# Reflexive closure

In mathematics, the reflexive closure of a binary relation R on a set X is the smallest reflexive relation on X that contains R.

For example, if X is a set of distinct numbers and x R y means "x is less than y", then the reflexive closure of R is the relation "x is less than or equal to y".

## Definition

The reflexive closure S of a relation R on a set X is given by

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

In words, the reflexive closure of R is the union of R with the identity relation on X.