Transitive relation

In mathematics, a homogeneous relation $R$ over a set $X$ 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 $c$ . Transitivity is a key property of both partial orders and equivalence relations.

Definition

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 $a,b$ and $S\neq \varnothing :$	${\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}$	${\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}$	${\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}$	${\begin{aligned}\min S\\{\text{exists}}\end{aligned}}$	${\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}$	${\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}$	$aRa$	${\text{not }}aRa$	${\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}$

indicates that the column's property is always true 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

in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.

All definitions tacitly require the homogeneous relation $R$ be transitive: for all $a,b,c,$ if $aRb$ and $bRc$ then $aRc.$
A term's definition may require additional properties that are not listed in this table.

A homogeneous relation $R$ on the set $X$ is a transitive relation if,^[1]

for all

a, b, c \in X

, if

a R b

and

b R c

, then

a R c

.

Or in terms of first-order logic:

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

where $a R b$ is the infix notation for $(a, b) \in R$ .

Examples

As a nonmathematical 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, too, is an ancestor of Carrie.

On the other hand, "is the birth parent of" is not a transitive relation, because if Alice is the birth parent of Brenda, and Brenda is the birth parent of Claire, then Alice is not the birth parent of Claire. What is more, it is antitransitive: Alice can never be the birth parent of Claire.

"Is greater than", "is at least as great as", and "is equal to" (equality) are transitive relations on various sets, for instance, 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 of" (set inclusion, a relation on sets)
"divides" (divisibility, a relation on natural numbers)
"implies" (implication, symbolized by "⇒", a relation on propositions)

Examples of non-transitive relations:

"is the successor of" (a relation on natural numbers)
"is a member of the set" (symbolized as "∈")^[2]
"is perpendicular to" (a relation on lines in Euclidean geometry)

The empty relation on any set $X$ is transitive^[3]^[4] because there are no elements $a,b,c\in X$ such that $aRb$ and $bRc$ , and hence the transitivity condition is vacuously true. A relation $R$ containing only one ordered 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 $aRc$ , 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 $R$ is vacuously transitive.

Properties

Closure properties

The inverse (converse) of a transitive relation is always transitive. For instance, knowing that "is a subset of" is transitive and "is a superset of" is its inverse, one can conclude that the latter is transitive as well.
The intersection of two transitive relations is always transitive. 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 to Franklin D. Roosevelt, which is in turn related to Franklin Pierce, while Hoover is not related to Franklin Pierce.
The complement of a transitive relation need not be transitive. 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 if and only if it is irreflexive.^[5]

A transitive relation need not be reflexive. When it is, it is called a 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.

Transitive extensions and 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 $(a, b) \in R$ and $(b, c) \in R$ then $(a, c) \in R 1$ .^[6] For example, suppose $X$ is a set of towns, some of which are connected by roads. Let $R$ be the relation on towns where $(A, B) \in R$ 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 $(A, C) \in R 1$ 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 $R \infty$ is the set union of $R$ , $R 1$ , $R 2$ , ... .^[7]

The transitive closure of a relation is a transitive relation.^[7]

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, $(A, C) \in R *$ provided you can travel between towns $A$ and $C$ using any number of roads.

Relation properties that require transitivity

Preorder – a reflexive transitive relation
Partial order – an antisymmetric preorder
Total preorder – a total preorder
Equivalence relation – a symmetric preorder
Strict weak ordering – a strict partial order in which incomparability is an equivalence relation
Total ordering – a total, antisymmetric transitive relation

Counting transitive relations

No general formula that counts the number of transitive relations on a finite set (sequence A006905 in the OEIS) is known.^[8] However, there is a formula for finding the number of relations that are simultaneously reflexive, symmetric, and transitive – in other words, equivalence relations – (sequence A000110 in the OEIS), those that are symmetric and transitive, those that are symmetric, transitive, and antisymmetric, and those that are total, transitive, and antisymmetric. Pfeiffer^[9] 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.^[10]

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	2^n²		2ⁿ⁽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 Stirling numbers of the second kind.

Related properties

Cycle diagram — The Rock–paper–scissors game is based on an intransitive and antitransitive relation "x beats y".

A relation R is called intransitive 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 if xRy and yRz always implies that xRz does not hold. For example, the relation defined by xRy if xy is an even number is intransitive,^[11] but not antitransitive.^[12] The relation defined by xRy if x is even and y is odd is both transitive and antitransitive.^[13] The relation defined by xRy if x is the successor number of y is both intransitive^[14] and antitransitive.^[15] Unexpected examples of intransitivity arise in situations such as political questions or group preferences.^[16]

Generalized to stochastic versions (stochastic transitivity), the study of transitivity finds applications of in decision theory, psychometrics and utility models.^[17]

A quasitransitive relation is another generalization; it is required to be transitive only on its non-symmetric part. Such relations are used in social choice theory or microeconomics.^[18]

Notes

^ Smith, Eggen & St. Andre 2006, p. 145
^ However, the class of von Neumann ordinals is constructed in a way such that ∈ is transitive when restricted to that class.
^ Smith, Eggen & St. Andre 2006, p. 146
^ https://courses.engr.illinois.edu/cs173/sp2011/Lectures/relations.pdf
^ 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 from the 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 ^b Liu 1985, p. 112
^ Steven R. Finch, "Transitive relations, topologies and partial orders", 2003.
^ Götz Pfeiffer, "Counting Transitive Relations", Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
^ Gunnar Brinkmann and Brendan D. McKay,"Counting unlabelled topologies and transitive relations"
^ since e.g. 3R4 and 4R5, but not 3R5
^ since e.g. 2R3 and 3R4 and 2R4
^ since xRy and yRz can never happen
^ since e.g. 3R2 and 2R1, but not 3R1
^ since, more generally, xRy and yRz implies x=y+1=z+2≠z+1, i.e. not xRz, for all x, y, z
^ Drum, Kevin (November 2018). "Preferences are not transitive". Mother Jones. Retrieved 2018-11-29.
^ 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: 381–393. doi:10.2307/2296434. Zbl 0181.47302. {{cite journal}}: Invalid |ref=harv (help)

References

Grimaldi, Ralph P. (1994), Discrete and Combinatorial Mathematics (3rd ed.), Addison-Wesley, ISBN 0-201-19912-2
Liu, C.L. (1985), Elements of Discrete Mathematics, McGraw-Hill, ISBN 0-07-038133-X
Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.
Smith, Douglas; Eggen, Maurice; St.Andre, Richard (2006), A Transition to Advanced Mathematics (6th ed.), Brooks/Cole, ISBN 978-0-534-39900-9

External links

"Transitivity", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Transitivity in Action at cut-the-knot

[1] Smith, Eggen & St. Andre 2006, p. 145

[2] However, the class of von Neumann ordinals is constructed in a way such that ∈ is transitive when restricted to that class.

[3] Smith, Eggen & St. Andre 2006, p. 146

[4] ttps://courses.engr.illinois.edu/cs173/sp2011/Lectures/relations.pdf

[5] 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 from the original (PDF) on 2013-11-02. Lemma 1.1 (iv). Note that this source refers to asymmetric relations as "strictly antisymmetric".

[6] Liu 1985, p. 111

[Liu112-7] Liu 1985, p. 112

[8] Steven R. Finch, "Transitive relations, topologies and partial orders", 2003.

[9] Götz Pfeiffer, "Counting Transitive Relations", Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

[10] Gunnar Brinkmann and Brendan D. McKay,"Counting unlabelled topologies and transitive relations"

[11] since e.g. 3R4 and 4R5, but not 3R5

[12] since e.g. 2R3 and 3R4 and 2R4

[13] since xRy and yRz can never happen

[14] since e.g. 3R2 and 2R1, but not 3R1

[15] since, more generally, xRy and yRz implies x=y+1=z+2≠z+1, i.e. not xRz, for all x, y, z

[16] Drum, Kevin (November 2018). "Preferences are not transitive". Mother Jones. Retrieved 2018-11-29.

[17] 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.

[18] Sen, A. (1969). "Quasi-transitivity, rational choice and collective decisions". Rev. Econ. Stud. 36: 381–393. doi:10.2307/2296434. Zbl 0181.47302. {{cite journal}}: Invalid |ref=harv (help)

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