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. 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'. The symmetric closure S of a relation R on a set X is given by In other words, the symmetric closure of R is the union of R with its converse relation, RT.