# Transitive relation

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Type of binary relation

Transitive relation Type Binary relation Field Elementary algebra Statement A relation R {\displaystyle R} on a set X {\displaystyle X} is transitive if, for all elements a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} in X {\displaystyle X} , whenever R {\displaystyle R} relates a {\displaystyle a} to b {\displaystyle b} and b {\displaystyle b} to c {\displaystyle c} , then R {\displaystyle R} also relates a {\displaystyle a} to c {\displaystyle c} . Symbolic statement ∀ a , b , c ∈ X : ( a R b ∧ b R c ) ⇒ a R c {\displaystyle \forall a,b,c\in X:(aRb\wedge bRc)\Rightarrow aRc}

In [mathematics](/source/Mathematics), a [binary relation](/source/Binary_relation) R on a [set](/source/Set_(mathematics)) X is **transitive** if, for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c.

Every [partial order](/source/Partial_order) and every [equivalence relation](/source/Equivalence_relation) is transitive. For example, less than and [equality](/source/Equality_(mathematics)) among [real numbers](/source/Real_number) are both transitive: If *a* < *b* and *b* < *c* then *a* < *c*; and if *x* = *y* and *y* = *z* then *x* = *z*.

## Definition

Transitive binary relations v t e 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 ≠ ∅ : {\displaystyle S\neq \varnothing :} a R b ⇒ b R a {\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}} a R b and b R a ⇒ a = b {\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}} a ≠ b ⇒ a R b or b R a {\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}} min S exists {\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}} a ∨ b exists {\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}} a ∧ b exists {\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}} a R a {\displaystyle aRa} not a R a {\displaystyle {\text{not }}aRa} a R b ⇒ not b R a {\displaystyle {\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 Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive: 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.

A [homogeneous relation](/source/Homogeneous_relation) R on the set X is a *transitive relation* if,[1]

- for all *a*, *b*, *c* ∈ *X*, if *a R b* and *b R c*, then *a R c*.

Or in terms of [first-order logic](/source/First-order_logic):

- ∀ a , b , c ∈ X : ( a R b ∧ b R c ) ⇒ a R c {\displaystyle \forall a,b,c\in X:(aRb\wedge bRc)\Rightarrow aRc} ,

where *a R b* is the [infix notation](/source/Infix_notation) for (*a*, *b*) ∈ *R*.

## Examples

As a non-mathematical example, the relation "is an ancestor of" is transitive. For example, if Amy is an ancestor of Becky, and Becky is an ancestor of Carrie, then Amy is also an ancestor of Carrie.

On the other hand, "is the birth mother of" is not a transitive relation, because if Alice is the birth mother of Brenda, and Brenda is the birth mother of Claire, then it does not follow that Alice is the birth mother of Claire. In fact, this relation is [antitransitive](/source/Antitransitive): Alice can *never* be the birth mother of Claire.

Non-transitive, non-antitransitive relations include sports fixtures (playoff schedules), 'knows' and 'talks to'.

The examples "is greater than", "is at least as great as", and "is equal to" ([equality](/source/Equality_(mathematics))) are transitive relations on various sets. As are the set of real numbers or the set of natural numbers:

- whenever *x* > *y* and *y* > *z*, then also *x* > *z*

- whenever *x* ≥ *y* and *y* ≥ *z*, then also *x* ≥ *z*

- whenever *x* = *y* and *y* = *z*, then also *x* = *z*.

More examples of transitive relations:

- "is a [subset](/source/Subset) of" (set inclusion, a relation on sets)

- "divides" ([divisibility](/source/Divisor), a relation on natural numbers)

- "implies" ([implication](/source/Material_conditional), symbolized by "⇒", a relation on [propositions](/source/Proposition))

Examples of non-transitive relations:

- "is the [successor](/source/Successor_function) of" (a relation on natural numbers)

- "is a member of the set" (symbolized as "∈")[2]

- "is [perpendicular](/source/Perpendicular) to" (a relation on lines in [Euclidean geometry](/source/Euclidean_geometry))

The [empty relation](/source/Empty_relation) on any set X {\displaystyle X} is transitive[3] because there are no elements a , b , c ∈ X {\displaystyle a,b,c\in X} such that a R b {\displaystyle aRb} and b R c {\displaystyle bRc} , and hence the transitivity condition is [vacuously true](/source/Vacuous_truth). A relation *R* containing only one [ordered pair](/source/Ordered_pair) is also transitive: if the ordered pair is of the form ( x , x ) {\displaystyle (x,x)} for some x ∈ X {\displaystyle x\in X} the only such elements a , b , c ∈ X {\displaystyle a,b,c\in X} are a = b = c = x {\displaystyle a=b=c=x} , and indeed in this case a R c {\displaystyle aRc} , while if the ordered pair is not of the form ( x , x ) {\displaystyle (x,x)} then there are no such elements a , b , c ∈ X {\displaystyle a,b,c\in X} and hence R {\displaystyle R} is vacuously transitive.

Vacuous transitivity is transitivity when in a relation there are no ordered pairs of the form (*a*,*b*) and (*b*,*c*).

## Properties

### Closure properties

- The [converse](/source/Converse_relation) (inverse) of a transitive relation is always transitive. For instance, knowing that "is a [subset](/source/Subset) of" is transitive and "is a [superset](/source/Superset) of" is its converse, one can conclude that the latter is transitive as well.

- The intersection of two transitive relations is always transitive.[4] For instance, knowing that "was born before" and "has the same first name as" are transitive, one can conclude that "was born before and also has the same first name as" is also transitive.

- The union of two transitive relations need not be transitive. For instance, "was born before or has the same first name as" is not a transitive relation, since e.g. [Herbert Hoover](/source/Herbert_Hoover) is related to [Franklin D. Roosevelt](/source/Franklin_D._Roosevelt), who is in turn related to [Franklin Pierce](/source/Franklin_Pierce), while Hoover is not related to Franklin Pierce.

- The complement of a transitive relation need not be transitive.[5] For instance, while "equal to" is transitive, "not equal to" is only transitive on sets with at most one element.

### Other properties

A transitive relation is [asymmetric](/source/Asymmetric_relation) if and only if it is [irreflexive](/source/Irreflexive_relation).[6]

A transitive relation need not be [reflexive](/source/Reflexive_relation). When it is, it is called a [preorder](/source/Preorder). For example, on set *X* = {1,2,3}:

- *R* = { (1,1), (2,2), (3,3), (1,3), (3,2) } is reflexive, but not transitive, as the pair (1,2) is absent,

- *R* = { (1,1), (2,2), (3,3), (1,3) } is reflexive as well as transitive, so it is a preorder,

- *R* = { (1,1), (2,2), (3,3) } is reflexive as well as transitive, another preorder,

- *R* = { (1,2), (2,3), (1,3) } is transitive, but not reflexive.

As a counter example, the relation < {\displaystyle <} on the real numbers is transitive, but not reflexive.

## Transitive extensions and transitive closure

Main article: [Transitive closure](/source/Transitive_closure)

Let R be a binary relation on set X. The *transitive extension* of R, denoted *R*1, is the smallest binary relation on X such that *R*1 contains R, and if (*a*, *b*) ∈ *R* and (*b*, *c*) ∈ *R* then (*a*, *c*) ∈ *R*1.[7] For example, suppose X is a set of towns, some of which are connected by roads. Let R be the relation on towns where (*A*, *B*) ∈ *R* if there is a road directly linking town A and town B. This relation need not be transitive. The transitive extension of this relation can be defined by (*A*, *C*) ∈ *R*1 if you can travel between towns A and C by using at most two roads.

If a relation is transitive then its transitive extension is itself, that is, if R is a transitive relation then *R*1 = *R*.

The transitive extension of *R*1 would be denoted by *R*2, and continuing in this way, in general, the transitive extension of *R**i* would be *R**i* + 1. The *transitive closure* of R, denoted by *R** or *R*∞ is the set union of R, *R*1, *R*2, ... .[8]

The transitive closure of a relation is a transitive relation.[8]

The relation "is the birth parent of" on a set of people is not a transitive relation. However, in biology the need often arises to consider birth parenthood over an arbitrary number of generations: the relation "is a birth ancestor of" *is* a transitive relation and it is the transitive closure of the relation "is the birth parent of".

For the example of towns and roads above, (*A*, *C*) ∈ *R** provided you can travel between towns A and C using any number of roads.

## Relation types that require transitivity

- [Preorder](/source/Preorder) – a [reflexive](/source/Reflexive_relation) and transitive relation

- [Partial order](/source/Partially_ordered_set) – an [antisymmetric](/source/Antisymmetric_relation) preorder

- [Total preorder](/source/Total_preorder) – a [connected](/source/Connected_relation) (formerly called total) preorder

- [Equivalence relation](/source/Equivalence_relation) – a [symmetric](/source/Symmetric_relation) preorder

- [Strict weak ordering](/source/Strict_weak_ordering) – a strict partial order in which incomparability is an equivalence relation

- [Total ordering](/source/Total_ordering) – a connected (total), antisymmetric, and transitive relation

## Counting transitive relations

No general formula that counts the number of transitive relations on a finite set (sequence [A006905](https://oeis.org/A006905) in the [OEIS](/source/On-Line_Encyclopedia_of_Integer_Sequences)) is known.[9] However, there is a formula for finding the number of relations that are simultaneously reflexive, symmetric, and transitive – in other words, [equivalence relations](/source/Equivalence_relation) – (sequence [A000110](https://oeis.org/A000110) in the [OEIS](/source/On-Line_Encyclopedia_of_Integer_Sequences)), those that are symmetric and transitive, those that are symmetric, transitive, and antisymmetric, and those that are total, transitive, and antisymmetric. Pfeiffer[10] has made some progress in this direction, expressing relations with combinations of these properties in terms of each other, but still calculating any one is difficult. See also Brinkmann and McKay (2005)[11] and Mala (2022).[12]

Since the reflexivization of any transitive relation is a [preorder](/source/Preorder), the number of transitive relations an on *n*-element set is at most 2*n* time more than the number of preorders, thus it is asymptotically 2 ( 1 / 4 + o ( 1 ) ) n 2 {\displaystyle 2^{(1/4+o(1))n^{2}}} by results of Kleitman and Rothschild.[13]

Number of n-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 2n2 2n(n−1) 2n(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 [Stirling numbers of the second kind](/source/Stirling_numbers_of_the_second_kind).

## Related properties

The [Rock–paper–scissors](/source/Rock%E2%80%93paper%E2%80%93scissors) game is based on an intransitive and antitransitive relation "*x* beats *y*".

A relation *R* is called *[intransitive](/source/Intransitivity)* if it is not transitive, that is, if *xRy* and *yRz*, but not *xRz*, for some *x*, *y*, *z*. In contrast, a relation *R* is called *[antitransitive](/source/Antitransitive)* if *xRy* and *yRz* always implies that *xRz* does not hold. For example, the relation defined by *xRy* if *xy* is an [even number](/source/Even_number) is intransitive,[14] but not antitransitive.[15] The relation defined by *xRy* if *x* is even and *y* is [odd](/source/Odd_number) is both transitive and antitransitive.[16] The relation defined by *xRy* if *x* is the [successor](/source/Successor_function) number of *y* is both intransitive[17] and antitransitive.[18] Unexpected examples of intransitivity arise in situations such as political questions or group preferences.[19]

Generalized to stochastic versions (*[stochastic transitivity](/source/Stochastic_transitivity)*), the study of transitivity finds applications of in [decision theory](/source/Decision_theory), [psychometrics](/source/Psychometrics) and [utility models](/source/Utilitarianism).[20]

A *[quasitransitive relation](/source/Quasitransitive_relation)* is another generalization;[5] it is required to be transitive only on its non-symmetric part. Such relations are used in [social choice theory](/source/Social_choice_theory) or [microeconomics](/source/Microeconomics).[21]

**Proposition:** If *R* is a [univalent](/source/Univalent_relation), then R;RT is transitive.

- proof: Suppose x R ; R T y R ; R T z . {\displaystyle xR;R^{T}yR;R^{T}z.} Then there are *a* and *b* such that x R a R T y R b R T z . {\displaystyle xRaR^{T}yRbR^{T}z.} Since *R* is univalent, *yRb* and *aR*T*y* imply *a*=*b*. Therefore *x*R*a*RT*z*, hence *x*R;RT*z* and R;RT is transitive.

**Corollary**: If *R* is univalent, then R;RT is an [equivalence relation](/source/Equivalence_relation) on the domain of *R*.

- proof: R;RT is symmetric and reflexive on its domain. With univalence of *R*, the transitive requirement for equivalence is fulfilled.

## See also

- [Transitive reduction](/source/Transitive_reduction)

- [Intransitive dice](/source/Intransitive_dice)

- [Rational choice theory](/source/Rational_choice_theory#Formal_statement)

- [Hypothetical syllogism](/source/Hypothetical_syllogism) — transitivity of the material conditional

## Notes

1. **[^](#cite_ref-1)** [Smith, Eggen & St. Andre 2006](#CITEREFSmithEggenSt._Andre2006), p. 145

1. **[^](#cite_ref-2)** However, the class of [von Neumann ordinals](/source/Von_Neumann_ordinal) is constructed in a way such that ∈ *is* transitive when restricted to that class.

1. **[^](#cite_ref-3)** [Smith, Eggen & St. Andre 2006](#CITEREFSmithEggenSt._Andre2006), p. 146

1. **[^](#cite_ref-4)** Bianchi, Mariagrazia; Mauri, Anna Gillio Berta; Herzog, Marcel; Verardi, Libero (2000-01-12), ["On finite solvable groups in which normality is a transitive relation"](https://www.degruyter.com/document/doi/10.1515/jgth.2000.012/html), *Journal of Group Theory*, **3** (2), [doi](/source/Doi_(identifier)):[10.1515/jgth.2000.012](https://doi.org/10.1515%2Fjgth.2000.012), [ISSN](/source/ISSN_(identifier)) [1433-5883](https://search.worldcat.org/issn/1433-5883), [archived](https://web.archive.org/web/20230204151127/https://www.degruyter.com/document/doi/10.1515/jgth.2000.012/html) from the original on 2023-02-04, retrieved 2022-12-29

1. ^ [***a***](#cite_ref-Derek.1964_5-0) [***b***](#cite_ref-Derek.1964_5-1) Robinson, Derek J. S. (January 1964), ["Groups in which normality is a transitive relation"](https://www.cambridge.org/core/product/identifier/S0305004100037403/type/journal_article), *Mathematical Proceedings of the Cambridge Philosophical Society*, **60** (1): 21–38, [Bibcode](/source/Bibcode_(identifier)):[1964PCPS...60...21R](https://ui.adsabs.harvard.edu/abs/1964PCPS...60...21R), [doi](/source/Doi_(identifier)):[10.1017/S0305004100037403](https://doi.org/10.1017%2FS0305004100037403), [ISSN](/source/ISSN_(identifier)) [0305-0041](https://search.worldcat.org/issn/0305-0041), [S2CID](/source/S2CID_(identifier)) [119707269](https://api.semanticscholar.org/CorpusID:119707269), [archived](https://web.archive.org/web/20230204151127/https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/abs/groups-in-which-normality-is-a-transitive-relation/E1EECC9F60124437962FBF9FDD8E81BA) from the original on 2023-02-04, retrieved 2022-12-29

1. **[^](#cite_ref-6)** Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007), [*Transitive Closures of Binary Relations I*](https://web.archive.org/web/20131102214049/http://www.karlin.mff.cuni.cz/~jezek/120/transitive1.pdf) (PDF), Prague: School of Mathematics - Physics Charles University, p. 1, archived from [the original](http://www.karlin.mff.cuni.cz/~jezek/120/transitive1.pdf) (PDF) on 2013-11-02 Lemma 1.1 (iv). Note that this source refers to asymmetric relations as "strictly antisymmetric".

1. **[^](#cite_ref-7)** [Liu 1985](#CITEREFLiu1985), p. 111

1. ^ [***a***](#cite_ref-Liu112_8-0) [***b***](#cite_ref-Liu112_8-1) [Liu 1985](#CITEREFLiu1985), p. 112

1. **[^](#cite_ref-9)** Finch, Steven R. (2003), [*Transitive relations, topologies and partial orders*](https://web.archive.org/web/20160304111410/http://www.people.fas.harvard.edu/~sfinch/csolve/posets.pdf) (PDF), archived from [the original](http://www.people.fas.harvard.edu/~sfinch/csolve/posets.pdf) (PDF) on 2016-03-04

1. **[^](#cite_ref-10)** Pfeiffer, Götz (2004), ["Counting transitive relations"](https://www.cs.uwaterloo.ca/journals/JIS/VOL7/Pfeiffer/pfeiffer6.html), *Journal of Integer Sequences*, **7** (3) 04.3.2: 1–11, [MR](/source/MR_(identifier)) [2085342](https://mathscinet.ams.org/mathscinet-getitem?mr=2085342)

1. **[^](#cite_ref-11)** Brinkmann, Gunnar; McKay, Brendan D. (2005), ["Counting unlabelled topologies and transitive relations"](https://cs.uwaterloo.ca/journals/JIS/VOL8/McKay/mckay170.html), *Journal of Integer Sequences*, **8** (2) 05.2.1: 1–7, [MR](/source/MR_(identifier)) [2134160](https://mathscinet.ams.org/mathscinet-getitem?mr=2134160)

1. **[^](#cite_ref-12)** Mala, Firdous Ahmad (2022), "On the number of transitive relations on a set", *Indian Journal of Pure and Applied Mathematics*, **53** (1): 228–232, [doi](/source/Doi_(identifier)):[10.1007/s13226-021-00100-0](https://doi.org/10.1007%2Fs13226-021-00100-0), [MR](/source/MR_(identifier)) [4387391](https://mathscinet.ams.org/mathscinet-getitem?mr=4387391)

1. **[^](#cite_ref-13)** Kleitman, D.; Rothschild, B. (1970), "The number of finite topologies", *Proceedings of the American Mathematical Society*, **25** (2): 276–282, [doi](/source/Doi_(identifier)):[10.1090/S0002-9939-1970-0253944-9](https://doi.org/10.1090%2FS0002-9939-1970-0253944-9), [JSTOR](/source/JSTOR_(identifier)) [2037205](https://www.jstor.org/stable/2037205)

1. **[^](#cite_ref-14)** since e.g. 3*R*4 and 4*R*5, but not 3*R*5

1. **[^](#cite_ref-:0_15-0)** since e.g. 2*R*3 and 3*R*4 and 2*R*4

1. **[^](#cite_ref-16)** since *xRy* and *yRz* can never happen

1. **[^](#cite_ref-17)** since e.g. 3*R*2 and 2*R*1, but not 3*R*1

1. **[^](#cite_ref-18)** since, more generally, *xRy* and *yRz* implies *x*=*y*+1=*z*+2≠*z*+1, i.e. not *xRz*, for all *x*, *y*, *z*

1. **[^](#cite_ref-19)** Drum, Kevin (November 2018), ["Preferences are not transitive"](https://www.motherjones.com/kevin-drum/2018/11/preferences-are-not-transitive/), *Mother Jones*, [archived](https://web.archive.org/web/20181129113105/https://www.motherjones.com/kevin-drum/2018/11/preferences-are-not-transitive/) from the original on 2018-11-29, retrieved 2018-11-29

1. **[^](#cite_ref-20)** Oliveira, I.F.D.; Zehavi, S.; Davidov, O. (August 2018), "Stochastic transitivity: Axioms and models", *Journal of Mathematical Psychology*, **85**: 25–35, [doi](/source/Doi_(identifier)):[10.1016/j.jmp.2018.06.002](https://doi.org/10.1016%2Fj.jmp.2018.06.002), [ISSN](/source/ISSN_(identifier)) [0022-2496](https://search.worldcat.org/issn/0022-2496)

1. **[^](#cite_ref-21)** [Sen, A.](/source/Amartya_Sen) (1969), "Quasi-transitivity, rational choice and collective decisions", *Rev. Econ. Stud.*, **36** (3): 381–393, [doi](/source/Doi_(identifier)):[10.2307/2296434](https://doi.org/10.2307%2F2296434), [JSTOR](/source/JSTOR_(identifier)) [2296434](https://www.jstor.org/stable/2296434), [Zbl](/source/Zbl_(identifier)) [0181.47302](https://zbmath.org/?format=complete&q=an:0181.47302)

## References

- Liu, C.L. (1985), [*Elements of Discrete Mathematics*](https://archive.org/details/elementsofdiscre00liuc), McGraw-Hill, [ISBN](/source/ISBN_(identifier)) [0-07-038133-X](https://en.wikipedia.org/wiki/Special:BookSources/0-07-038133-X)

- Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006), *A Transition to Advanced Mathematics* (6th ed.), Brooks/Cole, [ISBN](/source/ISBN_(identifier)) [978-0-534-39900-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-534-39900-9)

## Further reading

- [Grimaldi, Ralph P.](/source/Ralph_Grimaldi) (1994), *Discrete and Combinatorial Mathematics* (3rd ed.), Addison-Wesley, [ISBN](/source/ISBN_(identifier)) [0-201-19912-2](https://en.wikipedia.org/wiki/Special:BookSources/0-201-19912-2)

- [Gunther Schmidt](/source/Gunther_Schmidt), 2010. *Relational Mathematics*. Cambridge University Press, [ISBN](/source/ISBN_(identifier)) [978-0-521-76268-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-76268-7).

## External links

- ["Transitivity"](https://www.encyclopediaofmath.org/index.php?title=Transitivity), *[Encyclopedia of Mathematics](/source/Encyclopedia_of_Mathematics)*, [EMS Press](/source/European_Mathematical_Society), 2001 [1994]

- [Transitivity in Action](http://www.cut-the-knot.org/triangle/remarkable.shtml) at [cut-the-knot](/source/Cut-the-knot)

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