# Transitive relation

From Citizendium, the Citizens' Compendium

In set theory, a **transitive relation** on a set is a relation with the property that if *x*→*y* and *y*→*z* then *x*→*z*.

## Examples

- An equivalence relation is transitive:
- Equality is transitive: if
*x*=*y*and*y*=*z*then*x*=*z*; - The trivial (always-true) relation is transitive;

- Equality is transitive: if
- An order relation is transitive:
- The usual order on the integers is transitive: if
*x*>*y*and*y*>*z*then*x*>*z*; - Divisibility on the natural numbers is transitive: if
*x*divides*y*and*y*divides*z*then*x*divides*z*; - Inclusion on subsets of a set is transitive: if
*x*is a subset of*y*and*y*is a subset of*z*then*x*is a subset of*z*.

- The usual order on the integers is transitive: if

## Properties

- 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*.

## Transitive closure

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.