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In `F33f`_`[mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematics]`_`f, a `F33f`_`[binary relation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Binary_relation]`_`f R {\\displaystyle R} on a `F33f`_`[set`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Set_(mathematics)]`_`f X {\\displaystyle X} is `!reflexive`! if it relates every element of X {\\displaystyle X} to itself.`:cite-ref-footnotelevy197974-1-0[`F5bf`_`[1`#cite-note-footnotelevy197974-1]`_`f]`:cite-ref-footnoteschmidt2010-2-0[`F5bf`_`[2`#cite-note-footnoteschmidt2010-2]`_`f]

An example of a reflexive relation is the relation "`F33f`_`[is equal to`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Equality_(mathematics)]`_`f" on the set of `F33f`_`[real numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Real_number]`_`f, 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 `F33f`_`[symmetry`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Symmetric_relation]`_`f and `F33f`_`[transitivity`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Transitive_relation]`_`f, reflexivity is one of three properties defining `F33f`_`[equivalence relations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Equivalence_relation]`_`f.

>>Contents

• `F0af`_`[Etymology`#etymology]`_`f

• `F0af`_`[Definitions`#definitions]`_`f

• `F0af`_`[Related definitions`#related-definitions]`_`f

• `F0af`_`[Examples`#examples]`_`f

• `F0af`_`[Number of reflexive relations`#number-of-reflexive-relations]`_`f

• `F0af`_`[Philosophical logic`#philosophical-logic]`_`f

• `F0af`_`[Notes`#notes]`_`f

• `F0af`_`[References`#references]`_`f

• `F0af`_`[External links`#external-links]`_`f

-─

>>Etymology

The word `*reflexive`* is originally derived from the `F33f`_`[Medieval Latin`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Medieval_Latin]`_`f `*reflexivus`* ('recoiling' [cf. `*`F33f`_`[reflex`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Reflex]`_`f`*], or 'directed upon itself') (c. 1250 AD) from the `F33f`_`[classical Latin`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Classical_Latin]`_`f `*reflexus-`* ('turn away', 'reflection') + `*-īvus`* (suffix). The word entered `F33f`_`[Early Modern English`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Early_Modern_English]`_`f 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. `*`F33f`_`[Reflexive verb`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Reflexive_verb]`_`f`* and `*`F33f`_`[Reflexive pronoun`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Reflexive_pronoun]`_`f`*).`:cite-ref-3[`F5bf`_`[3`#cite-note-3]`_`f]`:cite-ref-4[`F5bf`_`[4`#cite-note-4]`_`f]

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 `F33f`_`[Giuseppe Peano`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Giuseppe_Peano]`_`f in his `*`F33f`_`[Arithmetices principia`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Arithmetices_principia,_nova_methodo_exposita]`_`f`* (1889), wherein he defines one of the fundamental properties of `F33f`_`[equality`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Equality_(mathematics)]`_`f being a = a {\\displaystyle a=a} .`:cite-ref-5[`F5bf`_`[5`#cite-note-5]`_`f]`:cite-ref-0-6-0[`F5bf`_`[6`#cite-note-0-6]`_`f] The first use of the word `*reflexive`* in the sense of mathematics and logic was by `F33f`_`[Bertrand Russell`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bertrand_Russell]`_`f in his `*`F33f`_`[Principles of Mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Principles_of_Mathematics]`_`f`* (1903).`:cite-ref-0-6-1[`F5bf`_`[6`#cite-note-0-6]`_`f]`:cite-ref-7[`F5bf`_`[7`#cite-note-7]`_`f]

>>Definitions

A relation R {\\displaystyle R} on the set X {\\displaystyle X} is said to be `*reflexive`* if for every x ∈ ∈ X {\\displaystyle x\\in X} , ( x , x ) ∈ ∈ R {\\displaystyle (x,x)\\in R} .

Equivalently, letting I X := { ( x , x ) : x ∈ ∈ X } {\\displaystyle \\operatorname {I} _{X}:=\\{(x,x)~:~x\\in X\\}} denote the `F33f`_`[identity relation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Identity_relation]`_`f on X {\\displaystyle X} , the relation R {\\displaystyle R} is reflexive if I X ⊆ ⊆ R {\\displaystyle \\operatorname {I} _{X}\\subseteq R} .

The `*`F33f`_`[reflexive closure`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Reflexive_closure]`_`f`* of R {\\displaystyle R} is the union R ∪ ∪ I X , {\\displaystyle R\\cup \\operatorname {I} _{X},} which can equivalently be defined as the smallest (with respect to ⊆ ⊆ {\\displaystyle \\subseteq } ) reflexive relation on X {\\displaystyle X} that is a `F33f`_`[superset`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Superset]`_`f of R . {\\displaystyle R.} A relation R {\\displaystyle R} is reflexive if and only if it is equal to its reflexive closure.

The `*reflexive reduction`* or `*irreflexive kernel`* of R {\\displaystyle R} is the smallest (with respect to ⊆ ⊆ {\\displaystyle \\subseteq } ) relation on X {\\displaystyle X} that has the same reflexive closure as R . {\\displaystyle R.} It is equal to R ∖ ∖ I X = { ( x , y ) ∈ ∈ R : x ≠ ≠ y } . {\\displaystyle 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 < {\\displaystyle <} on the `F33f`_`[reals`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Real_number]`_`f R {\\displaystyle \\mathbb {R} } is the usual non-strict inequality ≤ ≤ {\\displaystyle \\leq } whereas the reflexive reduction of ≤ ≤ {\\displaystyle \\leq } is < . {\\displaystyle <.}

>>>Related definitions

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

`!irreflexive`!, `!anti-reflexive`! or `!aliorelative`! `:cite-ref-8[`F5bf`_`[8`#cite-note-8]`_`f] if it does not relate any element to itself; that is, if x R x {\\displaystyle xRx} holds for no x ∈ ∈ X . {\\displaystyle x\\in X.} A relation is irreflexive `F33f`_`[if and only if`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=If_and_only_if]`_`f its `F33f`_`[complement`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complementary_relation]`_`f in X × × X {\\displaystyle X\\times X} is reflexive. An `F33f`_`[asymmetric relation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Asymmetric_relation]`_`f is necessarily irreflexive. A transitive and irreflexive relation is necessarily asymmetric. `!left quasi-reflexive`! if whenever x , y ∈ ∈ X {\\displaystyle x,y\\in X} are such that x R y , {\\displaystyle xRy,} then necessarily x R x . {\\displaystyle xRx.} `:cite-ref-britannica-9-0[`F5bf`_`[9`#cite-note-britannica-9]`_`f] `!right quasi-reflexive`! if whenever x , y ∈ ∈ X {\\displaystyle 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 ∈ ∈ X {\\displaystyle 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 its `F33f`_`[symmetric closure`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Symmetric_closure]`_`f R ∪ ∪ R T {\\displaystyle R\\cup R^{\\operatorname {T} }} is left (or right) quasi-reflexive. `!`F33f`_`[antisymmetric`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Antisymmetric_relation]`_`f`! if whenever x , y ∈ ∈ X {\\displaystyle x,y\\in X} are such that x R y and y R x , {\\displaystyle xRy{\\text{ and }}yRx,} then necessarily x = y . {\\displaystyle x=y.} `!coreflexive`! if whenever x , y ∈ ∈ X {\\displaystyle x,y\\in X} are such that x R y , {\\displaystyle xRy,} then necessarily x = y . {\\displaystyle x=y.} `:cite-ref-footnotefonseca-de-oliveirapereira-cunha-rodrigues2004337-10-0[`F5bf`_`[10`#cite-note-footnotefonseca-de-oliveirapereira-cunha-rodrigues2004337-10]`_`f] A relation R {\\displaystyle R} is coreflexive if and only if its symmetric closure is `F33f`_`[anti-symmetric`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Antisymmetric_relation]`_`f.

A reflexive relation on a nonempty set X {\\displaystyle X} can neither be irreflexive, nor `F33f`_`[asymmetric`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Asymmetric_relation]`_`f ( R {\\displaystyle R} is called `*asymmetric`* if x R y {\\displaystyle xRy} implies not y R x {\\displaystyle yRx} ), nor `F33f`_`[antitransitive`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Antitransitive]`_`f ( R {\\displaystyle R} is `*antitransitive`* if x R y and y R z {\\displaystyle xRy{\\text{ and }}yRz} implies not x R z {\\displaystyle xRz} ).

>>Examples

Examples of reflexive relations include:

• "is equal to" (`F33f`_`[equality`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Equality_(mathematics)]`_`f)

• "is a `F33f`_`[subset`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Subset]`_`f of" (set inclusion)

• "divides" (`F33f`_`[divisibility`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Divisor]`_`f)

• "is greater than or equal to"

• "is less than or equal to"

Examples of irreflexive relations include:

• "is not equal to"

• "is `F33f`_`[coprime`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Coprime]`_`f to" on the integers larger than 1

• "is a `F33f`_`[proper subset`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Proper_subset]`_`f 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 {\\displaystyle x>y} ) on the `F33f`_`[real numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Real_number]`_`f. 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 of `F33f`_`[even numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Even_number]`_`f, irreflexive on the set of odd numbers, and neither reflexive nor irreflexive on the set of `F33f`_`[natural numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Natural_number]`_`f.

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 left `F33f`_`[Euclidean relation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Euclidean_relation]`_`f, 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 `F33f`_`[integers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Integer]`_`f 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 n {\\displaystyle n} -element set is 2 n 2 − − n . {\\displaystyle 2^{n^{2}-n}.} `:cite-ref-11[`F5bf`_`[11`#cite-note-11]`_`f]

|---|---|---|---|---|---|---|---|---|---|

| 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 n 2 | | 2 n ( n −1) | 2 n ( n +1)/2 | | | ∑ n k =0 k ! S ( n , k ) | n ! | ∑ n k =0 S ( n , k ) |

| OEIS | A002416 | A006905 | A053763 | A006125 | A000798 | A001035 | A000670 | A000142 | A000110 |

Note that `*S`*(`*n`*, `*k`*) refers to `F33f`_`[Stirling numbers of the second kind`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Stirling_numbers_of_the_second_kind]`_`f.

>>Philosophical logic

Authors in `F33f`_`[philosophical logic`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Philosophical_logic]`_`f often use different terminology. Reflexive relations in the mathematical sense are called `!totally reflexive`! in philosophical logic, and quasi-reflexive relations are called `!reflexive`!.`:cite-ref-footnotehausmankahanetidman2013327-328-12-0[`F5bf`_`[12`#cite-note-footnotehausmankahanetidman2013327-328-12]`_`f]`:cite-ref-footnoteclarkebehling1998187-13-0[`F5bf`_`[13`#cite-note-footnoteclarkebehling1998187-13]`_`f]

>>Notes

`:cite-note-footnotelevy197974-1`!1.`! `F0af`_`[↑`#cite-ref-footnotelevy197974-1-0]`_`f `F33f`_`[Levy 1979`#citereflevy1979]`_`f, p. 74

`:cite-note-footnoteschmidt2010-2`!2.`! `F0af`_`[↑`#cite-ref-footnoteschmidt2010-2-0]`_`f `F33f`_`[Schmidt 2010`#citerefschmidt2010]`_`f

`:cite-note-3`!3.`! `F0af`_`[↑`#cite-ref-3]`_`f "reflexive | Etymology of reflexive by etymonline". `*www.etymonline.com`*. Retrieved 2024-12-22.

`:cite-note-4`!4.`! `F0af`_`[↑`#cite-ref-4]`_`f `*`F33f`_`[Oxford English Dictionary`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Oxford_English_Dictionary]`_`f`*, s.v. “Reflexive (`*adj.`* & `*n.`*), Etymology,” September 2024.

`:cite-note-5`!5.`! `F0af`_`[↑`#cite-ref-5]`_`f `:citerefpeano1889`a`F33f`_`[Peano, Giuseppe`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Giuseppe_Peano]`_`f (1889). `*Arithmetices principia: nova methodo`* (in Latin). Fratres Bocca. pp. XIII. Archived from the original on 2009-07-15.

`:cite-note-0-6`!6.`! `F0af`_`[↑`#cite-ref-0-6-0]`_`f `:citerefrussell1903`a`F33f`_`[Russell, Bertrand`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bertrand_Russell]`_`f (1903). `*Principles of Mathematics`*. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.4324/9780203864760. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-1-135-22311-3. `B100`F9d9{{cite book}}`f`b: ISBN / Date incompatibility (help)

`:cite-note-7`!7.`! `F0af`_`[↑`#cite-ref-7]`_`f `*`F33f`_`[Oxford English Dictionary`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Oxford_English_Dictionary]`_`f`*, s.v. “Reflexive (`*adj.`*), sense 7 - `*Mathematics and Logic`*”, "`!1903–`!", September 2024.

`:cite-note-8`!8.`! `F0af`_`[↑`#cite-ref-8]`_`f This term is due to `F33f`_`[C S Peirce`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=C_S_Peirce]`_`f; see `F33f`_`[Russell 1920`#citerefrussell1920]`_`f, p. 32. Russell also introduces two equivalent terms `*to be contained in`* or `*imply diversity`*.

`:cite-note-britannica-9`!9.`! `F0af`_`[↑`#cite-ref-britannica-9-0]`_`f The Encyclopædia Britannica calls this property quasi-reflexivity.

`:cite-note-footnotefonseca-de-oliveirapereira-cunha-rodrigues2004337-10`!10.`! `F0af`_`[↑`#cite-ref-footnotefonseca-de-oliveirapereira-cunha-rodrigues2004337-10-0]`_`f `F33f`_`[Fonseca de Oliveira & Pereira Cunha Rodrigues 2004`#citereffonseca-de-oliveirapereira-cunha-rodrigues2004]`_`f, p. 337

`:cite-note-11`!11.`! `F0af`_`[↑`#cite-ref-11]`_`f On-Line Encyclopedia of Integer Sequences A053763

`:cite-note-footnotehausmankahanetidman2013327-328-12`!12.`! `F0af`_`[↑`#cite-ref-footnotehausmankahanetidman2013327-328-12-0]`_`f `F33f`_`[Hausman, Kahane & Tidman 2013`#citerefhausmankahanetidman2013]`_`f, pp. 327–328

`:cite-note-footnoteclarkebehling1998187-13`!13.`! `F0af`_`[↑`#cite-ref-footnoteclarkebehling1998187-13-0]`_`f `F33f`_`[Clarke & Behling 1998`#citerefclarkebehling1998]`_`f, p. 187

>>References

• `:citerefclarkebehling1998`aClarke, D.S.; Behling, Richard (1998). `*Deductive Logic – An Introduction to Evaluation Techniques and Logical Theory`*. University Press of America. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-7618-0922-8.

• `:citereffonseca-de-oliveirapereira-cunha-rodrigues2004`aFonseca 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, `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1007/978-3-540-27764-4_18, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-3-540-22380-1

• `:citerefhausmankahanetidman2013`aHausman, Alan; Kahane, Howard; Tidman, Paul (2013). `*Logic and Philosophy – A Modern Introduction`*. Wadsworth. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-1-133-05000-1.

• `:citereflevy1979`aLevy, A. (1979), `*Basic Set Theory`*, Perspectives in Mathematical Logic, Dover, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-486-42079-5

• `:citereflidlpilz1998`aLidl, R.; Pilz, G. (1998), `*Applied abstract algebra`*, `F33f`_`[Undergraduate Texts in Mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Undergraduate_Texts_in_Mathematics]`_`f, Springer-Verlag, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-387-98290-6

• `:citerefquine1951`aQuine, W. V. (1951), `*Mathematical Logic`*, Revised Edition, Reprinted 2003, Harvard University Press, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-674-55451-5 `B100`F9d9{{citation}}`f`b: ISBN / Date incompatibility (help)

• `:citerefrussell1920`a`F33f`_`[Russell, Bertrand`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bertrand_Russell]`_`f (1920). `*Introduction to Mathematical Philosophy`* (PDF) (2nd ed.). London: George Allen & Unwin, Ltd. (Online corrected edition, Feb 2010)

• `:citerefschmidt2010`aSchmidt, Gunther (2010), `*Relational Mathematics`*, Cambridge University Press, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-521-76268-7

>>External links

• "Reflexivity", `*`F33f`_`[Encyclopedia of Mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Encyclopedia_of_Mathematics]`_`f`*, `F33f`_`[EMS Press`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=European_Mathematical_Society]`_`f, 2001 [1994]

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