Symmetric closure

In mathematics, the symmetric closure of a binary relation ${\displaystyle R}$ on a set ${\displaystyle X}$ is the smallest symmetric relation on ${\displaystyle X}$ that contains ${\displaystyle R.}$

For example, if ${\displaystyle X}$ is a set of airports and ${\displaystyle xRy}$ means "there is a direct flight from airport ${\displaystyle x}$ to airport ${\displaystyle y}$", then the symmetric closure of ${\displaystyle R}$ is the relation "there is a direct flight either from ${\displaystyle x}$ to ${\displaystyle y}$ or from ${\displaystyle y}$ to ${\displaystyle x}$". Or, if ${\displaystyle X}$ is the set of humans and ${\displaystyle R}$ is the relation 'parent of', then the symmetric closure of ${\displaystyle R}$ is the relation "${\displaystyle x}$ is a parent or a child of ${\displaystyle y}$".

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

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