Transitive relation

In mathematics, a homogeneous relation over a set is transitive if for all elements , , in , whenever relates to and to , then also relates to . Each partial order as well as each equivalence relation needs to be transitive.
Definition.
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" (equality) are transitive relations on various sets, for instance, the set of real numbers or the set of natural numbers:
More examples of transitive relations:
Examples of non-transitive relations:
The empty relation on any set formula_2 is transitive because there are no elements formula_3 such that formula_4 and formula_5, 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 formula_6 for some formula_7 the only such elements formula_3 are formula_9, and indeed in this case formula_10, while if the ordered pair is not of the form formula_6 then there are no such elements formula_3 and hence formula_13 is vacuously transitive.
Properties.
Other properties.
A transitive relation is asymmetric if and only if it is irreflexive.
A transitive relation need not be reflexive. When it is, it is called a preorder. For example, on set "X" = {1,2,3}:
Transitive extensions and transitive closure.
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.
Counting transitive relations.
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 ("stochastic transitivity"), 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.