mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...

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 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

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& ...

R^.

References

Franz Baader Franz Baader (15 June 1959, Spalt) is a German computer scientist at Dresden University of Technology. He received his PhD in Computer Science in 1989 from the University of Erlangen-Nuremberg, Germany, where he was a teaching and research assi ...

and

Tobias Nipkow Tobias Nipkow (born 1958) is a German computer scientist. Career Nipkow received his Diplom (MSc) in computer science from the Department of Computer Science of the Technische Hochschule Darmstadt in 1982, and his Ph.D. from the University of ...

,
Term Rewriting and All That
', Cambridge University Press, 1998, p. 8 Binary relations Closure operators Rewriting systems

Definition

See also

References