In
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 reflexive 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 in ...
''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 ...
on ''X'' that contains ''R''.
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 to ''y''".
Definition
The reflexive closure ''S'' of a relation ''R'' on a set ''X'' is given by
:
In English, the reflexive closure of ''R'' is the union of ''R'' with the
identity relation on ''X''.
Example
As an example, if
:
:
then the relation
is already reflexive by itself, so it does not differ from its reflexive closure.
However, if any of the pairs in
was absent, it would be inserted for the reflexive closure.
For example, if on the same set
:
then the reflexive closure is
:
See also
*
Transitive closure
In mathematics, the transitive closure of a binary relation on a set is the smallest relation on that contains and is transitive. For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinite ...
*
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 ...
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 assis ...
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 o ...
,
Term Rewriting and All That', Cambridge University Press, 1998, p. 8
Binary relations
Closure operators
Rewriting systems
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