Transitivement

Transitivement refers to the mathematical property of a binary relation being transitive. If R is a relation on a set X, R is transitive when, for all elements a, b, and c in X, the implications a R b and b R c together imply a R c. In symbols: if a R b and b R c, then a R c.

Transitivity is one of the core closure properties of relations, alongside reflexivity (every element relates to itself) and symmetry (a R b implies b R a). When a relation is reflexive, transitive, and antisymmetric, it is called a partial order; if reflexive and transitive but not necessarily antisymmetric, it is called a preorder. Transitivity also plays a key role in many mathematical structures such as equivalence relations (which are reflexive, symmetric, and transitive) and order relations.

Examples illustrate the concept. The "less than or equal to" relation on numbers is transitive: if a ≤ b and b ≤ c, then a ≤ c. The ancestor relation in genealogy is transitive: if A is an ancestor of B and B is an ancestor of C, then A is an ancestor of C. The relation "is a parent of" is not transitive. The subset relation ⊆ is transitive: if A ⊆ B and B ⊆ C, then A ⊆ C.

In many contexts, one considers the transitive closure of a relation—the smallest transitive relation that contains the original one. This concept is central in graph theory and computer science, where algorithms like Warshall’s algorithm or Floyd–Warshall compute reachability in directed graphs.

Transitively is a foundational concept used across logic, discrete mathematics, and theoretical computer science to reason about indirect connections and implications.