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 ".

Definition edit

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  

Example edit

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 edit

  • Symmetric closure – operation on binary relations
  • Transitive closure – Smallest transitive relation containing a given binary relation

References edit