- An equivalence relation is transitive:
- Equality is transitive: if x=y and y=z then x=z;
- The trivial (always-true) relation is transitive;
- An order relation is transitive:
- The intersection of transitive relations is transitive. That is, if R and S are transitive relations on a set X, then the relation R&S, defined by x R&S y if x R y and x S y, is also transitive. The same holds for intersections of arbitrary families of transitive relations: indeed, the transitive relations on a set form a closure system.
Transitivity may be defined in terms of relation composition. A relation R is transitive if the composite R.R implies (is contained in) R.
The transitive closure of a relation R may be defined as the intersection R* of all transitive relations containing R (one always exists, namely the always-true relation): loosely the "smallest" transitive relation containing R. The closure may also be constructed as
where denotes the composition of R with itself n times.