A homogeneous relation on the set is a transitive relation if, Or in terms of first-order logic: where is the infix notation for.
Examples
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" are transitive relations on various sets, for instance, the set of real numbers or the set of natural numbers: More examples of transitive relations:
The empty relation on any set is transitive because there are no elements such that and, and hence the transitivity condition is vacuously true. A relation 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.
Properties
Closure properties
The inverse 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.
Let be a binary relation on set. The transitive extension of, denoted, is the smallest binary relation on such that contains, and if and then. For example, suppose is a set of towns, some of which are connected by roads. Let be the relation on towns where if there is a road directly linking town and town. This relation need not be transitive. The transitive extension of this relation can be defined by if you can travel between towns and by using at most two roads. If a relation is transitive then its transitive extension is itself, that is, if is a transitive relation then. The transitive extension of would be denoted by, and continuing in this way, in general, the transitive extension of would be. The transitive closure of, denoted by or is the set union of,,,.... The transitive closure of a relation is a transitive relation. 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, provided you can travel between towns and using any number of roads.
No general formula that counts the number of transitive relations on a finite set is known. However, there is a formula for finding the number of relations that are simultaneously reflexive, symmetric, and transitive – in other words, equivalence relations –, those that are symmetric and transitive, those that are symmetric, transitive, and antisymmetric, and those that are total, transitive, and antisymmetric. Pfeiffer 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.
Related properties
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, but not antitransitive. The relation defined by xRy if x is even and y is odd is both transitive and antitransitive. The relation defined by xRy if x is the successor number of y is both intransitive and antitransitive. Unexpected examples of intransitivity arise in situations such as political questions or group preferences. Generalized to stochastic versions, the study of transitivity finds applications of in decision theory, psychometrics and utility models. 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.
