In mathematics, a homogeneous binary relation ''R'' over 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 reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations.


Let R be a homogeneous binary relation over a set X, which by definition is just a subset of X \times X. For any x, y \in X, the notation x R y means that (x, y) \in R while "not x R y" means that (x, y) \not\in R. The relation R is called if x R x for every x \in X or equivalently, if \operatorname_X \subseteq R where \operatorname_X := \ denotes the identity relation on X. The reflexive closure of R is the union R \cup \operatorname_, which can equivalently be defined as the smallest (with respect to \,\subseteq) reflexive relation on X that is a superset of R. A relation R is reflexive if and only if it is equal to its reflexive closure. The reflexive reduction or irreflexive kernel of R is the smallest (with respect to \,\subseteq) relation on X that has the same reflexive closure as R. It is equal to R \setminus \operatorname_X = \. The irreflexive kernel of R can, in a sense, be seen as a construction that is the "opposite" of the reflexive closure of R. For example, the reflexive closure of the canonical strict inequality \,<\, on the reals \mathbb is the usual non-strict inequality \,\leq\, whereas the reflexive reduction of \,\leq\, is \,<.

Related definitions

There are several definitions related to the reflexive property. The relation R is called: ; or : If it does not relate any element to itself; that is, if not x R x for every x \in X. A relation is irreflexive if and only if its complement in X \times X is reflexive. An asymmetric relation is necessarily irreflexive. A transitive and irreflexive relation is necessarily asymmetric. ;: If whenever x, y \in X are such that x R y, then necessarily x R x.Th
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calls this property quasi-reflexivity.
;: If whenever x, y \in X are such that x R y, then necessarily y R y. ;: If every element that is related to some element is also related to itself. Explicitly, this means that whenever x, y \in X are such that x R y, then necessarily x R x y R y. Equivalently, a binary relation is quasi-reflexive if and only if it is both left quasi-reflexive and right quasi-reflexive. A relation R is quasi-reflexive if and only if its symmetric closure R \cup R^ is left (or right) quasi-reflexive. ;Antisymmetric: If whenever x, y \in X are such that x R y and y R x, then necessarily x = y. ;: If whenever x, y \in X are such that x R y, then necessarily x = y. A relation R is coreflexive if and only if its symmetric closure is anti-symmetric. A reflexive relation on a nonempty set X can neither be irreflexive, nor asymmetric (R is called asymmetric if x R y implies not y R x), nor antitransitive (R is antitransitive if x R y and y R z implies not x R z).


Examples of reflexive relations include: * "is equal to" (equality) * "is a subset of" (set inclusion) * "divides" (divisibility) * "is greater than or equal to" * "is less than or equal to" 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" * "is less than" An example of an irreflexive relation, which means that it does not relate any element to itself, is the "greater than" relation (x > y) on the real numbers. 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 (i.e., neither all nor none are). 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 reflexive nor irreflexive on the set of natural numbers. An example of a quasi-reflexive relation R is "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. An example of a left quasi-reflexive relation is a left Euclidean relation, which is always left quasi-reflexive but not necessarily right quasi-reflexive, and thus not necessarily quasi-reflexive. An example of a coreflexive relation is the relation on integers in which each odd number is related to itself and there are no other relations. The equality relation is the only example of a both reflexive and coreflexive relation, and any coreflexive relation is a subset of the identity relation. The union of a coreflexive relation and a transitive relation on the same set is always transitive.

Number of reflexive relations

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

Philosophical logic

Authors in philosophical logic often use different terminology. Reflexive relations in the mathematical sense are called totally reflexive in philosophical logic, and quasi-reflexive relations are called reflexive. Here: p.187



* Levy, A. (1979) ''Basic Set Theory'', Perspectives in Mathematical Logic, Springer-Verlag. Reprinted 2002, Dover. * Lidl, R. and Pilz, G. (1998). ''Applied abstract algebra'', Undergraduate Texts in Mathematics, Springer-Verlag. * Quine, W. V. (1951). ''Mathematical Logic'', Revised Edition. Reprinted 2003, Harvard University Press. * Gunther Schmidt, 2010. ''Relational Mathematics''. Cambridge University Press, .

External links

* {{springer|title=Reflexivity|id=p/r080590