In mathematics, the symmetric closure of a binary relation on a set is the smallest symmetric relation on that contains
For example, if is a set of airports and means "there is a direct flight from airport to airport ", then the symmetric closure of is the relation "there is a direct flight either from to or from to ". Or, if is the set of humans and is the relation 'parent of', then the symmetric closure of is the relation " is a parent or a child of ".
Definition
The symmetric closure of a relation on a set is given by
In other words, the symmetric closure of is the union of with its converse relation,
See also
References
Franz Baader and Tobias Nipkow, Term Rewriting and All That, Cambridge University Press, 1998, p. 8