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 reflexive 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 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

reflexive relation In mathematics, a binary relation R on a set X is reflexive if it relates every element of X to itself. An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal to itself. A ...

X

that contains

R

, i.e. the set

R \cup \

. For example, if

X

is a set of distinct numbers and

x R y

means "

x

is less than

y

", then the reflexive closure of

R

is the relation "

x

is less than or equal

Definition

The reflexive closure

S

of a relation

R

on a set

X

is given by

S = R \cup \

In plain English, the reflexive closure of

R

is the union of

R

with the

identity relation In mathematics, a homogeneous relation (also called endorelation) on a set ''X'' is a binary relation between ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation ...

X.

Example

As an example, if

X = \

R = \

then the relation

R

is already reflexive by itself, so it does not differ from its reflexive closure. However, if any of the reflexive pairs in

R

was absent, it would be inserted for the reflexive closure. For example, if on the same set

X

R = \

then the reflexive closure is

S = R \cup \ = \ .

References

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

Definition

Example

See also

References