distributieve

Distributieve, in mathematics and logic, refers to the distributive property, a rule describing how an operation interacts with addition and, more generally, with other operations. In algebra, a binary operation × is distributive over + if for all a, b, c in a set, a × (b + c) = (a × b) + (a × c) (left distributive). The dual notion, right distributive, states (a + b) × c = (a × c) + (b × c). When an operation is distributive on both sides, it is simply called distributive.

In common algebraic structures, multiplication distributes over addition on both sides. For example, in the integers, reals, polynomials, and matrices with standard addition and multiplication, a × (b + c) = (a × b) + (a × c) and (a + b) × c = (a × c) + (b × c). An illustration: 3 × (4 + 5) = 27 and (3 × 4) + (3 × 5) = 12 + 15 = 27.

Not all operations are distributive. Exponentiation, for instance, distributes over multiplication (a^(b + c) = a^b · a^c) rather than over addition, so a^(b + c) ≠ a^b + a^c in general. Distributive properties also arise in other domains: in logic, conjunction distributes over disjunction (p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)); in lattice theory, a distributive lattice has meet distributing over join and vice versa.

The distributive law is fundamental for simplifying expressions, expanding products, and proving algebraic identities, and it underpins many areas of mathematics, computer science, and logic.