Reflexive relation explained

on a set

is reflexive if it relates every element of

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

A relation

on the set

is said to be if for every

x\inX

(x,x)\inR

Equivalently, letting

\operatorname{I}_X:=\{(x,x)~:~x\inX\}

denote the identity relation on

, the relation

is reflexive if

\operatorname{I}_X\subseteqR

The of

is the union

R\cup\operatorname{I}_X,

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

\subseteq

) reflexive relation on

that is a superset of

A relation

is reflexive if and only if it is equal to its reflexive closure.

The or of

is the smallest (with respect to

\subseteq

) relation on

that has the same reflexive closure as

It is equal to

R\setminus\operatorname{I}_X=\{(x,y)\inR~:~x ≠ y\}.

The reflexive reduction 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

on the reals

is the usual non-strict inequality

\leq

whereas the reflexive reduction of

\leq

Related definitions

There are several definitions related to the reflexive property. The relation

is called:

, or :^[1] if it does not relate any element to itself; that is, if

xRx

holds for no

x\inX.

A relation is irreflexive if and only if its complement in

X x X

is reflexive. An asymmetric relation is necessarily irreflexive. A transitive and irreflexive relation is necessarily asymmetric.

: if whenever

x,y\inX

are such that

xRy,

then necessarily

xRx.

^[2]

: if whenever

x,y\inX

are such that

xRy,

then necessarily

yRy.

: if every element that is part of some relation is related to itself. Explicitly, this means that whenever

x,y\inX

are such that

xRy,

then necessarily

xRx

and

yRy.

Equivalently, a binary relation is quasi-reflexive if and only if it is both left quasi-reflexive and right quasi-reflexive. A relation

is quasi-reflexive if and only if its symmetric closure

R\cupR^{\operatorname{T}

} is left (or right) quasi-reflexive.

antisymmetric : if whenever

x,y\inX

are such that

xRyandyRx,

then necessarily

x=y.

: if whenever

x,y\inX

are such that

xRy,

then necessarily

x=y.

A relation

is coreflexive if and only if its symmetric closure is anti-symmetric.

A reflexive relation on a nonempty set

can neither be irreflexive, nor asymmetric (

is called if

xRy

implies not

yRx

), nor antitransitive (

is if

xRyandyRz

implies not

xRz

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" 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 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

and

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.

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

-element set is

	n^2-n
2

^[3]

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.

References

Book: D.S. . Clarke . Richard . Behling . 1998 . Deductive Logic – An Introduction to Evaluation Techniques and Logical Theory . University Press of America . 0-7618-0922-8 .
- Book: Alan . Hausman . Howard . Kahane . Paul . Tidman . 2013 . Logic and Philosophy – A Modern Introduction . Wadsworth . 1-133-05000-X .
  - - Book: Bertrand . Russell . Bertrand Russell . 1920 . Introduction to Mathematical Philosophy . London . George Allen & Unwin, Ltd. . 2nd . (Online corrected edition, Feb 2010)

Notes and References

This term is due to C S Peirce; see . Russell also introduces two equivalent terms to be contained in or imply diversity.
The Encyclopedia Britannica calls this property quasi-reflexivity.
On-Line Encyclopedia of Integer Sequences A053763