Definition:Reflexive Closure

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, it follows that $$\mathcal{R}^=$$ is the smallest reflexive relation on $$S$$ which contains $$\mathcal{R}$$.

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

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