Reflexive relation

File:Arithmetices principia Properties of equality.png introduction of the reflexive property, along with symmetry and transitivity.]]

The word reflexive is originally derived from the Medieval Latin reflexivus ('recoiling' [c.f. 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 (c.f. Reflexive verb and Reflexive pronoun).{{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}}Oxford English Dictionary, s.v. “Reflexive (adj. & n.), Etymology,” September 2024.

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 (1889), wherein he defines one of the fundamental properties of equality being $a\; =\; a$.{{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}}{{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 }} The first use of the word reflexive in the sense of mathematics and logic was by Bertrand Russell in his Principles of Mathematics (1903).Oxford English Dictionary, s.v. “Reflexive (adj.), sense 7 - Mathematics and Logic”, "1903–", September 2024.

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

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 {{em|reflexive closure}} 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 {{em|reflexive reduction}} or {{em|irreflexive kernel}} 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 reals $\backslash mathbb\{R\}$ is the usual non-strict inequality $\backslash leq$ whereas the reflexive reduction of $\backslash leq$ is

There are several definitions related to the reflexive property.

The relation $R$ is called:

:; {{visible anchor|irreflexive|Irreflexivity|Irreflexive relation}}, {{visible anchor|anti-reflexive|Anti-reflexivity|Anti-reflexive relation}} or {{visible anchor|aliorelative}}: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. 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.

:; {{visible anchor|left quasi-reflexive|Left quasi-reflexivity}} : if whenever $x,\; y\; \backslash in\; X$ are such that $x\; R\; y,$ then necessarily $x\; R\; x.$The [https://www.britannica.com/topic/formal-logic/Logical-manipulations-in-LPC#ref534730 Encyclopædia Britannica] calls this property quasi-reflexivity.

:; {{visible anchor|right quasi-reflexive|Right quasi-reflexivity|Right quasi-reflexive relation}} : if whenever $x,\; y\; \backslash in\; X$ are such that $x\; R\; y,$ then necessarily $y\; R\; y.$

:; {{visible anchor|quasi-reflexive|Quasi-reflexivity}} : 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 $x\; =\; y.$

:; {{visible anchor|coreflexive|Coreflexivity|Coreflexive relation}} : if whenever $x,\; y\; \backslash in\; X$ are such that $x\; R\; y,$ then necessarily $x\; =\; y.${{sfn|ps=|Fonseca de Oliveira|Pereira Cunha Rodrigues|2004|p=337}} 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 {{em|asymmetric}} if $x\; R\; y$ implies not $y\; R\; x$), nor antitransitive ($R$ is {{em|antitransitive}} if $x\; R\; y\; \backslash text\{\; and\; \}\; y\; R\; z$ implies not $x\; R\; z$).

{{multiple image

| image1=GreaterThanOrEqualTo.png

| width1=250

| image2=GreaterThan.png

| width2=200

}}

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"

{{clear}}

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 $x$ and $y$ 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 $R$ 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 $n$-element set is $2^\{n^2-n\}.$On-Line Encyclopedia of Integer Sequences A053763

{{Number of relations}}

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}}

{{reflist|group=note}}

{{reflist}}

{{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}}

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

Category:Properties of binary relations