Distributivity

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Distributivity

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, and in particular in abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as group , ring , field , module , vector spaces, and algebra over a field....
, distributivity is a property of binary operation

Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator....
s that generalises the distributive law from elementary algebra

Elementary algebra

Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic....
. For example:

2 x (1 + 3) = (2 x 1) + (2 x 3).

In the left-hand side of the above equation, the 2 multiplies the sum of 1 and 3; on the right-hand side, it multiplies the 1 and the 3 individually, with the results added afterwards. Because these give the same final answer (8), we say that multiplication by 2 distributes over addition of 1 and 3. Since we could have put any real number

Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s in place of 2, 1, and 3 above, and still have obtained a true equation, we say that multiplication

Multiplication

Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
of real numbers distributes over addition

Addition

Addition is the mathematics process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples?meaning three apples and two other apples?which is the same as five apples, since 3 + 2 = 5....
of real numbers.

· (y + z) = (x · y) + (x · z);

· x = (y · x) + (z · x);

Notice that when · is commutative, then the three above conditions are logically equivalent

Logical equivalence

In logic, statements p and q are logically equivalent if they have the same logical content.Syntax , p and q are equivalent if each can be proof from the other....
.

Examples

Multiplication of number
Number

A number is a mathematical object used in counting and measurement. A notational symbol which represents a number is called a Numeral system, but in common usage the word number is used for both the abstract object and the symbol, as well as for the numeral for the number....
s is distributive over addition of numbers, for a broad class of different kinds of numbers ranging from natural number
Natural number

In mathematics, a natural number can mean either an element of the Set = *n = = ? = ? ...
s to complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
s and cardinal number
Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of Set ....
s.
Multiplication of ordinal number
Ordinal number

In set theory, an ordinal number, or just ordinal, is the order type of a well-order. They are usually identified with hereditarily transitive sets....
s, in contrast, is only left-distributive, not right-distributive.
Matrix
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
multiplication
Matrix multiplication

In mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix. This article gives an overview of the various ways to perform matrix multiplication....
is distributive over matrix addition
Matrix addition

In mathematics, matrix addition is the operation of adding two matrix by adding the corresponding entries together. However, there is another operation which could also be considered as a kind of addition for matrices....
, even though it's not commutative.

The union

Union (set theory)

In set theory, the term Union refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets....
of sets is distributive over intersection

Intersection (set theory)

In mathematics, the intersection of two Set A and B is the set that contains all elements of A that also belong to B , but no other elements....
, and intersection is distributive over union.

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Encyclopedia

In mathematics

Mathematics

Abstract algebra

Binary operation

Elementary algebra

2 x (1 + 3) = (2 x 1) + (2 x 3).

Real number

Multiplication

Addition

Definition

Given a set S and two binary operation

Binary operation

is left-distributive over + if, given any elements x, y, and z of S,

x · (y + z) = (x · y) + (x · z);

is right-distributive over + if, given any elements x, y, and z of S:

· x = (y · x) + (z · x);

is distributive over + if it is both left- and right-distributive.

Notice that when · is commutative, then the three above conditions are logically equivalent

Logical equivalence

In logic, statements p and q are logically equivalent if they have the same logical content.Syntax , p and q are equivalent if each can be proof from the other....
.

Examples

Multiplication of number
Number

A number is a mathematical object used in counting and measurement. A notational symbol which represents a number is called a Numeral system, but in common usage the word number is used for both the abstract object and the symbol, as well as for the numeral for the number....
s is distributive over addition of numbers, for a broad class of different kinds of numbers ranging from natural number
Natural number

In mathematics, a natural number can mean either an element of the Set = *n = = ? = ? ...
s to complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
s and cardinal number
Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of Set ....
s.
Multiplication of ordinal number
Ordinal number

In set theory, an ordinal number, or just ordinal, is the order type of a well-order. They are usually identified with hereditarily transitive sets....
s, in contrast, is only left-distributive, not right-distributive.
Matrix
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
multiplication
Matrix multiplication

In mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix. This article gives an overview of the various ways to perform matrix multiplication....
is distributive over matrix addition
Matrix addition

In mathematics, matrix addition is the operation of adding two matrix by adding the corresponding entries together. However, there is another operation which could also be considered as a kind of addition for matrices....
, even though it's not commutative.
The union
Union (set theory)

In set theory, the term Union refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets....
of sets is distributive over intersection
Intersection (set theory)

In mathematics, the intersection of two Set A and B is the set that contains all elements of A that also belong to B , but no other elements....
, and intersection is distributive over union. Also, intersection is distributive over the symmetric difference
Symmetric difference

In mathematics, the symmetric difference of two Set is the set of elements which are in one of the sets, but not in both. This operation is the set-theoretic kin of the exclusive disjunction in Boolean logic....
.
Logical disjunction
Logical disjunction

File:ORGate2.pngIn logic and mathematics, or, also known as logical disjunction or inclusive disjunction is a logical operator that results in true whenever one or more of its operands are true....
("or") is distributive over logical conjunction
Logical conjunction

In logic and/or mathematics, logical conjunction or and is a two-place logical operation that results in a value of true if both of its operands are true, otherwise a value of false....
("and"), and conjunction is distributive over disjunction. Also, conjunction is distributive over exclusive disjunction
Exclusive disjunction

The Logical connective exclusive disjunction, also called exclusive or , is a type of logical disjunction on two operands that results in a value of true if and only if exactly one of the operands has a value of true....
("xor").
For real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s (or for any totally ordered set), the maximum operation is distributive over the minimum operation, and vice versa: max(a,min(b,c)) = min(max(a,b),max(a,c)) and min(a,max(b,c)) = max(min(a,b),min(a,c)).
For integer
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
s, the greatest common divisor
Greatest common divisor

In mathematics, the greatest common divisor , sometimes known as the greatest common factor or highest common factor , of two non-zero integers, is the largest positive integer that divisor both numbers without remainder....
is distributive over the least common multiple
Least common multiple

In arithmetic and number theory, the least common multiple or lowest common multiple or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple both of a and of b....
, and vice versa: gcd(a,lcm(b,c)) = lcm(gcd(a,b),gcd(a,c)) and lcm(a,gcd(b,c)) = gcd(lcm(a,b),lcm(a,c)).
For real numbers, addition distributes over the maximum operation, and also over the minimum operation: a + max(b,c) = max(a+b,a+c) and a + min(b,c) = min(a+b,a+c).

Distributivity and rounding

In practice, the distributive property of multiplication (and division) over addition is lost around the limits of arithmetic precision. For example, the identity ?+?+? = (1+1+1)/3 appears to fail if conducted in decimal arithmetic; however many significant digits are used, the calculation will take the form 0.33333+0.33333+0.33333 = 0.99999 ? 1. Even where fractional numbers are representable exactly, errors will be introduced if rounding too far; for example, buying two books each priced at £14.99 before a tax

Vat

Vat and VAT may refer to:* Value added tax* A type of Packaging and labelling such as a barrel , storage tank, or tub, often constructed of welded sheet stainless steel, and used for holding, storing, and processing liquids such as milk, wine, and beer...
of 17.5% in two separate transactions will actually save £0.01 over buying them together: £14.99×1.175 = £17.61 to the nearest £0.01, giving a total expenditure of £35.22, but £29.98×1.175 = £35.23. Methods such as banker's rounding

Rounding

Rounding involves reducing the number of significant digits in a number. The result of rounding is a "shorter" number having fewer non-zero digits yet similar in magnitude....
may help in some cases, as may increasing the precision used, but ultimately some calculation errors are inevitable.

Distributivity in rings

Distributivity is most commonly found in rings and distributive lattice

Distributive lattice

In mathematics, distributive lattices are lattice for which the operations of join and meet distributivity over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection ....
s.

A ring has two binary operations (commonly called "+" and "*"), and one of the requirements of a ring is that * must distribute over +. Most kinds of numbers (example 1) and matrices (example 3) form rings. A lattice

Lattice (order)

In mathematics, a lattice is a partially ordered set in which subsets of any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain Axiom identity ....
is another kind of algebraic structure

Algebraic structure

In algebra, a branch of pure mathematics, an algebraic structure consists of one or more Set Closure under one or more Operation , satisfying some axiom....
with two binary operations, ^ and v. If either of these operations (say ^) distributes over the other (v), then v must also distribute over ^, and the lattice is called distributive. See also the article on distributivity (order theory)

Distributivity (order theory)

In the mathematics area of order theory, there are various notions of the common concept of distributivity, applied to the formation of supremum and infimum....
.

Examples 4 and 5 are Boolean algebras, which can be interpreted either as a special kind of ring (a Boolean ring

Boolean ring

In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R; that is, R consists only of idempotent elements....
) or a special kind of distributive lattice (a Boolean lattice). Each interpretation is responsible for different distributive laws in the Boolean algebra. Examples 6 and 7 are distributive lattices which are not Boolean algebras.

Rings and distributive lattices are both special kinds of rigs, certain generalisations of rings. Those numbers in example 1 that don't form rings at least form rigs. Near-rigs are a further generalisation of rigs that are left-distributive but not right-distributive; example 2 is a near-rig.

Generalizations of distributivity

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

Order theory

Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another....
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 implication operator of Heyting algebra

Heyting algebra

In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebra s, named after Arend Heyting....
s. Details of the according definitions and their relations are given in the article distributivity (order theory)

Distributivity (order theory)

In the mathematics area of order theory, there are various notions of the common concept of distributivity, applied to the formation of supremum and infimum....
. This also includes the notion of a completely distributive lattice
Completely distributive lattice

In the mathematical area of order theory, a completely distributive lattice is a complete lattice in which arbitrary joins distributivity over arbitrary meets....
.

In the presence of an ordering relation, one can also weaken the above equalities by replacing = by either = or =. 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 intervals

Interval (mathematics)

In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set....
.

In category theory

Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
, if (S, µ, ?) and (S', µ', ?') are monad

Monad (category theory)

In category theory, a monad or triple is an functor, together with two associated natural transformations. They are important in the theory of pairs of adjoint functors, and they generalize closure operators on posets to arbitrary categories....
s on a category

Category (mathematics)

In mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships. A category is composed of a collection of abstract "objects" of any kind, linked together by a collection of abstract "morphism" of any kind that have a few basic properties ....
C, a distributive law S.S' ? S'.S is a natural transformation

Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved....
? : S.S' ? S'.S such that (S' , ?) is a lax map of monads S ? S and (S, ?) is a colax map of monads S' ? S' . This is exactly the data needed to define a monad structure on S'.S: the multiplication map is S'µ.µ'S².S'?S and the unit map is ?'S.?. See: distributive law between monads

Distributive law between monads

In category theory, an abstract branch of mathematics, distributive laws between monads are a way to express abstractly that two algebraic structures distribute one over the other one....
.

External links

for integer arithmetic (from cut-the-knot
Cut-the-knot

Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics....
)