Transitive relation

In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b and b is related to an element c then a is also related to c. Transitivity (or transitiveness) is a key property of both partial order relations and equivalence relations.

Formal definition

In terms of set theory, the binary relation R defined 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 symbolic form,

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

Examples

"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.

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

More examples of transitive relations:

The empty relation on any non-empty set X is transitive,[2][3] because the conditional defining a transitive relation is logically true if the antecedent is false, resulting in the statement being true (vacuous truth).

A relation R containing only one ordered pair is transitive for the same reason.

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.[4]

A transitive relation need not be reflexive. When it is, it is called a preorder. For example, on set X ={1,2,3}, the relations;

  • R = {(1,1),(2,2),(3,3),(1,3),(3,2)} is reflexive, but not transitive, as element (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 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.[5] 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, ... .[6]

The transitive closure of a relation is a transitive relation.[6]

The relation "is the mother of" on a set of people is not a transitive relation. However, in biology the need often arises to consider motherhood over an arbitrary number of generations: the relation "is a matrilinear ancestor of" is a transitive relation and it is the transitive closure of the relation "is the mother 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

Counting transitive relations

No general formula that counts the number of transitive relations on a finite set (sequence A006905 in the OEIS) is known.[7] 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[8] 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.[9]

Number of n-element binary relations of different types
nalltransitivereflexivepreorderpartial ordertotal preordertotal orderequivalence relation
011111111
122111111
21613443322
35121716429191365
46553639944096355219752415
n2n22n2−nΣn
k=0  k! S(n, k)		n!Σn
k=0  S(n, k)
OEISA002416A006905A053763A000798A001035A000670A000142A000110

See also

Notes

  1. ↑ Smith, Eggen & St. Andre 2006, p. 145
  2. ↑ Smith, Eggen & St. Andre 2006, p. 146
  3. ↑ https://courses.engr.illinois.edu/cs173/sp2011/Lectures/relations.pdf
  4. ↑ 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".
  5. ↑ Liu 1985, p. 111
  6. 1 2 Liu 1985, p. 112
  7. ↑ Steven R. Finch, "Transitive relations, topologies and partial orders", 2003.
  8. ↑ Götz Pfeiffer, "Counting Transitive Relations", Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
  9. ↑ Gunnar Brinkmann and Brendan D. McKay,"Counting unlabelled topologies and transitive relations"

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
  • 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
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