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In mathematics, a binary relation R over a set X is transitive if for all elements a, b, c in X, whenever a is related to b and b is related to c, then a is also related to c. Transitivity is a key property of both partial orders and equivalence relations.

DefinitionEdit

A binary 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.

ExamplesEdit

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 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][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.

PropertiesEdit

Closure propertiesEdit

  • 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 propertiesEdit

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 closureEdit

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.[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) ∈ 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, ... .[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 transitivityEdit

Counting transitive relationsEdit

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 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 2n2 2n2n n
k=0
 
k! S(n, k)
n! n
k=0
 
S(n, k)
OEIS A002416 A006905 A053763 A000798 A001035 A000670 A000142 A000110

Related propertiesEdit

 
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]

See alsoEdit

NotesEdit

  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. ^ https://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
  7. ^ a b 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.

ReferencesEdit

External linksEdit