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

References

Franz Baader and Tobias Nipkow, [https://books.google.com/books?id=N7BvXVUCQk8C&q=%22Symmetric+closure%22 Term Rewriting and All That], Cambridge University Press, 1998, p. 8

Category:Binary relations

Category:Closure operators

Category:Rewriting systems

symmetric closure

Definition

See also

References