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 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.

Definitions

A relation R on the set X is said to be if for every x \in X, (x,x) \in R. Equivalently, letting denote the identity relation on X, the relation R is reflexive if. The of R is the union 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 or of R is the smallest (with respect to \subseteq) relation on X that has the same reflexive closure as R. It is equal to The reflexive reduction 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{R} 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: A reflexive relation on a nonempty set X can neither be irreflexive, nor asymmetric (R is called if x R y implies not y R x), nor antitransitive (R is if implies not x R z).

Examples

Examples of reflexive relations include: Examples of irreflexive relations include: 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 (that is, 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.