This basic property of numbers is part of the definition of most algebraic structures that have two operations called addition and multiplication, such as complex numbers, polynomials, matrices, rings, and fields.
The operators used for examples in this section are those of the usual addition
In either case, the distributive property can be described in words as: To multiply a sum (or difference) by a factor, each summand (or minuend and subtrahend) is multiplied by this factor and the resulting products are added (or subtracted).
If the operation outside the parentheses (in this case, the multiplication) is commutative, then left-distributivity implies right-distributivity and vice versa, and one talks simply of distributivity.
One example of an operation that is "only" right-distributive is division, which is not commutative:
In the following examples, the use of the distributive law on the set of real numbers
From the point of view of algebra, the real numbers form a field, which ensures the validity of the distributive law.
The distributive law is valid for matrix multiplication.
In standard truth-functional propositional logic, distribution[3][4] in logical proofs uses two valid rules of replacement to expand individual occurrences of certain logical connectives, within some formula, into separate applications of those connectives across subformulas of the given formula.
is a metalogical symbol representing "can be replaced in a proof with" or "is logically equivalent to".
The following logical equivalences demonstrate that distributivity is a property of particular connectives.
fails in decimal arithmetic, regardless of the number of significant digits.
Methods such as banker's rounding may help in some cases, as may increasing the precision used, but ultimately some calculation errors are inevitable.
A semiring has two binary operations, commonly denoted
A lattice is another kind of algebraic structure with two binary operations,
Each interpretation is responsible for different distributive laws in the Boolean algebra.
The operations are usually defined to be distributive on the right but not on the left.
In several mathematical areas, generalized distributivity laws are considered.
This may involve the weakening of the above conditions or the extension to infinitary operations.
Especially in order theory one finds numerous important variants of distributivity, some of which include infinitary operations, such as the infinite distributive law; others being defined in the presence of only one binary operation, such as the according definitions and their relations are given in the article distributivity (order theory).
In the presence of an ordering relation, one can also weaken the above equalities by replacing
Naturally, this will lead to meaningful concepts only in some situations.
An application of this principle is the notion of sub-distributivity as explained in the article on interval arithmetic.
This is exactly the data needed to define a monad structure on
A generalized distributive law has also been proposed in the area of information theory.
The ubiquitous identity that relates inverses to the binary operation in any group, namely
which is taken as an axiom in the more general context of a semigroup with involution, has sometimes been called an antidistributive property (of inversion as a unary operation).
[5] In the context of a near-ring, which removes the commutativity of the additively written group and assumes only one-sided distributivity, one can speak of (two-sided) distributive elements but also of antidistributive elements.
[6] In the study of propositional logic and Boolean algebra, the term antidistributive law is sometimes used to denote the interchange between conjunction and disjunction when implication factors over them:[7]
These two tautologies are a direct consequence of the duality in De Morgan's laws.