Reflexive closure

The reflexive closure ${\displaystyle S}$ of a relation ${\displaystyle R}$ on a set ${\displaystyle X}$ is given by

$${\displaystyle S=R\cup \{(x,x):x\in X\}}$$

In plain English, the reflexive closure of ${\displaystyle R}$ is the union of ${\displaystyle R}$ with the identity relation on ${\displaystyle X.}$

As an example, if

$${\displaystyle X=\{1,2,3,4\}}$$

$${\displaystyle R=\{(1,1),(2,2),(3,3),(4,4)\}}$$

then the relation ${\displaystyle R}$ is already reflexive by itself, so it does not differ from its reflexive closure.

However, if any of the pairs in ${\displaystyle R}$ was absent, it would be inserted for the reflexive closure. For example, if on the same set ${\displaystyle X}$

$${\displaystyle R=\{(1,1),(2,2),(4,4)\}}$$

then the reflexive closure is

$${\displaystyle S=R\cup \{(x,x):x\in X\}=\{(1,1),(2,2),(3,3),(4,4)\}.}$$

• Symmetric closure – operation on binary relationsPages displaying wikidata descriptions as a fallback
• Transitive closure – Smallest transitive relation containing a given binary relation

• Franz Baader and Tobias Nipkow, Term Rewriting and All That, Cambridge University Press, 1998, p. 8