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, $$R^{\operatorname{T}}.$$