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:


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

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

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

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

Also see

 * Reflexive Closure is Reflexive
 * Reflexive Closure is Smallest Reflexive Extension
 * Reflexive Closure is Intersection of Reflexive Extensions
 * Equivalence of Reflexive Closure Definitions
 * Reflexive Closure of Reflexive Relation is Identical


 * Definition:Reflexive Reduction