Transitive relation

In mathematics, a binary relation R on 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.

Transitive relation

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

Every partial order and every equivalence relation is transitive. For example, less than and equality among real numbers 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 ${\displaystyle a,b}$ and ${\displaystyle S\neq \varnothing$ :} ${\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}$ ${\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}$ ${\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}$ ${\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}$ ${\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}$ ${\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}$ ${\displaystyle aRa}$ ${\displaystyle {\text{not }}aRa}$ ${\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}$
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 ${\displaystyle R}$ be transitive: for all ${\displaystyle a,b,c,}$ if ${\displaystyle aRb}$ and ${\displaystyle bRc}$ then ${\displaystyle aRc.}$ 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 ∈ X, if a R b and b R c, then a R c.

Or in terms of first-order logic:

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

where a R b is the 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: 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) 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 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 ${\displaystyle X}$ is transitive^[3] because there are no elements ${\displaystyle a,b,c\in X}$ such that ${\displaystyle aRb}$ and ${\displaystyle bRc}$, 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 ${\displaystyle (x,x)}$ for some ${\displaystyle x\in X}$ the only such elements ${\displaystyle a,b,c\in X}$ are ${\displaystyle a=b=c=x}$, and indeed in this case ${\displaystyle aRc}$, while if the ordered pair is not of the form ${\displaystyle (x,x)}$ then there are no such elements ${\displaystyle a,b,c\in X}$ and hence ${\displaystyle R}$ is vacuously transitive.

Properties

Closure properties

• The converse (inverse) of a transitive relation is always transitive. For instance, knowing that "is a subset of" is transitive and "is a 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 is related to Franklin D. Roosevelt, who 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.^[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 if and only if it is irreflexive.^[6]

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.

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

Let R be a binary relation on set X. The transitive extension of R, denoted R₁, is the smallest binary relation on X such that R₁ contains R, and if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R₁.^[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) ∈ R₁ 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 R₁ = R.

The transitive extension of R₁ would be denoted by R₂, 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, 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 towns A and C using any number of roads.

Relation types that require transitivity

• Preorder – a reflexive and transitive relation
• Partial order – an antisymmetric preorder
• Total preorder – a connected (formerly called total) preorder
• Equivalence relation – a symmetric 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

No general formula that counts the number of transitive relations on a finite set (sequence A006905 in the OEIS) 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 – (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^[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]

Since the reflexivization of any transitive relation is a preorder, the number of transitive relations an on n-element set is at most 2ⁿ time more than the number of preorders, thus it is asymptotically ${\displaystyle 2^{(1/4+o(1))n^{2}}}$ by results of Kleitman and Rothschild.^[12]

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.

Related properties

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,^[13] but not antitransitive.^[14] The relation defined by xRy if x is even and y is odd is both transitive and antitransitive.^[15] The relation defined by xRy if x is the successor number of y is both intransitive^[16] and antitransitive.^[17] Unexpected examples of intransitivity arise in situations such as political questions or group preferences.^[18]

Generalized to stochastic versions (stochastic transitivity), the study of transitivity finds applications of in decision theory, psychometrics and utility models.^[19]

A 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 or microeconomics.^[20]

Proposition: If R is a univalent, then R;R^T is transitive.

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

Corollary: If R is univalent, then R;R^T is an equivalence relation on the domain of R.

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

See also

• Transitive reduction
• Intransitive dice
• Rational choice theory
• Hypothetical syllogism — transitivity of the material conditional

Notes

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. 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. Retrieved 2022-12-29.
5. Robinson, 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. Retrieved 2022-12-29.
6. 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".
7. Liu 1985, p. 111
8. Liu 1985, p. 112
9. Steven R. Finch, "Transitive relations, topologies and partial orders" Archived 2016-03-04 at the Wayback Machine, 2003.
10. Götz Pfeiffer, "Counting Transitive Relations Archived 2023-02-04 at the Wayback Machine", Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
11. Gunnar Brinkmann and Brendan D. McKay,"Counting unlabelled topologies and transitive relations Archived 2005-07-20 at the Wayback Machine"
12. Kleitman, D.; Rothschild, B. (1970), "The number of finite topologies", Proceedings of the American Mathematical Society, 25 (2): 276–282, JSTOR 2037205
13. since e.g. 3R4 and 4R5, but not 3R5
14. since e.g. 2R3 and 3R4 and 2R4
15. since xRy and yRz can never happen
16. since e.g. 3R2 and 2R1, but not 3R1
17. since, more generally, xRy and yRz implies x=y+1=z+2≠z+1, i.e. not xRz, for all x, y, z
18. Drum, Kevin (November 2018). "Preferences are not transitive". Mother Jones. Archived from the original on 2018-11-29. Retrieved 2018-11-29.
19. 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.
20. 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

• 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
• Pfeiffer, G. (2004). Counting transitive relations. Journal of Integer Sequences, 7(2), 3.

External links

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