Definition:Reflexive Closure/Union with Diagonal

Definition

Let $\RR$ be a relation on a set $S$.

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

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

That is:

$\RR^= := \RR \cup \Delta_S$

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

Also see

• Equivalence of Definitions of Reflexive Closure

• Results about reflexive closures can be found here.

Sources

• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.19$: Some Important Properties of Relations: Exercise $2$