# Definition:Reflexive Closure/Union with Diagonal

## 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:

$\mathcal R^= := \mathcal R \cup \set {\tuple {x, x}: x \in S}$

That is:

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

where $\Delta_S$ is the diagonal relation on $S$.

## Also see

• Results about reflexive closures can be found here.