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

${\displaystyle S=R\cup \left\{(x,x):x\in X\right\}}$

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

Example

As an example, if

${\displaystyle X=\left\{1,2,3,4\right\}}$

${\displaystyle R=\left\{(1,1),(2,2),(3,3),(4,4)\right\}}$

then the relation ${\displaystyle R}$ is already reflexive by itself, so it doesn't differ from its reflexive closure.

However, if any of the pairs in ${\displaystyle R}$ was absent, it would be inserted for the reflexive closure. For example, if

${\displaystyle X=\left\{1,2,3,4\right\}}$

${\displaystyle R=\left\{(1,1),(2,2),(4,4)\right\}}$

then reflexive closure is, by the definition of a reflexive closure:

${\displaystyle S=R\cup \left\{(x,x):x\in X\right\}=\left\{(1,1),(2,2),(3,3),(4,4)\right\}}$.

See also

• Transitive closure
• Symmetric closure

References

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