Symmetric closure

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

Definition

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

In other words, the symmetric closure of [math]\displaystyle{ R }[/math] is the union of [math]\displaystyle{ R }[/math] with its converse relation, [math]\displaystyle{ R^{\operatorname{T}}. }[/math]

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