In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the distributive property of
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
s generalizes the distributive law, which asserts that the equality
is always true in
elementary algebra.
For example, in
elementary arithmetic, one has
One says that
multiplication
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
''distributes'' over
addition
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
.
This basic property of numbers is part of the definition of most
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...
s that have two operations called addition and multiplication, such as
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s,
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s,
matrices,
rings, and
fields. It is also encountered in
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
and
mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...
, where each of the
logical and (denoted
) and the
logical or (denoted
) distributes over the other.
Definition
Given a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
and two
binary operator
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary o ...
s
and
on
*the operation
is over (or with respect to)
if,
given any elements
of
*the operation
is over
if, given any elements
of
*and the operation
is over
if it is left- and right-distributive.
When
is
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, the three conditions above are
logically equivalent.
Meaning
The operators used for examples in this section are those of the usual
addition
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
and
multiplication
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
If the operation denoted
is not commutative, there is a distinction between left-distributivity and right-distributivity:
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 .
One example of an operation that is "only" right-distributive is division, which is not commutative:
In this case, left-distributivity does not apply:
The distributive laws are among the axioms for
rings (like the ring of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s) and
fields (like the field of
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s). Here multiplication is distributive over addition, but addition is not distributive over multiplication. Examples of structures with two operations that are each distributive over the other are
Boolean algebras such as the
algebra of sets
In mathematics, the algebra of sets, not to be confused with the mathematical structure of ''an'' algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the ...
or the
switching algebra.
Multiplying sums can be put into words as follows: When a sum is multiplied by a sum, multiply each summand of a sum with each summand of the other sum (keeping track of signs) then add up all of the resulting products.
Examples
Real numbers
In the following examples, the use of the distributive law on the set of real numbers
is illustrated. When multiplication is mentioned in elementary mathematics, it usually refers to this kind of multiplication. From the point of view of algebra, the real numbers form a
field, which ensures the validity of the distributive law.
Matrices
The distributive law is valid for
matrix multiplication. More precisely,
for all
-matrices
and
-matrices
as well as
for all
-matrices
and
-matrices
Because the commutative property does not hold for matrix multiplication, the second law does not follow from the first law. In this case, they are two different laws.
Other examples
*
Multiplication
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
of
ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the leas ...
s, in contrast, is only left-distributive, not right-distributive.
* The
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
is left- and right-distributive over
vector addition, though not commutative.
* The
union of sets is distributive over
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
, and intersection is distributive over union.
*
Logical disjunction
In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor ...
("or") is distributive over
logical conjunction
In logic, mathematics and linguistics, And (\wedge) is the truth-functional operator of logical conjunction; the ''and'' of a set of operands is true if and only if ''all'' of its operands are true. The logical connective that represents thi ...
("and"), and vice versa.
* For
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s (and for any
totally ordered set
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive) ...
), the maximum operation is distributive over the minimum operation, and vice versa:
* For
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s, the
greatest common divisor is distributive over the
least common multiple
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by ...
, and vice versa:
* For real numbers, addition distributes over the maximum operation, and also over the minimum operation:
* For
binomial
Binomial may refer to:
In mathematics
*Binomial (polynomial), a polynomial with two terms
*Binomial coefficient, numbers appearing in the expansions of powers of binomials
*Binomial QMF, a perfect-reconstruction orthogonal wavelet decomposition
* ...
multiplication, distribution is sometimes referred to as the
FOIL Method (First terms
Outer
Inner
and Last
) such as:
* In all
semirings, including the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, the
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s,
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s, and
matrices, multiplication distributes over addition:
* In all
algebras over a field, including the
octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s and other
non-associative algebras, multiplication distributes over addition.
Propositional logic
Rule of replacement
In standard truth-functional propositional logic, in logical proofs uses two valid
rules of replacement to expand individual occurrences of certain
logical connective
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...
s, within some
formula
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
, into separate applications of those connectives across subformulas of the given formula. The rules are
where "
", also written
is a
metalogical
symbol
A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...
representing "can be replaced in a proof with" or "is
logically equivalent to".
Truth functional connectives
is a property of some logical connectives of truth-functional
propositional logic. The following logical equivalences demonstrate that distributivity is a property of particular connectives. The following are truth-functional
tautologies.
;Double distribution:
Distributivity and rounding
In approximate arithmetic, such as
floating-point arithmetic
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be ...
, the distributive property of multiplication (and division) over addition may fail because of the limitations of
arithmetic precision
Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something.
If a number expres ...
. For example, the identity
fails in
decimal arithmetic
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
, regardless of the number of
significant digit
Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something.
If a number expres ...
s. Methods such as
banker's rounding may help in some cases, as may increasing the precision used, but ultimately some calculation errors are inevitable.
In rings and other structures
Distributivity is most commonly found in
semirings, notably the particular cases of
rings and
distributive lattices.
A semiring has two binary operations, commonly denoted
and
and requires that
must distribute over
A ring is a semiring with additive inverses.
A
lattice is another kind of
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...
with two binary operations,
If either of these operations distributes over the other (say
distributes over
), then the reverse also holds (
distributes over
), and the lattice is called distributive. See also .
A
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
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. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a ge ...
). Each interpretation is responsible for different distributive laws in the Boolean algebra.
Similar structures without distributive laws are
near-rings and
near-fields instead of rings and
division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
s. The operations are usually defined to be distributive on the right but not on the left.
Generalizations
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 is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
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 binary operation, such as 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 infima. Most of these apply to partially ordered sets that are at least lattices, but the concep ...
. 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.
Formally, a complete lattice ''L'' is said to be completely distributive if, for any doub ...
.
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
interval arithmetic
Interval arithmetic (also known as interval mathematics, interval analysis, or interval computation) is a mathematical technique used to put bounds on rounding errors and measurement errors in mathematical computation. Numerical methods using ...
.
In
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, if
and
are
monads on a
category a distributive law
is a
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 (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
such that
is a
lax map of monads and
is a
colax map of monads This is exactly the data needed to define a monad structure on
: the multiplication map is
and the unit map is
See:
distributive law between monads.
A
generalized distributive law The generalized distributive law (GDL) is a generalization of the distributive property which gives rise to a general message passing algorithm. It is a synthesis of the work of many authors in the information theory, digital communications, sign ...
has also been proposed in the area of
information theory
Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. ...
.
Antidistributivity
The ubiquitous
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), an ...
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 In mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism, which—roughly speaking—brings it closer to a group because this involution, consider ...
, has sometimes been called an antidistributive property (of inversion as a
unary operation
In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation o ...
).
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. The latter reverse the order of (the non-commutative) addition; assuming a left-nearring (i.e. one which all elements distribute when multiplied on the left), then an antidistributive element
reverses the order of addition when multiplied to the right:
In the study of
propositional logic and
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
, the term antidistributive law is sometimes used to denote the interchange between conjunction and disjunction when implication factors over them:
These two
tautologies are a direct consequence of the duality in
De Morgan's laws.
Notes
External links
{{Wiktionary, distributivity
A demonstration of the Distributive Lawfor integer arithmetic (from
cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
)
Properties of binary operations
Elementary algebra
Rules of inference
Theorems in propositional logic