Transitivity
Transitivity
A dyadic relation R is transitive if, whenever xRy and yRz both hold, xRz also holds. Important examples of transitive relations are the relation of identity or equality; the relation less than among whole numbers, or among rational numbers, or among real numbers, the relation precedes among instants of time (as usually taken); the relation of class inclusion, ? (see logic, formal, 7); the relations of material implication and material equivalence among propositions, the relations of formal implication and formal equivalence among monadic propositional functions. In the propositional calculus, the laws of transitivity of material implication and material equivalence (the conditional and biconditional) are
[p ? q][q ? r] ? [p ? r]
[p = q][q = r] ? [p = r]
Similar laws of transitivity may be formulated for equality (e.g., in the functional calculus of first order with equality), class inclusion (e.g., in the Zermelo set theory), formal implication (e.g., in the pure functional calculus of first order), etc. — A.C.