Transitieve

Transitieve is a term used to describe the property of transitivity in various systems, most notably in mathematics, logic, and linguistics. In abstract terms, a relation R on a set is transitieve if for all elements a, b, and c, whenever a is related to b and b is related to c, then a is related to c. This simple rule has wide-ranging implications for structure and reasoning.

In mathematics and computer science, transitivity underpins orderings, hierarchies, and connectivity. Classic examples include the less-than relation on numbers (if a < b and b < c, then a < c), divisibility (if a divides b and b divides c, then a divides c), and the subset relation among sets (A ⊆ B and B ⊆ C implies A ⊆ C). Not all relations are transitive; for example, the “is a parent of” relation is not transitive do not imply a grandparent relationship. When a relation is not transitive, one may consider its transitive closure, the smallest transitive relation containing the original one.

In linguistics, transitiveness describes whether a verb takes a direct object. A transitive verb requires an object (for example, “eat an apple”), while an intransitive verb does not (for example, “sleep”). Some verbs are ambitransitive and can function as either transitive or intransitive, depending on usage.

Historically, the term derives from Latin transitivus, via medieval and modern languages. Transitieve as a concept highlights how a simple rule—if A relates to B and B relates to C, then A relates to C—organizes reasoning, language, and formal structures across disciplines.