This article needs additional citations for verification. (October 2013) (Learn how and when to remove this template message) |

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.

## Contents

## Definition[edit]

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:

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

## Examples[edit]

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 is transitive^{[3]}^{[4]} 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.

## Properties[edit]

### Closure properties[edit]

- 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[edit]

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* = :

*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[edit]

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}.^{[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*) ∈ *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}, ... .^{[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*) ∈ *R** provided you can travel between towns A and C using any number of roads.

## Relation properties that require transitivity[edit]

- 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[edit]

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]}

Elements | Any | Transitive | Reflexive | Preorder | Partial order | Total preorder | Total order | Equivalence relation |
---|---|---|---|---|---|---|---|---|

0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 |

2 | 16 | 13 | 4 | 4 | 3 | 3 | 2 | 2 |

3 | 512 | 171 | 64 | 29 | 19 | 13 | 6 | 5 |

4 | 65,536 | 3,994 | 4,096 | 355 | 219 | 75 | 24 | 15 |

n | 2^{n2} | 2^{n2−n} | ∑nk=0 k! S(n, k) | n! | ∑nk=0 S(n, k) | |||

OEIS | A002416 | A006905 | A053763 | A000798 | A001035 | A000670 | A000142 | A000110 |

## Related properties[edit]

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]}

## See also[edit]

## Notes[edit]

**^**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. 3*R*4 and 4*R*5, but not 3*R*5**^**since e.g. 2*R*3 and 3*R*4 and 2*R*4**^**since*xRy*and*yRz*can never happen**^**since e.g. 3*R*2 and 2*R*1, but not 3*R*1**^**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.

## References[edit]

- 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[edit]

- Hazewinkel, Michiel, ed. (2001) [1994], "Transitivity",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Transitivity in Action at cut-the-knot