Editing Reflexive relation

{{short description|Binary relation that relates every element to itself}}
{{Binary relations}}
In [[mathematics]], a [[binary relation]] <math>R</math> on a [[Set (mathematics)|set]] <math>X</math> is '''reflexive''' if it relates every element of <math>X</math> to itself.{{sfn|ps=|Levy|1979|p=74}}{{sfn|ps=|Schmidt|2010}} 

An example of a reflexive relation is the relation "[[Equality (mathematics)|is equal to]]" on the set of [[real number]]s, 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 [[Symmetric relation|symmetry]] and [[Transitive relation|transitivity]], reflexivity is one of three properties defining [[equivalence relation]]s.

== Etymology ==
[[File:Arithmetices principia Properties of equality.png|thumb|242x242px|[[Giuseppe Peano|Giuseppe Peano's]] introduction of the reflexive property, along with symmetry and transitivity.]]
The word ''reflexive'' is originally derived from the [[Medieval Latin]] ''reflexivus'' ('recoiling' [cf. ''[[reflex]]''], or 'directed upon itself') (c. 1250 AD) from the [[classical Latin]] ''reflexus-'' ('turn away', 'reflection') + ''-īvus'' (suffix). The word entered [[Early 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]]'' and ''[[Reflexive pronoun]]'').<ref>{{Cite web |title=reflexive {{!}} Etymology of reflexive by etymonline |url=https://www.etymonline.com/word/reflexive#etymonline_v_37000 |access-date=2024-12-22 |website=www.etymonline.com |language=en}}</ref><ref>''[[Oxford English Dictionary]]'', s.v. “[[doi:10.1093/OED/7005855243|Reflexive (''adj.'' & ''n.''), Etymology]],” September 2024.</ref> 

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 to [[Giuseppe Peano]] in his ''[[Arithmetices principia, nova methodo exposita|Arithmetices principia]]'' (1889), wherein he defines one of the fundamental properties of [[Equality (mathematics)|equality]] being <math>a = a</math>.<ref>{{Cite book |last=Peano |first=Giuseppe |author-link=Giuseppe Peano |url=https://books.google.com/books?id=z80GAAAAYAAJ |title=Arithmetices principia: nova methodo |date=1889 |publisher=Fratres Bocca |pages=XIII |language=la |archive-url=https://archive.org/details/arithmeticespri00peangoog/page/n18/mode/2up |archive-date=2009-07-15}}</ref><ref name=":0">{{Cite journal |last=Russell |first=Bertrand |author-link=Bertrand Russell |date=1903 |title=Principles of Mathematics |url=https://doi.org/10.4324/9780203864760 |journal=[[Routledge]] |doi=10.4324/9780203864760|isbn=978-1-135-22311-3 |url-access=subscription }}</ref> The first use of the word ''reflexive'' in the sense of mathematics and logic was by [[Bertrand Russell]] in his ''[[Principles of Mathematics]]'' (1903).<ref name=":0" /><ref>''[[Oxford English Dictionary]]'', s.v. “[[doi:10.1093/OED/4192548146|Reflexive (''adj.''), sense 7 - ''Mathematics and Logic'']]”, "'''1903–'''", September 2024.</ref>

== Definitions ==

A relation <math>R</math> on the set <math>X</math> is said to be {{em|reflexive}} if for every <math>x \in X</math>, <math>(x,x) \in R</math>.

Equivalently, letting <math>\operatorname{I}_X := \{ (x, x) ~:~ x \in X \}</math> denote the [[identity relation]] on <math>X</math>, the relation <math>R</math> is reflexive if <math>\operatorname{I}_X \subseteq R</math>.

The {{em|[[reflexive closure]]}} of <math>R</math> is the union <math>R \cup \operatorname{I}_X,</math> which can equivalently be defined as the smallest (with respect to <math>\subseteq</math>) reflexive relation on <math>X</math> that is a [[superset]] of <math>R.</math> A relation <math>R</math> is reflexive if and only if it is equal to its reflexive closure. 

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

=== Related definitions ===

There are several definitions related to the reflexive property. 
The relation <math>R</math> is called:
:; '''{{visible anchor|irreflexive|Irreflexivity|Irreflexive relation}}''', '''{{visible anchor|anti-reflexive|Anti-reflexivity|Anti-reflexive relation}}''' or '''{{visible anchor|aliorelative}}''':<ref>This term is due to [[C S Peirce]]; see {{harvnb|Russell|1920|p=32}}. Russell also introduces two equivalent terms ''to be contained in'' or ''imply diversity''.</ref> if it does not relate any element to itself; that is, if <math>x R x</math> holds for no <math>x \in X.</math> A relation is irreflexive [[if and only if]] its [[Complementary relation|complement]] in <math>X \times X</math> is reflexive. An [[asymmetric relation]] is necessarily irreflexive. A transitive and irreflexive relation is necessarily asymmetric. 
:; '''{{visible anchor|left quasi-reflexive|Left quasi-reflexivity}}''' : if whenever <math>x, y \in X</math> are such that <math>x R y,</math> then necessarily <math>x R x.</math><ref name="Britannica">The [https://www.britannica.com/topic/formal-logic/Logical-manipulations-in-LPC#ref534730 Encyclopædia Britannica] calls this property quasi-reflexivity.</ref>
:; '''{{visible anchor|right quasi-reflexive|Right quasi-reflexivity|Right quasi-reflexive relation}}''' : if whenever <math>x, y \in X</math> are such that <math>x R y,</math> then necessarily <math>y R y.</math>
:; '''{{visible anchor|quasi-reflexive|Quasi-reflexivity}}''' : if every element that is part of some relation is related to itself. Explicitly, this means that whenever <math>x, y \in X</math> are such that <math>x R y,</math> then necessarily <math>x R x</math> and <math>y R y.</math> Equivalently, a binary relation is quasi-reflexive if and only if it is both left quasi-reflexive and right quasi-reflexive. A relation <math>R</math> is quasi-reflexive if and only if its [[symmetric closure]] <math>R \cup R^{\operatorname{T}}</math> is left (or right) quasi-reflexive.
:; '''[[Antisymmetric relation|antisymmetric]]''' : if whenever <math>x, y \in X</math> are such that <math>x R y \text{ and } y R x,</math> then necessarily <math>x = y.</math> 
:; '''{{visible anchor|coreflexive|Coreflexivity|Coreflexive relation}}''' : if whenever <math>x, y \in X</math> are such that <math>x R y,</math> then necessarily <math>x = y.</math>{{sfn|ps=|Fonseca de Oliveira|Pereira Cunha Rodrigues|2004|p=337}} A relation <math>R</math> is coreflexive if and only if its symmetric closure is [[Antisymmetric relation|anti-symmetric]].

A reflexive relation on a nonempty set <math>X</math> can neither be irreflexive, nor [[Asymmetric relation|asymmetric]] (<math>R</math> is called {{em|asymmetric}} if <math>x R y</math> implies not <math>y R x</math>), nor [[antitransitive]] (<math>R</math> is {{em|antitransitive}} if <math>x R y \text{ and } y R z</math> implies not <math>x R z</math>).

== Examples ==
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Examples of reflexive relations include:
* "is equal to" ([[Equality (mathematics)|equality]])
* "is a [[subset]] of" (set inclusion)
* "divides" ([[Divisor|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"

{{clear}}

An example of an irreflexive relation, which means that it does not relate any element to itself, is the "greater than" relation (<math>x > y</math>) on the [[real number]]s. 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 <math>x</math> and <math>y</math> is even" is reflexive on the set of [[even number]]s, irreflexive on the set of odd numbers, and neither reflexive nor irreflexive on the set of [[natural number]]s. 

An example of a quasi-reflexive relation <math>R</math> 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 [[integer]]s 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 <math>n</math>-element set is <math>2^{n^2-n}.</math><ref>On-Line Encyclopedia of Integer Sequences [[OEIS:A053763|A053763]]</ref>

{{Number of relations}}

== 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'''.{{sfn|ps=|Hausman|Kahane|Tidman|2013|pp=327–328}}{{sfn|ps=|Clarke|Behling|1998|p=187}}

== Notes ==

{{reflist|group=note}}
{{reflist}}

== References ==
{{refbegin}}
* {{cite book |first1=D.S. |last1=Clarke |first2=Richard |last2=Behling |year=1998 |title=Deductive Logic – An Introduction to Evaluation Techniques and Logical Theory |publisher=University Press of America |isbn=0-7618-0922-8 }}
* {{citation |last1=Fonseca de Oliveira |first1=José Nuno |last2=Pereira Cunha Rodrigues |first2=César de Jesus |year=2004 |title=Transposing relations: from Maybe functions to hash tables |journal=Mathematics of Program Construction |series=Lecture Notes in Computer Science |volume=3125 |publisher=Springer |pages=334–356 |doi=10.1007/978-3-540-27764-4_18 |isbn=978-3-540-22380-1 }}
* {{cite book |first1=Alan |last1=Hausman |first2=Howard |last2=Kahane |first3=Paul |last3=Tidman |year=2013 |title=Logic and Philosophy – A Modern Introduction |publisher=Wadsworth |isbn=978-1-133-05000-1 }}
* {{citation |last1=Levy |first1=A. |year=1979 |title=Basic Set Theory |series=Perspectives in Mathematical Logic |publisher=Dover |isbn=0-486-42079-5 }}
* {{citation |last1=Lidl |first1=R. |last2=Pilz |first2=G. |year=1998 |title=Applied abstract algebra |series=[[Undergraduate Texts in Mathematics]] |publisher=Springer-Verlag |isbn=0-387-98290-6 }}
* {{citation |last1=Quine |first1=W. V. |year=1951 |title=Mathematical Logic |others=Revised Edition, Reprinted 2003 |publisher=Harvard University Press |isbn=0-674-55451-5 }}
* {{cite book |first1=Bertrand |last1=Russell |author1-link=Bertrand Russell |year=1920 |title=Introduction to Mathematical Philosophy |location=London |publisher=George Allen & Unwin, Ltd. |edition=2nd |url=https://people.umass.edu/klement/imp/imp-ebk.pdf | isbn= }}  (Online corrected edition, Feb 2010)
* {{citation |first1=Gunther |last1=Schmidt |year=2010 |title=Relational Mathematics |publisher=Cambridge University Press |isbn=978-0-521-76268-7 }}
{{refend}}

== External links ==

* {{springer|title=Reflexivity|id=p/r080590}}

[[Category:Properties of binary relations]]
[[Category:Reflexive relations| ]]