Symmetric closure

In mathematics, the symmetric closure of a binary relation $R$ on a set $X$ is the smallest symmetric relation on $X$ that contains $R.$ For example, if $X$ is a set of airports and $xRy$ means "there is a direct flight from airport $x$ to airport $y$", then the symmetric closure of $R$ is the relation "there is a direct flight either from $x$ to $y$ or from $y$ to $x$". Or, if $X$ is the set of humans and $R$ is the relation 'parent of', then the symmetric closure of $R$ is the relation "$x$ is a parent or a child of $y$".

Definition

The symmetric closure $S$ of a relation $R$ on a set $X$ is given by $$S=R\cup \{(y,x):(x,y)\in R\}.$$

In other words, the symmetric closure of $R$ is the union of $R$ with its converse relation, $R^{\mathrm{T}}.$

See also

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

References

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