Transitive relation - Definition and Overview

In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c.

In mathematical notation, this is:

<math>\forall a, b, c \in X,\ a R b \and b R c \; \Rightarrow a R c<math>

For example, "is greater than" and "is equal to" are transitive relations: if a = b and b = c, then a = c.

On the other hand, "is the mother of" is not a transitive relation, because if Alice is the mother of Brenda, and Brenda is the mother of Claire, then Alice is not the mother of Claire.

Examples of transitive relations include:

• "is equal to" (equality)
• "is a subset of" (set inclusion)
• "is less than" and "is less than or equal to" (inequality)
• "divides" (divisibility)

A transitive relation that is also reflexive is a preorder. A preorder that is antisymmetric is a partial order. A preorder that is symmetric, is an equivalence relation.

See also transitive closure, Intransitivity