In mathematics, the symmetric closure of a

binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and i ...

R

on a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

X

is the smallest

symmetric relation A symmetric relation is a type of binary relation. An example is the relation "is equal to", because if ''a'' = ''b'' is true then ''b'' = ''a'' is also true. Formally, a binary relation ''R'' over a set ''X'' is symmetric if: :\forall a, b \in X ...

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

y

or from

y

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 \.

In other words, the symmetric closure of

R

is the union of

R

with its

converse relation In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent&n ...

R^.

References

* Franz Baader and Tobias Nipkow,
Term Rewriting and All That
', Cambridge University Press, 1998, p. 8 Binary relations Closure operators Rewriting systems

Definition

See also

References