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.

** Definitions **

Let $R$ be a homogeneous binary relation over a set $X,$ which by definition is just a subset of $X\; \backslash times\; X.$
For any $x,\; y\; \backslash in\; X,$ the notation $x\; R\; y$ means that $(x,\; y)\; \backslash in\; R$ while "not $x\; R\; y$" means that $(x,\; y)\; \backslash not\backslash in\; R.$
The relation $R$ is called if $x\; R\; x$ for every $x\; \backslash in\; X$ or equivalently, if $\backslash operatorname\_X\; \backslash subseteq\; R$ where $\backslash operatorname\_X\; :=\; \backslash $ denotes the identity relation on $X.$
The reflexive closure of $R$ is the union $R\; \backslash cup\; \backslash operatorname\_,$ which can equivalently be defined as the smallest (with respect to $\backslash ,\backslash subseteq$) 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 $\backslash ,\backslash subseteq$) relation on $X$ that has the same reflexive closure as $R.$ It is equal to $R\; \backslash setminus\; \backslash operatorname\_X\; =\; \backslash .$ 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 $\backslash ,<\backslash ,$ on the reals $\backslash mathbb$ is the usual non-strict inequality $\backslash ,\backslash leq\backslash ,$ whereas the reflexive reduction of $\backslash ,\backslash leq\backslash ,$ is $\backslash ,<.$

** 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 $x\; \backslash in\; X.$ A relation is irreflexive if and only if its complement in $X\; \backslash times\; X$ is reflexive. An asymmetric relation is necessarily irreflexive. A transitive and irreflexive relation is necessarily asymmetric.
;: If whenever $x,\; y\; \backslash in\; X$ are such that $x\; R\; y,$ then necessarily $x\; R\; x.$Th

Encyclopedia Britannica

calls this property quasi-reflexivity. ;: If whenever $x,\; y\; \backslash in\; X$ 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 $x,\; y\; \backslash in\; X$ 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 $R\; \backslash cup\; R^$ is left (or right) quasi-reflexive. ;Antisymmetric: If whenever $x,\; y\; \backslash in\; X$ are such that $x\; R\; y$ and $y\; R\; x,$ then necessarily $x\; =\; y.$ ;: If whenever $x,\; y\; \backslash in\; X$ 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 **

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

** Notes **

** References **

* 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

Encyclopedia Britannica

calls this property quasi-reflexivity. ;: If whenever $x,\; y\; \backslash in\; X$ 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 $x,\; y\; \backslash in\; X$ 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 $R\; \backslash cup\; R^$ is left (or right) quasi-reflexive. ;Antisymmetric: If whenever $x,\; y\; \backslash in\; X$ are such that $x\; R\; y$ and $y\; R\; x,$ then necessarily $x\; =\; y.$ ;: If whenever $x,\; y\; \backslash in\; X$ 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 $x\; R\; z$).