Distributivitetin

Distributivitetin, or distributivity, is a mathematical property describing how one binary operation interacts with another. If a binary operation ∘ distributes over ⊕, then for all elements a, b, c in the domain, a ∘ (b ⊕ c) = (a ∘ b) ⊕ (a ∘ c). The analogous right-distributive and left-distributive forms may also hold; when both hold, ∘ is distributive over ⊕.

In arithmetic, multiplication distributes over addition and subtraction: a(b+c) = ab+ac and a(b−c) = ab−ac; (a+b)c = ac+bc. In set theory and logic, several operations are distributive as well: Cartesian product distributes over union, A×(B∪C) = (A×B)∪(A×C); intersection distributes over union, A∩(B∪C) = (A∩B)∪(A∩C); conjunction distributes over disjunction in Boolean logic: p∧(q∨r) = (p∧q)∨(p∧r).

In algebra, multiplication distributes over addition in rings and fields: a(b+c) = ab+ac and (a+b)c = ac+bc. Some structures are not distributive. For example, the lattice of subspaces of a vector space with dimension at least two is not distributive; classic counterexamples in lattice theory include the lattices M3 and N5.

Distributivity thus constrains how elements combine and is foundational to many algebraic frameworks, enabling predictable simplifications and manipulations across mathematics.