In mathematics, the reflexive closure of a binary relation on a set is the smallest reflexive relation on that contains A relation is called reflexive if it relates every element of to itself.

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


The reflexive closure of a relation on a set is given by

In plain English, the reflexive closure of is the union of with the identity relation on


As an example, if

then the relation is already reflexive by itself, so it does not differ from its reflexive closure.

However, if any of the pairs in was absent, it would be inserted for the reflexive closure. For example, if on the same set

then the reflexive closure is

See also