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 in turn 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 transitive relation can be defined as:

Examples

For example, "is greater than", "is at least as great as," and "is equal to" (equality) are transitive relations:

whenever A > B and B > C, then also A > C
whenever A B and B C, then also A C
whenever A = B and B = C, then also A = C.

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.

Then again, in biology we often need to consider motherhood over an arbitrary number of generations: the relation "is a matrilinear ancestor of". This is a transitive relation. More precisely, it is the transitive closure of the relation "is the mother of".

More examples of transitive relations:

Properties

Closure properties

The converse of a transitive relation is always transitive: e.g. knowing that "is a subset of" is transitive and "is a superset of" is its converse, we can conclude that the latter is transitive as well.

The intersection of two transitive relations is always transitive: knowing that "was born before" and "has the same first name as" are transitive, we can conclude that "was born before and also has the same first name as" is also transitive.

The union of two transitive relations is not always transitive. For instance "was born before or has the same first name as" is not generally a transitive relation.

The complement of a transitive relation is not always 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.[1]

Counting transitive relations

No general formula that counts the number of transitive relations on a finite set (sequence A006905 in the OEIS) is known.[2] 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[3] 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.[4]

Number of n-element binary relations of different types
nalltransitivereflexivepreorderpartial ordertotal preordertotal orderequivalence relation
011111111
122111111
21613443322
35121716429191365
46553639944096355219752415
n2n2 2n2-n Σn
k=0

k! S(n,k)
n!Σn
k=0

S(n,k)
OEIS A002416 A006905 A053763 A000798 A001035 A000670 A000142 A000110

Sources

References

1. 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. Lemma 1.1 (iv). Note that this source refers to asymmetric relations as "strictly antisymmetric".
2. Steven R. Finch, "Transitive relations, topologies and partial orders", 2003.
3. Götz Pfeiffer, "Counting Transitive Relations", Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
4. Gunnar Brinkmann and Brendan D. McKay,"Counting unlabelled topologies and transitive relations"

Bibliography

• Ralph P. Grimaldi, Discrete and Combinatorial Mathematics, ISBN 0-201-19912-2.
• Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.