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 (alive or dead) 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

$S = R \cup \left\{ (x, y) : (y, x) \in R \right\}. \,$

In other words, the symmetric closure of R is the union of R with its inverse relation, R ^-1.

See also

• Transitive closure
• Reflexive closure

References

• Franz Baader and Tobias Nipkow, Term Rewriting and All That, Cambridge University Press, 1998, p. 8