Reflexive relation

 ( Properties Of Binary Relations ) 

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

The of $R$ is the union $R\; \backslash cup\; \backslash operatorname\{I\}\_X,$ which can equivalently be defined as the smallest (with respect to $\backslash subseteq$) 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 or of $R$ is the smallest (with respect to $\backslash subseteq$) relation on $X$ that has the same reflexive closure as $R.$ It is equal to $R\; \backslash setminus\; \backslash operatorname\{I\}\_X\; =\; \backslash \{\; (x,\; y)\; \backslash in\; R\; ~:~\; x\; \backslash neq\; y\; \backslash \}.$ The reflexive reduction 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 on the Real number $\backslash mathbb\{R\}$ is the usual non-strict inequality $\backslash leq$ whereas the reflexive reduction of $\backslash leq$ is

; , or :This term is due to C S Peirce; see . Russell also introduces two equivalent terms to be contained in or imply diversity. if it does not relate any element to itself; that is, if $x\; R\; x$ holds for no $x\; \backslash in\; X.$ 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 $x\; R\; x.$The Encyclopædia 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 part of some relation is related to itself. Explicitly, this means that whenever $x,\; y\; \backslash in\; X$ are such that $x\; R\; y,$ then necessarily $x\; R\; x$ and $y\; R\; y.$ 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 $R\; \backslash cup\; R^\{\backslash operatorname\{T\}\}$ is left (or right) quasi-reflexive.

; antisymmetric: if whenever $x,\; y\; \backslash in\; X$ are such that $x\; R\; y\; \backslash text\{\; and\; \}\; y\; R\; x,$ 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 if $x\; R\; y$ implies not $y\; R\; x$), nor antitransitive ($R$ is if $x\; R\; y\; \backslash text\{\; and\; \}\; y\; R\; z$ implies not $x\; R\; z$).

• "is equal to" (equality)
• "is a subset of" (set inclusion)
• "divides" (Divisor)
• "is greater than or equal to"
• "is less than or equal to"

• "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 ($x\; >\; y$) on the . 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 $x$ and $y$ is even" is reflexive on the set of , irreflexive on the set of odd numbers, and neither reflexive nor irreflexive on the set of .

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 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.

• (1998). 9780761809227, University Press of America. ISBN 9780761809227
• (2025). 9781133050001, Wadsworth. ISBN 9781133050001
• (Online corrected edition, Feb 2010)