Reflexive relation

In mathematics, a binary relation ${\displaystyle R}$ on a set ${\displaystyle X}$ is reflexive if it relates every element of ${\displaystyle X}$ to itself.^[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. 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.

A relation ${\displaystyle R}$ on the set ${\displaystyle X}$ is said to be reflexive if for every ${\displaystyle x\in X}$, ${\displaystyle (x,x)\in R}$.

Equivalently, letting ${\displaystyle \operatorname {I} _{X}:=\{(x,x)~:~x\in X\}}$ denote the identity relation on ${\displaystyle X}$, the relation ${\displaystyle R}$ is reflexive if ${\displaystyle \operatorname {I} _{X}\subseteq R}$.

The reflexive closure of ${\displaystyle R}$ is the union ${\displaystyle R\cup \operatorname {I} _{X},}$ which can equivalently be defined as the smallest (with respect to ${\displaystyle \subseteq }$) reflexive relation on ${\displaystyle X}$ that is a superset of ${\displaystyle R.}$ A relation ${\displaystyle R}$ is reflexive if and only if it is equal to its reflexive closure.

The reflexive reduction or irreflexive kernel of ${\displaystyle R}$ is the smallest (with respect to ${\displaystyle \subseteq }$) relation on ${\displaystyle X}$ that has the same reflexive closure as ${\displaystyle R.}$ It is equal to ${\displaystyle R\setminus \operatorname {I} _{X}=\{(x,y)\in R~:~x\neq y\}.}$ The reflexive reduction of ${\displaystyle R}$ can, in a sense, be seen as a construction that is the "opposite" of the reflexive closure of ${\displaystyle R.}$ For example, the reflexive closure of the canonical strict inequality ${\displaystyle <}$ on the reals ${\displaystyle \mathbb {R} }$ is the usual non-strict inequality ${\displaystyle \leq }$ whereas the reflexive reduction of ${\displaystyle \leq }$ is ${\displaystyle <.}$

There are several definitions related to the reflexive property. The relation ${\displaystyle R}$ is called:

irreflexive, anti-reflexive or aliorelative

^[3] if it does not relate any element to itself; that is, if ${\displaystyle xRx}$ holds for no ${\displaystyle x\in X.}$ A relation is irreflexive if and only if its complement in ${\displaystyle X\times X}$ is reflexive. An asymmetric relation is necessarily irreflexive. A transitive and irreflexive relation is necessarily asymmetric.

left quasi-reflexive

if whenever ${\displaystyle x,y\in X}$ are such that ${\displaystyle xRy,}$ then necessarily ${\displaystyle xRx.}$^[4]

right quasi-reflexive

if whenever ${\displaystyle x,y\in X}$ are such that ${\displaystyle xRy,}$ then necessarily ${\displaystyle yRy.}$

quasi-reflexive

if every element that is part of some relation is related to itself. Explicitly, this means that whenever ${\displaystyle x,y\in X}$ are such that ${\displaystyle xRy,}$ then necessarily ${\displaystyle xRx}$ and ${\displaystyle yRy.}$ Equivalently, a binary relation is quasi-reflexive if and only if it is both left quasi-reflexive and right quasi-reflexive. A relation ${\displaystyle R}$ is quasi-reflexive if and only if its symmetric closure ${\displaystyle R\cup R^{\operatorname {T} }}$ is left (or right) quasi-reflexive.

antisymmetric

if whenever ${\displaystyle x,y\in X}$ are such that ${\displaystyle xRy{\text{ and }}yRx,}$ then necessarily ${\displaystyle x=y.}$

coreflexive

if whenever ${\displaystyle x,y\in X}$ are such that ${\displaystyle xRy,}$ then necessarily ${\displaystyle x=y.}$^[5] A relation ${\displaystyle R}$ is coreflexive if and only if its symmetric closure is anti-symmetric.

A reflexive relation on a nonempty set ${\displaystyle X}$ can neither be irreflexive, nor asymmetric (${\displaystyle R}$ is called asymmetric if ${\displaystyle xRy}$ implies not ${\displaystyle yRx}$), nor antitransitive (${\displaystyle R}$ is antitransitive if ${\displaystyle xRy{\text{ and }}yRz}$ implies not ${\displaystyle xRz}$).

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"

An example of an irreflexive relation, which means that it does not relate any element to itself, is the "greater than" relation (${\displaystyle x>y}$) 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 ${\displaystyle x}$ and ${\displaystyle y}$ 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 ${\displaystyle R}$ 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.

The number of reflexive relations on an ${\displaystyle n}$-element set is ${\displaystyle 2^{n^{2}-n}.}$^[6]

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.^[7][8]

• "Reflexivity", Encyclopedia of Mathematics, EMS Press, 2001 [1994]