Transitive relation

From Citizendium, the Citizens' Compendium
Jump to: navigation, search
This article is developing and not approved.
Main Article
Talk
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and not meant to be cited; by editing it you can help to improve it towards a future approved, citable version. These unapproved articles are subject to a disclaimer.

In set theory, a transitive relation on a set is a relation with the property that if xy and yz then xz.

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;
  • 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.

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.