distributiviteti

Distributiviteti, or distributivity, is a basic property of binary operations in mathematics. An operation ⊗ is said to distribute over another operation ⊕ if applying ⊗ to a sum of elements equals the sum of the results of applying ⊗ to each element. In symbols, a ⊗ (b ⊕ c) = (a ⊗ b) ⊕ (a ⊗ c) and (b ⊕ c) ⊗ a = (b ⊗ a) ⊕ (c ⊗ a).

The most familiar example is multiplication over addition in the real numbers: a(b+c) = ab + ac. The distributive laws hold for both left and right forms, and they underpin expansion rules for polynomials and arithmetic simplification. In algebraic structures such as rings, multiplication distributes over addition, i.e., a(b+c) = ab + ac and (a+b)c = ac + bc.

In set theory and combinatorics, the Cartesian product distributes over union and intersection: A×(B∪C) = (A×B) ∪ (A×C) and A×(B∩C) = (A×B) ∩ (A×C). In propositional logic, conjunction distributes over disjunction and vice versa in classical logic: P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R) and P ∨ (Q ∧ R) ≡ (P ∨ Q) ∧ (P ∨ R). Structures that satisfy these laws for all elements are called distributive lattices.

Not every pair of operations is distributive. For example, subtraction is not distributive over addition in general, and division is not distributive over multiplication. Some contexts exhibit partial or limited distributivity, but it does not hold universally.