR \circ \RR \subseteq \RR$

Let $\tuple {x, y} \in \RR$.

By definition of reflexive relation:
 * $\tuple {y, y} \in \RR$

By definition of composition of relations:
 * $\tuple {x, y} \in \RR \circ \RR$

Hence:
 * $\RR \subseteq \RR \circ \RR$

Thus by definition of set equality:
 * $\RR \circ \RR = \RR$