Definition:Reflexive Closure

Definition
Let $$\mathcal R$$ be a relation on a set $$S$$.

The reflexive closure of $$\mathcal R$$ is denoted $$\mathcal R^=$$, and is defined as:


 * $$\mathcal R^= \ \stackrel {\mathbf {def}} {=\!=} \ \left\{{\left({x, x}\right): x \in S}\right\} \cup \mathcal R$$

From Union Smallest: General Result, it follows that $$\mathcal R^=$$ is the smallest reflexive relation on $$S$$ which contains $$\mathcal R$$.

It also follows from Intersection with Subset is Subset‎ that $$\mathcal R^=$$ is the intersection of all reflexive relations which contain $$\mathcal R$$.

Thus if $$\mathcal R$$ is reflexive, then $$\mathcal R = \mathcal R^=$$.