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 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:

$\forall a,b,c \in X: (aRb \wedge bRc) \Rightarrow aRc$

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:

• "is a subset of" (set inclusion)
• "divides" (divisibility)
• "implies" (implication)

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]

Properties that require transitivity

• 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

No general formula that counts the number of transitive relations on a finite set (sequence A006905 in 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 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
n	all	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	65536	3994	4096	355	219	75	24	15
OEIS	A002416	A006905	A053763	A000798	A001035	A000670	A000142	A000110

See also

• Transitive closure
• Transitive reduction
• Intransitivity
• Reflexive relation
• Symmetric relation
• Antisymmetric relation
• Quasitransitive relation
• Nontransitive dice
• Rational choice theory

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.

External links

• Hazewinkel, Michiel, ed. (2001), "Transitivity", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Transitivity in Action at cut-the-knot