Transitively

Transitively is the adverb form of transitive. It describes a process, relation, or action that operates through a transitive mechanism. The term appears in mathematics, logic, and linguistics to indicate that a connection can be derived by chaining through intermediate elements or that a verb takes a direct object.

In mathematics and logic, a relation R on a set is transitive if whenever aRb and bRc, then aRc. This allows conclusions to be drawn by chaining steps. For example, the usual order ≤ on real numbers is transitive: if a ≤ b and b ≤ c, then a ≤ c. The transitive property underpins many orderings and implications; the transitive closure of a relation is the smallest transitive relation that contains it, and is used to describe all indirect connections formed by repeated application of the relation.

In grammar, transitively describes verbs that take a direct object. A transitive verb appears in sentences like “She reads a book.” Some verbs can be used both transitively and intransitively depending on context; for example, “eat” can be used intransitively in “She eats” or transitively in “She eats an apple.” The term contrasts with intransitively, which describes verbs that do not take a direct object, as in “The dog slept.”

Related concepts include transitivity, transitive relations, and transitive verbs. These ideas appear across mathematics, logic, and linguistics, illustrating how a single term can describe a structural property in different domains.