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\$

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

Example

As an example, if
:$X = \left\$
:$R = \left\$
then the relation $R$ is already reflexive by itself, so it does not differ from its reflexive closure.

However, if any of the pairs in $R$ was absent, it would be inserted for the reflexive closure.
For example, if on the same set $X$
:$R = \left\$
then the reflexive closure is
:$S = R \cup \left\ = \left\ .$

See also

* Transitive closure
* Symmetric closure

References

* Franz Baader and Tobias Nipkow,
Term Rewriting and All That
', Cambridge University Press, 1998, p. 8

Binary relations
Closure operators
Rewriting systems

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