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Reflexive relation

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Binary relation that relates every element to itself

Transitive binary relations

	Symmetric	Antisymmetric	Connected	Well-founded	Has joins	Has meets	Reflexive	Irreflexive	Asymmetric
			Total, Semiconnex					Anti- reflexive
Equivalence relation	Y	✗	✗	✗	✗	✗	Y	✗	✗
Preorder(Quasiorder)	✗	✗	✗	✗	✗	✗	Y	✗	✗
Partial order	✗	Y	✗	✗	✗	✗	Y	✗	✗
Total preorder	✗	✗	Y	✗	✗	✗	Y	✗	✗
Total order	✗	Y	Y	✗	✗	✗	Y	✗	✗
Prewellordering	✗	✗	Y	Y	✗	✗	Y	✗	✗
Well-quasi-ordering	✗	✗	✗	Y	✗	✗	Y	✗	✗
Well-ordering	✗	Y	Y	Y	✗	✗	Y	✗	✗
Lattice	✗	Y	✗	✗	Y	Y	Y	✗	✗
Join-semilattice	✗	Y	✗	✗	Y	✗	Y	✗	✗
Meet-semilattice	✗	Y	✗	✗	✗	Y	Y	✗	✗
Strict partial order	✗	Y	✗	✗	✗	✗	✗	Y	Y
Strict weak order	✗	Y	✗	✗	✗	✗	✗	Y	Y
Strict total order	✗	Y	Y	✗	✗	✗	✗	Y	Y
	Symmetric	Antisymmetric	Connected	Well-founded	Has joins	Has meets	Reflexive	Irreflexive	Asymmetric
Definitions, for all $a, b {\displaystyle a,b}$ and $S\neq \varnothing :$	${\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}$	${\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}$	${\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}$	${\begin{aligned}\min S\\{\text{exists}}\end{aligned}}$	${\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}$	${\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}$	$a R a {\displaystyle aRa}$	${\text{not }}aRa$	${\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}$

Y indicates that the column's property is always true for the row's term (at the very left), while✗ indicates that the property is not guaranteed
in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric,
is indicated by Green tick

Y in the "Symmetric" column and✗ in the "Antisymmetric" column, respectively.

All definitions tacitly require thehomogeneous relation $R {\displaystyle R}$ betransitive: for all $a, b, c, {\displaystyle a,b,c,}$ if $a R b {\displaystyle aRb}$ and $b R c {\displaystyle bRc}$ then $a R c . {\displaystyle aRc.}$
A term's definition may require additional properties that are not listed in this table.

Inmathematics, abinary relation $R {\displaystyle R}$ on aset $X {\displaystyle X}$ isreflexive if it relates every element of $X {\displaystyle X}$ to itself.^[1]^[2]

An example of a reflexive relation is the relation "is equal to" on the set ofreal numbers, since every real number is equal to itself. A reflexive relation is said to have thereflexive property or is said to possessreflexivity. Along withsymmetry andtransitivity, reflexivity is one of three properties definingequivalence relations.

Etymology

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Giuseppe Peano's introduction of the reflexive property, along with symmetry and transitivity

The wordreflexive is originally derived from theMedieval Latinreflexivus ('recoiling' [cf.reflex], or 'directed upon itself') (c. 1250 AD) from theclassical Latinreflexus- ('turn away', 'reflection') +-īvus (suffix). The word enteredEarly 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 (cf.Reflexive verb andReflexive pronoun).^[3]^[4]

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 toGiuseppe Peano in hisArithmetices principia (1889), wherein he defines one of the fundamental properties ofequality being $a=a$ .^[5]^[6] The first use of the wordreflexive in the sense of mathematics and logic was byBertrand Russell in hisPrinciples of Mathematics (1903).^[6]^[7]

Definitions

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

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

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

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

Related definitions

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

irreflexive,anti-reflexive oraliorelative: ^[8] if it does not relate any element to itself; that is, if $x R x {\displaystyle xRx}$ holds for no $x\in X.$ A relation is irreflexiveif and only if itscomplement in $X\times X$ is reflexive. Anasymmetric relation is necessarily irreflexive. A transitive and irreflexive relation is necessarily asymmetric.
left quasi-reflexive: if whenever $x,y\in X$ are such that $x R y, {\displaystyle xRy,}$ then necessarily $x R x . {\displaystyle xRx.}$ ^[9]
right quasi-reflexive: if whenever $x,y\in X$ are such that $x R y, {\displaystyle xRy,}$ then necessarily $y R y . {\displaystyle yRy.}$
quasi-reflexive: if every element that is part of some relation is related to itself. Explicitly, this means that whenever $x,y\in X$ are such that $x R y, {\displaystyle xRy,}$ then necessarily $x R x {\displaystyle xRx}$ and $y R y . {\displaystyle yRy.}$ Equivalently, a binary relation is quasi-reflexive if and only if it is both left quasi-reflexive and right quasi-reflexive. A relation $R {\displaystyle R}$ is quasi-reflexive if and only if itssymmetric closure $R\cup R^{\operatorname {T} }$ is left (or right) quasi-reflexive.
antisymmetric: if whenever $x,y\in X$ are such that $xRy{\text{ and }}yRx,$ then necessarily $x=y.$
coreflexive: if whenever $x,y\in X$ are such that $x R y, {\displaystyle xRy,}$ then necessarily $x=y.$ ^[10] A relation $R {\displaystyle R}$ is coreflexive if and only if its symmetric closure isanti-symmetric.

A reflexive relation on a nonempty set $X {\displaystyle X}$ can neither be irreflexive, norasymmetric ( $R {\displaystyle R}$ is calledasymmetric if $x R y {\displaystyle xRy}$ implies not $y R x {\displaystyle yRx}$ ), norantitransitive ( $R {\displaystyle R}$ isantitransitive if $xRy{\text{ and }}yRz$ implies not $x R z {\displaystyle xRz}$ ).

Examples

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Examples of reflexive relations include:

"is equal to" (equality)
"is asubset 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"
"iscoprime to" on the integers larger than 1
"is aproper 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 thereal 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 $x {\displaystyle x}$ and $y {\displaystyle y}$ is even" is reflexive on the set ofeven numbers, irreflexive on the set of odd numbers, and neither reflexive nor irreflexive on the set ofnatural numbers.

An example of a quasi-reflexive relation $R {\displaystyle R}$ 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 leftEuclidean 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 onintegers 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

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The number of reflexive relations on an $n {\displaystyle n}$ -element set is $2^{n^{2}-n}.$ ^[11]

Number ofn-element binary relations of different types
Elements	Any	Transitive	Reflexive	Symmetric	Preorder	Partial order	Total preorder	Total order	Equivalence relation
0	1	1	1	1	1	1	1	1	1
1	2	2	1	2	1	1	1	1	1
2	16	13	4	8	4	3	3	2	2
3	512	171	64	64	29	19	13	6	5
4	65,536	3,994	4,096	1,024	355	219	75	24	15
n	2^n²		2ⁿ⁽ⁿ⁻¹⁾	2^n(n+1)/2			∑ⁿ _k=0k!S(n,k)	n!	∑ⁿ _k=0S(n,k)
OEIS	A002416	A006905	A053763	A006125	A000798	A001035	A000670	A000142	A000110

Note thatS(n,k) refers toStirling numbers of the second kind.

Philosophical logic

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Authors inphilosophical logic often use different terminology.Reflexive relations in the mathematical sense are calledtotally reflexive in philosophical logic, and quasi-reflexive relations are calledreflexive.^[12]^[13]

Notes

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^Levy 1979, p. 74
^Schmidt 2010
^"reflexive | Etymology of reflexive by etymonline".www.etymonline.com. Retrieved2024-12-22.
^Oxford English Dictionary, s.v. “Reflexive (adj. &n.), Etymology,” September 2024.
^Peano, Giuseppe (1889).Arithmetices principia: nova methodo (in Latin). Fratres Bocca. pp. XIII. Archived fromthe original on 2009-07-15.
^^a ^bRussell, Bertrand (1903).Principles of Mathematics.doi:10.4324/9780203864760.ISBN 978-1-135-22311-3.{{cite book}}:ISBN / Date incompatibility (help)
^Oxford English Dictionary, s.v. “Reflexive (adj.), sense 7 -Mathematics and Logic”, "1903–", September 2024.
^This term is due toC S Peirce; seeRussell 1920, p. 32. Russell also introduces two equivalent termsto be contained in orimply diversity.
^TheEncyclopædia Britannica calls this property quasi-reflexivity.
^Fonseca de Oliveira & Pereira Cunha Rodrigues 2004, p. 337
^On-Line Encyclopedia of Integer SequencesA053763
^Hausman, Kahane & Tidman 2013, pp. 327–328
^Clarke & Behling 1998, p. 187

References

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Clarke, D.S.; Behling, Richard (1998).Deductive Logic – An Introduction to Evaluation Techniques and Logical Theory. University Press of America.ISBN 0-7618-0922-8.
Fonseca de Oliveira, José Nuno; Pereira Cunha Rodrigues, César de Jesus (2004), "Transposing relations: from Maybe functions to hash tables",Mathematics of Program Construction, Lecture Notes in Computer Science,3125, Springer:334–356,doi:10.1007/978-3-540-27764-4_18,ISBN 978-3-540-22380-1{{citation}}: CS1 maint: work parameter with ISBN (link)
Hausman, Alan; Kahane, Howard; Tidman, Paul (2013).Logic and Philosophy – A Modern Introduction. Wadsworth.ISBN 978-1-133-05000-1.
Levy, A. (1979),Basic Set Theory, Perspectives in Mathematical Logic, Dover,ISBN 0-486-42079-5
Lidl, R.; Pilz, G. (1998),Applied abstract algebra,Undergraduate Texts in Mathematics, Springer-Verlag,ISBN 0-387-98290-6
Quine, W. V. (1951),Mathematical Logic, Revised Edition, Reprinted 2003, Harvard University Press,ISBN 0-674-55451-5{{citation}}:ISBN / Date incompatibility (help)
Russell, Bertrand (1920).Introduction to Mathematical Philosophy(PDF) (2nd ed.). London: George Allen & Unwin, Ltd. (Online corrected edition, Feb 2010)
Schmidt, Gunther (2010),Relational Mathematics, Cambridge University Press,ISBN 978-0-521-76268-7

External links

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

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