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 $R.$ A relation is called reflexive if it relates every element of $X$ to itself. For example, if $X$ is a set of distinct numbers and $xRy$ 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 $$S=R\cup \{(x,x):x\in X\}$$

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

Example

As an example, if $$X=\{1,2,3,4\}$$ $$R=\{(1,1),(2,2),(3,3),(4,4)\}$$ 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 $X$ $$R=\{(1,1),(2,2),(4,4)\}$$ then the reflexive closure is $$S=R\cup \{(x,x):x\in X\}=\{(1,1),(2,2),(3,3),(4,4)\}.$$

See also

• Symmetric closure
• Transitive closure – Smallest transitive relation containing a given binary relation

References

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