Reflexive relation

All definitions tacitly require the homogeneous relation

A term's definition may require additional properties that are not listed in this table.

In mathematics, a binary relation

is reflexive if it relates every element of

[1][2] 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.

Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations.

The word reflexive is originally derived from the Medieval Latin reflexivus ('recoiling' [c.f.

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 (c.f.

[3][4] 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 being

[5][6] The first use of the word reflexive in the sense of mathematics and logic was by Bertrand Russell in his Principles of Mathematics (1903).

denote the identity relation on

which can equivalently be defined as the smallest (with respect to

The reflexive reduction or irreflexive kernel of

can, in a sense, be seen as a construction that is the "opposite" of the reflexive closure of

For example, the reflexive closure of the canonical strict inequality

is the usual non-strict inequality

There are several definitions related to the reflexive property.

is called: A reflexive relation on a nonempty set

For example, the binary relation "the product of

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

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.

The union of a coreflexive relation and a transitive relation on the same set is always transitive.

The number of reflexive relations on an

[11] Note that S(n, k) refers to Stirling numbers of the second kind.

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.