Reflexive and Transitive Relation is Idempotent

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

Let $\mathcal R$ be both reflexive and transitive.

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

where $\circ$ denotes composition of relations.

Proof
Let $\mathcal R$ be both reflexive and transitive.

By definition of transitive relation:


 * $\mathcal R \circ \mathcal R \subseteq \mathcal R$

Let $\left({x, y}\right) \in \mathcal R$.

By definition of reflexive relation:
 * $\left({y, y}\right) \in \mathcal R$

By definition of composition of relations:
 * $\left({x, y}\right) \in \mathcal R \circ \mathcal R$

Hence:
 * $\mathcal R \subseteq \mathcal R \circ \mathcal R$

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