For example, less than and equality among real numbers are both transitive: If a < b and b < c then a < c; and if x = y and y = z then x = z.

All definitions tacitly require the homogeneous relation

A term's definition may require additional properties that are not listed in this table.

In fact, this relation is antitransitive: Alice can never be the birth mother of Claire.

Non-transitive, non-antitransitive relations include sports fixtures (playoff schedules), 'knows' and 'talks to'.

The transitive extension of R, denoted R1, is the smallest binary relation on X such that R1 contains R, and if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R1.

[7] For example, suppose X is a set of towns, some of which are connected by roads.

The transitive extension of this relation can be defined by (A, C) ∈ R1 if you can travel between towns A and C by using at most two roads.

The transitive closure of R, denoted by R* or R∞ is the set union of R, R1, R2, ...

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

Pfeiffer[10] 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.

[11] Since the reflexivization of any transitive relation is a preorder, the number of transitive relations an on n-element set is at most 2n time more than the number of preorders, thus it is asymptotically

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,[13] but not antitransitive.

[14] The relation defined by xRy if x is even and y is odd is both transitive and antitransitive.

[15] The relation defined by xRy if x is the successor number of y is both intransitive[16] and antitransitive.

[17] Unexpected examples of intransitivity arise in situations such as political questions or group preferences.

[18] Generalized to stochastic versions (stochastic transitivity), the study of transitivity finds applications of in decision theory, psychometrics and utility models.

[19] A quasitransitive relation is another generalization;[5] it is required to be transitive only on its non-symmetric part.

Such relations are used in social choice theory or microeconomics.

Corollary: If R is univalent, then R;RT is an equivalence relation on the domain of R.