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
The reflexive closure
of a relation
on a set
is given by![{\displaystyle S=R\cup \{(x,x):x\in X\}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
In plain English, the reflexive closure of
is the union of
with the identity relation on ![{\displaystyle X.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Example
As an example, if![{\displaystyle X=\{1,2,3,4\}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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 ![{\displaystyle X}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
then the reflexive closure is![{\displaystyle S=R\cup \{(x,x):x\in X\}=\{(1,1),(2,2),(3,3),(4,4)\}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
See also
- Symmetric closure – operation on binary relationsPages displaying wikidata descriptions as a fallback
- Transitive closure – Smallest transitive relation containing a given binary relation
References