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 ".

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,