Binary relation that relates every element to itself
In mathematics, a binary relation formula_1 on a set formula_2 is reflexive if it relates every element of formula_2 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' [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. "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 being formula_4. 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.
A relation formula_1 on the set formula_2 is said to be reflexive if for every formula_7, formula_8.
Equivalently, letting formula_9 denote the identity relation on formula_2, the relation formula_1 is reflexive if formula_12.
The reflexive closure of formula_1 is the union formula_14 which can equivalently be defined as the smallest (with respect to formula_15) reflexive relation on formula_2 that is a superset of formula_17 A relation formula_1 is reflexive if and only if it is equal to its reflexive closure. 
The reflexive reduction or irreflexive kernel of formula_1 is the smallest (with respect to formula_15) relation on formula_2 that has the same reflexive closure as formula_17 It is equal to formula_23 The reflexive reduction of formula_1 can, in a sense, be seen as a construction that is the "opposite" of the reflexive closure of formula_17 
For example, the reflexive closure of the canonical strict inequality formula_26 on the reals formula_27 is the usual non-strict inequality formula_28 whereas the reflexive reduction of formula_28 is formula_30 
Related definitions.
There are several definitions related to the reflexive property. 
The relation formula_1 is called:
; ;templatestyles src="Template:Visible anchor/styles.css" / ;irreflexive, ;templatestyles src="Template:Visible anchor/styles.css" / ;anti-reflexive or ;templatestyles src="Template:Visible anchor/styles.css" / ;aliorelative: if it does not relate any element to itself; that is, if formula_32 holds for no formula_33 A relation is irreflexive if and only if its complement in formula_34 is reflexive. An asymmetric relation is necessarily irreflexive. A transitive and irreflexive relation is necessarily asymmetric. 
; ;templatestyles src="Template:Visible anchor/styles.css" / ;left quasi-reflexive : if whenever formula_35 are such that formula_36 then necessarily formula_37
; ;templatestyles src="Template:Visible anchor/styles.css" / ;right quasi-reflexive : if whenever formula_35 are such that formula_36 then necessarily formula_40
; ;templatestyles src="Template:Visible anchor/styles.css" / ;quasi-reflexive : if every element that is part of some relation is related to itself. Explicitly, this means that whenever formula_35 are such that formula_36 then necessarily formula_32 and formula_40 Equivalently, a binary relation is quasi-reflexive if and only if it is both left quasi-reflexive and right quasi-reflexive. A relation formula_1 is quasi-reflexive if and only if its symmetric closure formula_46 is left (or right) quasi-reflexive.
; antisymmetric : if whenever formula_35 are such that formula_48 then necessarily formula_49 
; ;templatestyles src="Template:Visible anchor/styles.css" / ;coreflexive : if whenever formula_35 are such that formula_36 then necessarily formula_49 A relation formula_1 is coreflexive if and only if its symmetric closure is anti-symmetric.
A reflexive relation on a nonempty set formula_2 can neither be irreflexive, nor asymmetric (formula_1 is called asymmetric if formula_56 implies not formula_57), nor antitransitive (formula_1 is antitransitive if formula_59 implies not formula_60).
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 (formula_61) 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 formula_62 and formula_63 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 formula_1 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 formula_65-element set is formula_66
Note that "S"("n", "k") refers to Stirling numbers of the second kind.
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.
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References.
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