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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	$\forall a,b,c\in X:(aRb\wedge bRc)\Rightarrow aRc$

Inmathematics, abinary relationR on asetX istransitive if, for all elementsa,b,c inX, wheneverR relatesa tob andb toc, thenR also relatesa toc.

Everypartial order and everyequivalence relation is transitive. For example, less than andequality amongreal numbers are both transitive: Ifa <b andb <c thena <c; and ifx =y andy =z thenx =z.

Definition

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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.

Ahomogeneous relationR on the setX is atransitive relation if,^[1]

for alla,b,c ∈X, ifa R b andb R c, thena R c.

Or in terms offirst-order logic:

\forall a,b,c\in X:(aRb\wedge bRc)\Rightarrow aRc

wherea R b is theinfix notation for(a,b) ∈R.

Examples

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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 isantitransitive: Alice cannever 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) are transitive relations on various sets.As are the set of real numbers or the set of natural numbers:

wheneverx >y andy >z, then alsox >z

wheneverx ≥y andy ≥z, then alsox ≥z

wheneverx =y andy =z, then alsox =z.

More examples of transitive relations:

"is asubset of" (set inclusion, a relation on sets)
"divides" (divisibility, a relation on natural numbers)
"implies" (implication, symbolized by "⇒", a relation onpropositions)

Examples of non-transitive relations:

"is thesuccessor of" (a relation on natural numbers)
"is a member of the set" (symbolized as "∈")^[2]
"isperpendicular to" (a relation on lines inEuclidean geometry)

Theempty relation on any set $X {\displaystyle X}$ is transitive^[3] because there are no elements $a,b,c\in X$ such that $a R b {\displaystyle aRb}$ and $b R c {\displaystyle bRc}$ , and hence the transitivity condition isvacuously true. A relationR containing only oneordered pair is also transitive: if the ordered pair is of the form $(x,x)$ for some $x\in X$ the only such elements $a,b,c\in X$ are $a=b=c=x$ , and indeed in this case $a R c {\displaystyle aRc}$ , while if the ordered pair is not of the form $(x,x)$ then there are no such elements $a,b,c\in X$ 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

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Closure properties

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Theconverse (inverse) of a transitive relation is always transitive. For instance, knowing that "is asubset of" is transitive and "is asuperset 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 is related toFranklin D. Roosevelt, who is in turn related toFranklin 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

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A transitive relation isasymmetric if and only if it isirreflexive.^[6]

A transitive relation need not bereflexive. When it is, it is called apreorder. For example, on setX = {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 $<$ on the real numbers is transitive, but not reflexive.

Transitive extensions and transitive closure

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Main article:Transitive closure

LetR be a binary relation on setX. Thetransitive extension ofR, denotedR₁, is the smallest binary relation onX such thatR₁ containsR, and if(a,b) ∈R and(b,c) ∈R then(a,c) ∈R₁.^[7] For example, supposeX is a set of towns, some of which are connected by roads. LetR be the relation on towns where(A,B) ∈R if there is a road directly linking townA and townB. This relation need not be transitive. The transitive extension of this relation can be defined by(A,C) ∈R₁ if you can travel between townsA andC by using at most two roads.

If a relation is transitive then its transitive extension is itself, that is, ifR is a transitive relation thenR₁ =R.

The transitive extension ofR₁ would be denoted byR₂, and continuing in this way, in general, the transitive extension ofR_i would beR_{i + 1}. Thetransitive closure ofR, denoted byR* orR^∞ is the set union ofR,R₁,R₂, ... .^[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 townsA andC using any number of roads.

Relation types that require transitivity

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Preorder – areflexive and transitive relation
Partial order – anantisymmetric preorder
Total preorder – aconnected (formerly called total) preorder
Equivalence relation – asymmetric preorder
Strict weak ordering – a strict partial order in which incomparability is an equivalence relation
Total ordering – a connected (total), antisymmetric, and transitive relation

Counting transitive relations

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No general formula that counts the number of transitive relations on a finite set (sequenceA006905 in theOEIS) 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 – (sequenceA000110 in theOEIS), 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 apreorder, the number of transitive relations an onn-element set is at most 2ⁿ time more than the number of preorders, thus it is asymptotically $2^{(1/4+o(1))n^{2}}$ by results of Kleitman and Rothschild.^[13]

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.

Related properties

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Cycle diagram — TheRock–paper–scissors game is based on an intransitive and antitransitive relation "x beatsy".

A relationR is calledintransitive if it is not transitive, that is, ifxRy andyRz, but notxRz, for somex,y,z.In contrast, a relationR is calledantitransitive ifxRy andyRz always implies thatxRz does not hold.For example, the relation defined byxRy ifxy is aneven number is intransitive,^[14] but not antitransitive.^[15] The relation defined byxRy ifx is even andy isodd is both transitive and antitransitive.^[16] The relation defined byxRy ifx is thesuccessor number ofy 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), the study of transitivity finds applications of indecision theory,psychometrics andutility models.^[20]

Aquasitransitive relation is another generalization;^[5] it is required to be transitive only on its non-symmetric part. Such relations are used insocial choice theory ormicroeconomics.^[21]

Proposition: IfR is aunivalent, then R;R^T is transitive.

proof: Suppose

xR;R^{T}yR;R^{T}z.

Then there area andb such that

xRaR^{T}yRbR^{T}z.

SinceR is univalent,yRb andaR^Ty implya=b. ThereforexRaR^Tz, hencexR;R^Tz and R;R^T is transitive.

Corollary: IfR is univalent, then R;R^T is anequivalence relation on the domain ofR.

proof: R;R^T is symmetric and reflexive on its domain. With univalence ofR, the transitive requirement for equivalence is fulfilled.

Notes

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^Smith, Eggen & St. Andre 2006, p. 145
^However, the class ofvon Neumann ordinals is constructed in a way such that ∈is transitive when restricted to that class.
^Smith, Eggen & St. Andre 2006, p. 146
^Bianchi, Mariagrazia; Mauri, Anna Gillio Berta; Herzog, Marcel; Verardi, Libero (2000-01-12),"On finite solvable groups in which normality is a transitive relation",Journal of Group Theory,3 (2),doi:10.1515/jgth.2000.012,ISSN 1433-5883,archived from the original on 2023-02-04, retrieved2022-12-29
^^a ^bRobinson, Derek J. S. (January 1964),"Groups in which normality is a transitive relation",Mathematical Proceedings of the Cambridge Philosophical Society,60 (1):21–38,Bibcode:1964PCPS...60...21R,doi:10.1017/S0305004100037403,ISSN 0305-0041,S2CID 119707269,archived from the original on 2023-02-04, retrieved2022-12-29
^Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007),Transitive Closures of Binary Relations I(PDF), Prague: School of Mathematics - Physics Charles University, p. 1, archived fromthe original(PDF) on 2013-11-02 Lemma 1.1 (iv). Note that this source refers to asymmetric relations as "strictly antisymmetric".
^Liu 1985, p. 111
^^a ^bLiu 1985, p. 112
^Finch, Steven R. (2003),Transitive relations, topologies and partial orders(PDF), archived fromthe original(PDF) on 2016-03-04
^Pfeiffer, Götz (2004),"Counting transitive relations",Journal of Integer Sequences,7 (3) 04.3.2:1–11,MR 2085342
^Brinkmann, Gunnar; McKay, Brendan D. (2005),"Counting unlabelled topologies and transitive relations",Journal of Integer Sequences,8 (2) 05.2.1:1–7,MR 2134160
^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:10.1007/s13226-021-00100-0,MR 4387391
^Kleitman, D.; Rothschild, B. (1970), "The number of finite topologies",Proceedings of the American Mathematical Society,25 (2):276–282,doi:10.1090/S0002-9939-1970-0253944-9,JSTOR 2037205
^since e.g. 3R4 and 4R5, but not 3R5
^since e.g. 2R3 and 3R4 and 2R4
^sincexRy andyRz can never happen
^since e.g. 3R2 and 2R1, but not 3R1
^since, more generally,xRy andyRz impliesx=y+1=z+2≠z+1, i.e. notxRz, for allx,y,z
^Drum, Kevin (November 2018),"Preferences are not transitive",Mother Jones,archived from the original on 2018-11-29, retrieved2018-11-29
^Oliveira, I.F.D.; Zehavi, S.; Davidov, O. (August 2018), "Stochastic transitivity: Axioms and models",Journal of Mathematical Psychology,85:25–35,doi:10.1016/j.jmp.2018.06.002,ISSN 0022-2496
^Sen, A. (1969), "Quasi-transitivity, rational choice and collective decisions",Rev. Econ. Stud.,36 (3):381–393,doi:10.2307/2296434,JSTOR 2296434,Zbl 0181.47302

References

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Liu, C.L. (1985),Elements of Discrete Mathematics, McGraw-Hill,ISBN 0-07-038133-X
Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006),A Transition to Advanced Mathematics (6th ed.), Brooks/Cole,ISBN 978-0-534-39900-9