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