In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two 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 For example:
- 2 • (1 + 3) = (2 • 1) + (2 • 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 numbers in place of 2, 1, and 3 above, and still have obtained a true equation, we say that multiplication of real numbers distributes over addition of real numbers. In Mathematics, the real numbers may be described informally in several different ways Addition is the mathematical process of putting things together
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Definition
Given a set S and two binary operations • and + on S, we say that the operation •
- is left-distributive over + if, given any elements x, y, and z of S,
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- x • (y + z) = (x • y) + (x • z);
- is right-distributive over + if, given any elements x, y, and z of S:
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- (y + z) • x = (y • x) + (z • x);
- is distributive over + if it is both left- and right-distributive. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two In Predicate logic, universal quantification is an attempt to formalize the notion that something (a Logical predicate) is true for everything, or every In Predicate logic, universal quantification is an attempt to formalize the notion that something (a Logical predicate) is true for everything, or every
Notice that when • is commutative, then the three above conditions are logically equivalent. In Mathematics, commutativity is the ability to change the order of something without changing the end result In Logic, statements p and q are logically equivalent if they have the same logical content
Examples
- Multiplication of numbers is distributive over addition of numbers, for a broad class of different kinds of numbers ranging from natural numbers to complex numbers and cardinal numbers. A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English.
- Multiplication of ordinal numbers, in contrast, is only left-distributive, not right-distributive. In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set.
- Matrix multiplication is distributive over matrix addition, even though it's not commutative. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix In Mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together
- The union of sets is distributive over intersection, and intersection is distributive over union. In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets In Mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently Also, intersection is distributive over the symmetric difference. In Mathematics, the symmetric difference of two sets is the set of elements which are in one of the sets but not in both
- Logical disjunction ("or") is distributive over logical conjunction ("and"), and conjunction is distributive over disjunction. 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 Also, conjunction is distributive over exclusive disjunction ("xor").
- For real numbers (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)). In Mathematics, the real numbers may be described informally in several different ways In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation
- For integers, the greatest common divisor is distributive over the least common multiple, 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)). The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, the greatest common divisor (gcd, sometimes known as the greatest common factor (gcf or highest common factor (hcf, of two non-zero In Arithmetic and Number theory, the least common multiple or lowest common multiple ( lcm) or smallest common multiple of two
- 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. The precision of a value describes the number of digits that are used to express that value For example, the identity ⅓+⅓+⅓ = (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. Algorism is the technique of performing basic Arithmetic by writing numbers in Place value form and applying a set of memorized rules and facts to the digits The significant figures (also called significant digits and abbreviated sig figs) of a number are those digits that carry meaning contributing to its accuracy 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 of 17. Value added tax ( VAT) or goods and services tax ( GST) is a consumption Tax levied on value added. 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 may help in some cases, as may increasing the precision used, but ultimately some calculation errors are inevitable. For lip-rounding in phonetics see Labialisation and Roundedness.
Distributivity in rings
Distributivity is most commonly found in rings and distributive lattices. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other
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 is another kind of algebraic structure with two binary operations, ^ and v. In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, 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). In the mathematical area of Order theory, there are various notions of the common concept of Distributivity, applied to the formation of suprema and
Examples 4 and 5 are Boolean algebras, which can be interpreted either as a special kind of ring (a Boolean ring) or a special kind of distributive lattice (a Boolean lattice). In Abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. In Mathematics, a Boolean ring R is a ring (with identity for which x 2 = x for all x in R; that In Abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive 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. In Abstract algebra, a semiring is an Algebraic structure similar to a ring, but without the requirement that each element must have an Additive inverse 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 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 algebras. 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 In the mathematical area of Order theory, there are various notions of the common concept of Distributivity, applied to the formation of suprema and In Mathematics, Heyting algebras are special Partially ordered sets that constitute a generalization of Boolean algebras named after Arend Heyting Details of the according definitions and their relations are given in the article distributivity (order theory). In the mathematical area of Order theory, there are various notions of the common concept of Distributivity, applied to the formation of suprema and This also includes the notion of a completely distributive lattice. In the mathematical area of Order theory, a completely distributive lattice is a Complete lattice in which arbitrary joins distribute 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. In Mathematics, an 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, if (S, μ, η) and (S', μ', η') are monads on a category C, a distributive law S. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Category theory, a monad or triple is an (endo- Functor, together with two associated Natural transformations They are important in the theory In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships S' → S'. S is a natural transformation λ : S. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal S' → S'. S such that (S' , λ) is a lax map of monads S → S and (S, λ) 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'λS and the unit map is η'S. η. See: 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
External links
- A demonstration of the Distributive Law for integer arithmetic (from cut-the-knot)
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