In mathematics, a binary relation on a set is reflexive 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.

Etymology

The word reflexive is originally derived from the Medieval Latin reflexivus ('recoiling' [cf. reflex], or 'directed upon itself') (c. 1250 AD) from the classical Latin reflexus- ('turn away', 'reflection') + -īvus (suffix). The word entered Early Modern English in the 1580s. The sense of the word meaning 'directed upon itself', as now used in mathematics, surviving mostly by its use in philosophy and grammar (cf. Reflexive verb and Reflexive pronoun).

The first explicit use of "reflexivity", that is, describing a relation as having the property that every element is related to itself, is generally attributed to Giuseppe Peano in his Arithmetices principia (1889), wherein he defines one of the fundamental properties of equality. The first use of the word reflexive in the sense of mathematics and logic was by Bertrand Russell in his Principles of Mathematics (1903).

Definitions

is said to be .

denote the identity relation.

The that is a superset is reflexive if and only if it is equal to its reflexive closure.

The or on the reals

Related definitions

is called:

1. , or : A relation is irreflexive if and only if its complement is reflexive. An asymmetric relation is necessarily irreflexive. A transitive and irreflexive relation is necessarily asymmetric.

2. is quasi-reflexive if and only if its symmetric closure is left (or right) quasi-reflexive.

3. antisymmetric

4. is coreflexive if and only if its symmetric closure is anti-symmetric.

can neither be irreflexive, nor asymmetric is called ), nor antitransitive is ).

Examples

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" on the integers larger than 1

• "is a proper subset of"

• "is greater than"

• "is less than"

) on the real numbers 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.

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

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.

Notes

References

• (Online corrected edition, Feb 2010)

External links