distributiva

Distributiva is a term used in multiple disciplines to describe the distribution of an operation, relation, or element across others. In mathematics, it most often denotes the distributive property of one operation over another, a rule that allows one to expand or factor expressions.

In mathematics, there are left and right distributive properties. An operation * distributes over + on the left if a*(b+c) = a*b + a*c for all a, b, c. It distributes on the right if (b+c)*a = b*a + c*a. In many common algebraic structures, such as the real or complex numbers, both directions hold. Some less common or nonstandard algebraic systems may exhibit only one direction or fail distributivity in certain cases. Distributivity is a fundamental tool for simplifying expressions and for factoring polynomials or other algebraic objects.

Examples help illustrate the idea. In arithmetic, multiplication distributes over addition: 3*(4+5) = 3*4 + 3*5 = 27. In logic, the distributive law states that A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C). In set theory, union distributes over intersection: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C), and intersection distributes over union: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). These laws underlie many algebraic and logical manipulations.

In linguistics, the term distributiva is used to describe constructions that express distribution across members of a group, such as phrases indicating “each,” “per,” or “one by one.” These expressions differ from collective or unitary formulations and vary across languages.

See also: distributive property, Boolean algebra, ring theory, logic.