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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

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

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

See also

• Transitive closure
• Symmetric closure

References

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