distributivity

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OR:

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 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 ...
s generalizes the distributive law, which asserts that the equality $x \cdot (y + z) = x \cdot y + x \cdot z$ is always true in
elementary algebra Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variable (mathematics), variables (quantities without fixed values). This ...
. For example, in
elementary arithmetic The operators in elementary arithmetic Arithmetic () is an elementary part of 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 ...
, one has $2 \cdot (1 + 3) = (2 \cdot 1) + (2 \cdot 3).$ 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 and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
. This basic property of numbers is part of the definition of most
algebraic structure In mathematics, an algebraic structure consists of a nonempty Set (mathematics), set ''A'' (called the underlying set, carrier set or domain), a collection of operation (mathematics), operations on ''A'' (typically binary operations such as addit ...
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 form a ...
s,
polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
s,
matrices Matrix most commonly refers to: * The Matrix (franchise), ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within Th ...
, 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 variable (mathematics), variables are the truth values ''true'' and ''false'', usually denote ...
and
mathematical logic Mathematical logic is the study of formal logic within 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 quantiti ...
, where each of the logical and (denoted $\,\land\,$) and the logical or (denoted $\,\lor\,$) distributes over the other.

# Definition

Given a set $S$ and two
binary operator In mathematics, a binary operation or dyadic operation is a rule for combining two Element (mathematics), elements (called operands) to produce another element. More formally, a binary operation is an Operation (mathematics), operation of arity ...
s $\,*\,$ and $\,+\,$ on $S,$ *the operation $\,*\,$ is over (or with respect to) $\,+\,$ if, given any elements $x, y, \text z$ of $S,$ $x * (y + z) = (x * y) + (x * z);$ *the operation $\,*\,$ is over $\,+\,$ if, given any elements $x, y, \text z$ of $S,$ $(y + z) * x = (y * x) + (z * x);$ *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 Logic is the study of correct reason Reason is the capacity of Consciousness, consciously applying logic by Logical consequence, drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associat ...
.

# Meaning

The operators used for examples in this section are those of the usual
addition Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
$\,+\,$ 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 ...
$\,\cdot.\,$ If the operation denoted $\cdot$ is not commutative, there is a distinction between left-distributivity and right-distributivity: $a \cdot \left( b \pm c \right) = a \cdot b \pm a \cdot c \qquad \text$ $(a \pm b) \cdot c = a \cdot c \pm b \cdot c \qquad \text.$ 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: $(a \pm b) \div c = a \div c \pm b \div c.$ In this case, left-distributivity does not apply: $a \div(b \pm c) \neq a \div b \pm a \div c$ 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 language of ...
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 (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
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 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 ...
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 $\R$ 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 In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...
. More precisely, $(A + B) \cdot C = A \cdot C + B \cdot C$ for all $l \times m$-matrices $A, B$ and $m \times n$-matrices $C,$ as well as $A \cdot (B + C) = A \cdot B + A \cdot C$ for all $l \times m$-matrices $A$ and $m \times n$-matrices $B, C.$ 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 least n ...
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 Euclidean vector, vectors in a three-dimensional Orientation (vector space), oriented E ...
is left- and right-distributive over
vector addition In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has Magnitude (mathematics), magnitude (or euclidean norm, length) and Direction ( ...
, though not commutative. * The union of sets is distributive over intersection, and intersection is distributive over union. *
Logical disjunction In logic Logic is the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), deductively valid inferences or of logical truths. It is a formal science ...
("or") is distributive over logical conjunction ("and"), and vice versa. * For
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s (and for any totally ordered set), the maximum operation is distributive over the minimum operation, and vice versa: $\max(a, \min(b, c)) = \min(\max(a, b), \max(a, c)) \quad \text \quad \min(a, \max(b, c)) = \max(\min(a, b), \min(a, c)).$ * 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 language of ...
s, the
greatest common divisor 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 mo ...
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 bot ...
, and vice versa: $\gcd(a, \operatorname(b, c)) = \operatorname(\gcd(a, b), \gcd(a, c)) \quad \text \quad \operatorname(a, \gcd(b, c)) = \gcd(\operatorname(a, b), \operatorname(a, c)).$ * For real numbers, addition distributes over the maximum operation, and also over the minimum operation: $a + \max(b, c) = \max(a + b, a + c) \quad \text \quad a + \min(b, c) = \min(a + b, a + c).$ * For binomial multiplication, distribution is sometimes referred to as the FOIL Method (First terms $a c,$ Outer $a d,$ Inner $b c,$ and Last $b d$) such as: $\left(a + b\right) \cdot \left(c + d\right) = a c + a d + b c + b d.$ * 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 form a ...
s, the
quaternion 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 ...
s,
polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
s, and
matrices Matrix most commonly refers to: * The Matrix (franchise), ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within Th ...
, multiplication distributes over addition: $u \left(v + w\right) = u v + u w, \left(u + v\right)w = u w + v w.$ * In all algebras over a field, including the octonions 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 Mathematical logic, 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 (logic), syntax o ...
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 terminology, term ''formula'' in science refers to the Commensurability (philosophy o ...
, into separate applications of those connectives across subformulas of the given formula. The rules are $(P \land (Q \lor R)) \Leftrightarrow ((P \land Q) \lor (P \land R)) \qquad \text \qquad (P \lor (Q \land R)) \Leftrightarrow ((P \lor Q) \land (P \lor R))$ where "$\Leftrightarrow$", also written $\,\equiv,\,$ is a
metalogic Metalogic is the study of the metatheory of logic. Whereas ''logic'' studies how formal system, logical systems can be used to construct Validity (logic), valid and soundness, sound arguments, metalogic studies the properties of logical systems.Har ...
al
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 ...
representing "can be replaced in a proof with" or "is
logically equivalent Logic is the study of correct reason Reason is the capacity of Consciousness, consciously applying logic by Logical consequence, drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associat ...
to".

## Truth functional connectives

is a property of some logical connectives of truth-functional
propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
. The following logical equivalences demonstrate that distributivity is a property of particular connectives. The following are truth-functional tautologies. $\begin &(P &&\;\land &&(Q \lor R)) &&\;\Leftrightarrow\;&& ((P \land Q) &&\;\lor (P \land R)) && \quad\text && \text && \text && \text \\ &(P &&\;\lor &&(Q \land R)) &&\;\Leftrightarrow\;&& ((P \lor Q) &&\;\land (P \lor R)) && \quad\text && \text && \text && \text \\ &(P &&\;\land &&(Q \land R)) &&\;\Leftrightarrow\;&& ((P \land Q) &&\;\land (P \land R)) && \quad\text && \text && \text && \text \\ &(P &&\;\lor &&(Q \lor R)) &&\;\Leftrightarrow\;&& ((P \lor Q) &&\;\lor (P \lor R)) && \quad\text && \text && \text && \text \\ &(P &&\to &&(Q \to R)) &&\;\Leftrightarrow\;&& ((P \to Q) &&\to (P \to R)) && \quad\text && \text && \text && \text \\ &(P &&\to &&(Q \leftrightarrow R)) &&\;\Leftrightarrow\;&& ((P \to Q) &&\leftrightarrow (P \to R)) && \quad\text && \text && \text && \text \\ &(P &&\to &&(Q \land R)) &&\;\Leftrightarrow\;&& ((P \to Q) &&\;\land (P \to R)) && \quad\text && \text && \text && \text \\ &(P &&\;\lor &&(Q \leftrightarrow R)) &&\;\Leftrightarrow\;&& ((P \lor Q) &&\leftrightarrow (P \lor R)) && \quad\text && \text && \text && \text \\ \end$ ;Double distribution: $\begin &((P \land Q) &&\;\lor (R \land S)) &&\;\Leftrightarrow\;&& (((P \lor R) \land (P \lor S)) &&\;\land ((Q \lor R) \land (Q \lor S))) && \\ &((P \lor Q) &&\;\land (R \lor S)) &&\;\Leftrightarrow\;&& (((P \land R) \lor (P \land S)) &&\;\lor ((Q \land R) \lor (Q \land S))) && \\ \end$

# 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 (computer science), integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. ...
, the distributive property of multiplication (and division) over addition may fail because of the limitations of arithmetic precision. For example, the identity $1/3 + 1/3 + 1/3 = \left(1 + 1 + 1\right) / 3$ 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.

# In rings and other structures

Distributivity is most commonly found in
semiring In abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), ...
s, 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 (mathematics), set ''A'' (called the underlying set, carrier set or domain), a collection of operation (mathematics), operations on ''A'' (typically binary operations such as addit ...
with two binary operations, $\,\land \text \lor.$ If either of these operations distributes over the other (say $\,\land\,$ distributes over $\,\lor$), then the reverse also holds ($\,\lor\,$ distributes over $\,\land\,$), 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 variable (mathematics), variables are the truth values ''true'' and ''false'', usually denote ...
can be interpreted either as a special kind of ring (a
Boolean ring In mathematics, a Boolean ring ''R'' is a ring (mathematics), ring for which ''x''2 = ''x'' for all ''x'' in ''R'', that is, a ring that consists only of idempotent element (ring theory), idempotent elements. An example is the ring of modular arithm ...
) or a special kind of distributive lattice (a Boolean lattice). Each interpretation is responsible for different distributive laws in the Boolean algebra. Similar structures without distributive laws are
near-ring 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 mode ...
s and near-fields instead of rings and
division ring In algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost a ...
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 intro ...
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). This also includes the notion of a completely distributive lattice. In the presence of an ordering relation, one can also weaken the above equalities by replacing $\,=\,$ by either $\,\leq\,$ or $\,\geq.$ 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. 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, categ ...
, if $\left(S, \mu, \nu\right)$ and $\left\left(S^, \mu^, \nu^\right\right)$ are monads on a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being *Categories (Aristotle), ''Categories'' (Aristotle) *Category (Kant) ...
$C,$ a distributive law $S . S^ \to S^ . S$ is a
natural transformation 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 topo ...
$\lambda : S . S^ \to S^ . S$ such that $\left\left(S^, \lambda\right\right)$ is a lax map of monads $S \to S$ and $\left(S, \lambda\right)$ is a colax map of monads $S^ \to S^.$ This is exactly the data needed to define a monad structure on $S^ . S$: the multiplication map is $S^ \mu . \mu^ S^2 . S^ \lambda S$ and the unit map is $\eta^ S . \eta.$ See: distributive law between monads. A generalized distributive law has also been proposed in the area of
information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a ...
.

## Antidistributivity

The ubiquitous identity that relates inverses to the binary operation in any group, namely $\left(x y\right)^ = y^ x^,$ 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 In mathematics, an unary operation is an Operation (mathematics), 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 (mathematics), function , where ...
). In the context of a
near-ring 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 mode ...
, 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 $a$ reverses the order of addition when multiplied to the right: $\left(x + y\right) a = y a + x a.$ In the study of
propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
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 variable (mathematics), variables are the truth values ''true'' and ''false'', usually denote ...
, the term antidistributive law is sometimes used to denote the interchange between conjunction and disjunction when implication factors over them: $(a \lor b) \Rightarrow c \equiv (a \Rightarrow c) \land (b \Rightarrow c)$ $(a \land b) \Rightarrow c \equiv (a \Rightarrow c) \lor (b \Rightarrow c).$ These two tautologies are a direct consequence of the duality in De Morgan's laws.