# 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

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

## See also

## References

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

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