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.
In terms of set theory, the binary relation R defined on the set X is a transitive relation if,^{[1]}
Or, in symbolic form,
Where, for example, a R b is the infix notation for (a, b) ? R.
"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:
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.
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;
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}.^{[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) ? 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}, ... .^{[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.
No general formula that counts the number of transitive relations on a finite set (sequence 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 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]}
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 | 355 | 219 | 75 | 24 | 15 | |||
n | 2^{n2} | 2^{n2-n} | ?n k=0 k! S(n, k) |
n! | ?n k=0 S(n, k) | |||
OEIS | A002416 | A006905 | A053763 | A000798 | A001035 | A000670 | A000142 | A000110 |
Some binary relations are not transitive; they are called intransitive. Examples arise in situations such as political questions or group preferences.^{[10]}