Transitivity

Transitivity is a basic property of a binary relation on a set. A relation R is transitive if, for all elements a, b, and c, whenever a is related to b (aRb) and b is related to c (bRc), then a is related to c (aRc). Formally, this conditional must hold for every triple.

Common examples illustrate the concept. The less-than relation on real numbers is transitive: if a < b and b < c, then a < c. The equality relation is transitive: if a = b and b = c, then a = c. Some relations are not transitive, such as “is a sibling of”: if A is a sibling of B and B is a sibling of C, A is not necessarily a sibling of C.

In linguistics, transitivity refers to whether a verb takes a direct object. Transitive verbs, like eat or write, require an object (She reads a book). Intransitive verbs, like sleep, do not (He sleeps). Some verbs can alternate between transitive and intransitive uses depending on context or meaning.

In graph theory and computer science, transitivity leads to the transitive closure of a relation: if aRb and bRc, one can infer aRc and, in a directed graph, add the direct edge a→c. Transitivity also underpins reasoning about reachability and forms the basis for various algorithms in databases and knowledge representations.

Related concepts include reflexivity and symmetry; together with transitivity, these properties characterize different kinds of relations, such as total orders and equivalence relations.