Reflexive Closure is Idempotent

Theorem
Let $S$ be a set.

Let $R$ denote the set of all endorelations on $S$.

Then the reflexive closure operator is an idempotent mapping on $R$.

That is:


 * $\forall \mathcal R \in R: \mathcal R^= = \left ({\mathcal R^=}\right )^=$

where $\mathcal R^=$ denotes the reflexive closure of $\mathcal R$.

Proof
Let $\mathcal R \in R$.

By the definition of reflexive closure:


 * $\mathcal R^= = \mathcal R \cup \Delta_S$


 * $\left ({\mathcal R^=}\right )^= = \left( {\mathcal R \cup \Delta_S} \right ) \cup \Delta_S$

By Union is Associative:


 * $\left ({\mathcal R^=}\right )^= = \mathcal R \cup \left ( {\Delta_S \cup \Delta_S } \right )$

By Union is Idempotent Operation:


 * $\left ({\mathcal R^=}\right )^= = \mathcal R \cup \Delta_S$

Hence:


 * $\forall \mathcal R \in R: \mathcal R^= = \left ({\mathcal R^=}\right )^=$

$\blacksquare$