mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the symmetric closure of a

binary relation In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...

R

on a set

X

is the smallest

symmetric relation A symmetric relation is a type of binary relation. Formally, a binary relation ''R'' over a set ''X'' is symmetric if: : \forall a, b \in X(a R b \Leftrightarrow b R a) , where the notation ''aRb'' means that . An example is the relation "is equ ...

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,

R^.

References

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

Definition

See also

References