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".

DefinitionEdit

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

 

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

ExampleEdit

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 alsoEdit

ReferencesEdit