Transitive relation
TypeBinary relation
FieldElementary algebra
StatementA relation on a set is transitive if, for all elements , , in , whenever relates to and to , then also relates to .
Symbolic statement

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.

Every partial order and every equivalence relation is transitive. For example, inequality 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.


Transitive binary relations
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Total, Semiconnex Anti-
Equivalence relation Green tickY Green tickY
Preorder (Quasiorder) Green tickY
Partial order Green tickY Green tickY
Total preorder Green tickY Green tickY
Total order Green tickY Green tickY Green tickY
Prewellordering Green tickY Green tickY Green tickY
Well-quasi-ordering Green tickY Green tickY
Well-ordering Green tickY Green tickY Green tickY Green tickY
Lattice Green tickY Green tickY Green tickY Green tickY
Join-semilattice Green tickY Green tickY Green tickY
Meet-semilattice Green tickY Green tickY Green tickY
Strict partial order Green tickY Green tickY Green tickY
Strict weak order Green tickY Green tickY Green tickY
Strict total order Green tickY Green tickY Green tickY Green tickY
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Definitions, for all and
Green tickY 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 Green tickY in the "Symmetric" column and in the "Antisymmetric" column, respectively.

All definitions tacitly require the homogeneous relation be transitive: for all if and then
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, cX, if a R b and b R c, then a R c.

Or in terms of first-order logic:


where a R b is the infix notation for (a, b) ∈ R.


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, 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 this does not imply that Alice is the birth parent of Claire. What is more, it is antitransitive: Alice can never be the birth parent of Claire.

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

"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 xy and yz, then also xz
whenever x = y and y = z, then also x = z.

More examples of transitive relations:

Examples of non-transitive relations:

The empty relation on any set is transitive[3] because there are no elements such that and , 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 for some the only such elements are , and indeed in this case , while if the ordered pair is not of the form then there are no such elements and hence is vacuously transitive.


Closure properties

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}:

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 Ri would be Ri + 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

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 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!S(n, k)
n! n
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,[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;RT is transitive.

proof: Suppose Then there are a and b such that Since R is univalent, yRb and aRTy imply a=b. Therefore xRaRTz, hence xR;RTz and R;RT is transitive.

Corollary: If R is univalent, then R;RT is an 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


  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. ^ a b 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. ^ a b 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/ 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.