# Reflexive relation

Reflexive relation

In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S. For example, ~ could be "is equal to".

## Related terms

An irreflexive, or anti-reflexive, relation is the opposite of a reflexive relation. It is a binary relation on a set where no element is related to itself. An example is the "greater than" relation (x>y). Note that not every relation which is not reflexive is irreflexive; it is possible to define relations where some elements are related to themselves but others are not related to themselves (i.e. neither all nor none). For example, the binary relation "the product of x and y is even" is reflexive on the set of even numbers, irreflexive on the set of odd numbers, and neither on the set of natural numbers.

A relation is called quasi-reflexive if every element that is related to some element is related to itself. An example is the relation "has the same limit as" on the set of sequences of real numbers: Not every sequence has a limit, and thus the relation is not reflexive, but if a sequence has the same limit as some sequence, then it has the same limit as itself.

The reflexive closure of a binary relation ~ on a set S is the smallest relation ~′ such that ~′ is a superset of ~ and ~′ is reflexive on S. This is equivalent to the union of ~ and the identity relation on S. For example, the reflexive closure of x<y is x≤y.

The reflexive reduction of a binary relation ~ on a set S is the smallest relation ~′ such that ~′ shares the same reflexive closure as ~. It can be seen in a way as the opposite of the reflexive closure. It is equivalent to the complement of the identity relation on S with regard to ~. That is, it is equivalent to ~ except for where x~x is true. For example, the reflexive reduction of x≤y is x<y.

## Examples

Examples of reflexive relations include:

Examples of irreflexive relations include:

• "is not equal to"
• "is coprime to"(for the integers>1, since 1 is coprime to itself)
• "is a proper subset of"
• "is greater than"

## Number of reflexive relations

The number of reflexive relations on an n-element set is 2n2n.

Number of n-element binary relations of different types
n all transitive reflexive preorder partial order total preorder total order equivalence relation
0 1 1 1 1 1 1 1 1
1 2 2 1 1 1 1 1 1
2 16 13 4 4 3 3 2 2
3 512 171 64 29 19 13 6 5
4 65536 3994 4096 355 219 75 24 15
OEIS A002416 A006905 A053763 A000798 A001035 A000670 A000142 A000110

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Reflexive Relation — Drei reflexive Relationen, als gerichtete Graphen dargestellt Die Reflexivität einer zweistelligen Relation R auf einer Menge ist gegeben, wenn x R x für alle Elemente x der Menge gilt (also jedes Element in Relation zu sich selbst steht). Man… …   Deutsch Wikipedia

• reflexive relation — /rəˈflɛksɪv rəleɪʃən/ (say ruh fleksiv ruhlayshuhn) noun Mathematics, Logic a relation on a set such that every element is related to itself, as the equality relation, which satisfies x = x for every x …

• Reflexive — may refer to:In fiction: MetafictionIn grammar: *Reflexive pronoun, a pronoun with a reflexive relationship with its self identical antecedent *Reflexive verb, where a semantic agent and patient are the sameIn mathematics and computer science:… …   Wikipedia

• Relation réflexive — ● Relation réflexive relation binaire sur un ensemble telle que tout élément de cet ensemble soit en relation avec lui même …   Encyclopédie Universelle

• Relation binaire — En mathématiques, une relation binaire entre deux ensembles E et F (ou simplement relation entre E et F) est caractérisée par un sous ensemble du produit cartésien E × F, soit une collection de couples dont la première composante est dans E et la …   Wikipédia en Français

• Relation (Mengentheorie) — Eine Relation ist allgemein eine Beziehung, die zwischen Dingen bestehen kann. Relationen im Sinne der Mathematik sind ausschließlich diejenigen Beziehungen, bei denen stets klar ist, ob sie bestehen oder nicht. Zwei Gegenstände können also nicht …   Deutsch Wikipedia

• réflexive — ● réflexif, réflexive adjectif (de réflexion) Se dit, en philosophie, de la conscience qui se prend elle même pour objet. ● réflexif, réflexive (expressions) adjectif (de réflexion) Relation réflexive, relation binaire sur un ensemble telle que… …   Encyclopédie Universelle

• reflexive — 1. adjective a) Having a subject and object that are the same. Equals is a reflexive relation. b) Of a relation R on a set S, such that xRx for all members x of S (that is, the relation holds between any element of the set and itself) …   Wiktionary

• Relation (mathematics) — This article sets out the set theoretic notion of relation. For a more elementary point of view, see binary relations and triadic relations. : For a more combinatorial viewpoint, see theory of relations. In mathematics, especially set theory, and …   Wikipedia

• Relation antisymétrique — Relation binaire Une relation binaire est un concept mathématique qui systématise des notions comme « ... est supérieur ou égal à ... » en arithmétique, ou « ... est élément de l’ensemble ... » en théorie des ensembles. C’est… …   Wikipédia en Français