View source for Distributive property

{{Short description|Property involving two mathematical operations}}
{{redirect distinguish|Distributivity|Distributivism}}
{{Infobox mathematical statement
| name = Distributive property
| image = Illustration of distributive property with rectangles.svg
| caption = Visualization of distributive law for positive numbers
| type = [[Principle|Law]], [[rule of replacement]]
| field = {{Plainlist|
* [[Elementary algebra]]
* [[Boolean algebra]]
* [[Abstract algebra]]
* [[Set theory]]
* [[Propositional calculus]]
}}
| statement = 
| symbolic statement = {{Plainlist|
# Elementary algebra
#: <math display="block">x \cdot (y + z) = x \cdot y + x \cdot z</math>
# Propositional calculus:
## <math>(P \land (Q \lor R)) \Leftrightarrow ((P \land Q) \lor (P \land R))</math>
## <math>(P \lor (Q \land R)) \Leftrightarrow ((P \lor Q) \land (P \lor R))</math>
}}
}}

In [[mathematics]], the '''distributive property''' of [[binary operation]]s is a generalization of the '''distributive law''', which asserts that the equality
<math display="block">x \cdot (y + z) = x \cdot y + x \cdot z</math>
is always true in [[elementary algebra]].
For example, in [[elementary arithmetic]], one has
<math display="block">2 \cdot (1 + 3) = (2 \cdot 1) + (2 \cdot 3).</math>
Therefore, one would say that [[multiplication]] ''distributes'' over [[addition]].

This basic property of numbers is part of the definition of most [[algebraic structure]]s that have two operations called addition and multiplication, such as [[complex number]]s, [[polynomial]]s, [[Matrix (mathematics)|matrices]], [[Ring (mathematics)|rings]], and [[Field (mathematics)|fields]]. It is also encountered in [[Boolean algebra]] and [[mathematical logic]], where each of the [[logical and]] (denoted <math>\,\land\,</math>) and the [[logical or]] (denoted <math>\,\lor\,</math>) distributes over the other.

== Definition ==

Given a [[Set (mathematics)|set]] <math>S</math> and two [[binary operator]]s <math>\,*\,</math> and <math>\,+\,</math> on <math>S,</math>
*the operation <math>\,*\,</math> is {{em|left-distributive}} over (or with respect to) <math>\,+\,</math> if, [[given any]] elements <math>x, y, \text{ and } z</math> of <math>S,</math>
<math display=block>x * (y + z) = (x * y) + (x * z);</math>
*the operation <math>\,*\,</math> is {{em|right-distributive}} over <math>\,+\,</math> if, given any elements <math>x, y, \text{ and } z</math> of <math>S,</math>
<math display=block>(y + z) * x = (y * x) + (z * x);</math> 
*and the operation <math>\,*\,</math> is {{em|distributive}} over <math>\,+\,</math> if it is left- and right-distributive.<ref>[http://mathonline.wikidot.com/distributivity-of-binary-operations Distributivity of Binary Operations] from Mathonline</ref>

When <math>\,*\,</math> is [[commutative]], the three conditions above are [[Logical equivalence|logically equivalent]].

== Meaning ==

The operators used for examples in this section are those of the usual [[addition]] <math>\,+\,</math> and [[multiplication]] <math>\,\cdot.\,</math>

If the operation denoted <math>\cdot</math> is not commutative, there is a distinction between left-distributivity and right-distributivity:

<math display="block">a \cdot \left( b \pm c \right) = a \cdot b \pm a \cdot c \qquad \text{ (left-distributive) }</math>
<math display="block">(a \pm b) \cdot c = a \cdot c \pm b \cdot c \qquad \text{ (right-distributive) }.</math>

In either case, the distributive property can be described in words as:

To multiply a [[Summation|sum]] (or [[Difference (mathematics)|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 {{em|distributivity}}.

One example of an operation that is "only" right-distributive is division, which is not commutative:
<math display="block">(a \pm b) \div c = a \div c \pm b \div c.</math> 
In this case, left-distributivity does not apply:
<math display="block">a \div(b \pm c) \neq a \div b \pm a \div c</math>

The distributive laws are among the axioms for [[Ring (mathematics)|rings]] (like the ring of [[integer]]s) and [[Field (mathematics)|fields]] (like the field of [[rational number]]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]] 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 <math>\R</math> 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 (mathematics)|field]], which ensures the validity of the distributive law.

{{glossary}}
{{term|First example (mental and written multiplication)}}{{defn|During mental arithmetic, distributivity is often used unconsciously:
<math display="block">6 \cdot 16 = 6 \cdot (10 + 6) = 6\cdot 10 + 6 \cdot 6 = 60 + 36 = 96</math>

Thus, to calculate <math>6 \cdot 16</math> in one's head, one first multiplies <math>6 \cdot 10</math> and <math>6 \cdot 6</math> and add the intermediate results. Written multiplication is also based on the distributive law.
}}
{{term|Second example (with variables)}}{{defn|
<math display="block">3 a^2 b \cdot (4 a - 5 b) = 3 a^2 b \cdot 4a - 3 a^2 b \cdot 5 b = 12 a^3 b - 15 a^2 b^2</math>
}}
{{term|Third example (with two sums)}}{{defn|
<math display="block">\begin{align}
(a + b) \cdot (a - b) & = a \cdot (a - b) + b \cdot (a - b) = a^2 - ab + ba - b^2 = a^2 - b^2 \\
                      & = (a + b) \cdot a - (a + b) \cdot b = a^2 + ba - ab - b^2 = a^2 - b^2 \\
\end{align}</math>

Here the distributive law was applied twice, and it does not matter which bracket is first multiplied out.
}}
{{term|Fourth example}}{{defn|Here the distributive law is applied the other way around compared to the previous examples. Consider
<math display="block">12 a^3 b^2 - 30 a^4 b c + 18 a^2 b^3 c^2 \,.</math>

Since the factor <math>6 a^2 b</math> occurs in all summands, it can be factored out. That is, due to the distributive law one obtains
<math display="block">12 a^3 b^2 - 30 a^4 b c + 18 a^2 b^3 c^2 = 6 a^2 b \left(2 a b - 5 a^2 c + 3 b^2 c^2\right).</math>
}}
{{glossary end}}

=== Matrices ===

The distributive law is valid for [[matrix multiplication]]. More precisely,
<math display="block">(A + B) \cdot C = A \cdot C + B \cdot C</math>
for all  <math>l \times m</math>-matrices <math>A, B</math> and <math>m \times n</math>-matrices <math>C,</math> as well as
<math display="block">A \cdot (B + C) = A \cdot B + A \cdot C</math>
for all <math>l \times m</math>-matrices <math>A</math> and <math>m \times n</math>-matrices <math>B, C.</math> 
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 ===

* [[Ordinal arithmetic#Multiplication|Multiplication]] of [[ordinal number]]s, in contrast, is only left-distributive, not right-distributive.
* The [[cross product]] is left- and right-distributive over [[vector addition]], though not commutative.
* The [[Union (set theory)|union]] of sets is distributive over [[Intersection (set theory)|intersection]], and intersection is distributive over union.
* [[Logical disjunction]] ("or") is distributive over [[logical conjunction]] ("and"), and vice versa.
* For [[real number]]s (and for any [[totally ordered set]]), the [[maximum]] operation is distributive over the [[minimum]] operation, and vice versa: <math display="block">\max(a, \min(b, c)) = \min(\max(a, b), \max(a, c)) \quad \text{ and } \quad \min(a, \max(b, c)) = \max(\min(a, b), \min(a, c)).</math>
* For [[integer]]s, the [[greatest common divisor]] is distributive over the [[least common multiple]], and vice versa: <math display="block">\gcd(a, \operatorname{lcm}(b, c)) = \operatorname{lcm}(\gcd(a, b), \gcd(a, c)) \quad \text{ and } \quad \operatorname{lcm}(a, \gcd(b, c)) = \gcd(\operatorname{lcm}(a, b), \operatorname{lcm}(a, c)).</math>
* For real numbers, addition distributes over the maximum operation, and also over the minimum operation: <math display="block">a + \max(b, c) = \max(a + b, a + c) \quad \text{ and } \quad a + \min(b, c) = \min(a + b, a + c).</math>
* For [[Binomial (polynomial)|binomial]] multiplication, distribution is sometimes referred to as the [[FOIL Method]]<ref>Kim Steward (2011) [http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut28_multpoly.htm Multiplying Polynomials] from Virtual Math Lab at [[West Texas A&M University]]</ref> (First terms <math>a c,</math> Outer <math>a d,</math> Inner <math>b c,</math> and Last <math>b d</math>) such as: <math>(a + b) \cdot (c + d) = a c + a d + b c + b d.</math>
* In all [[semirings]], including the [[complex number]]s, the [[quaternion]]s, [[polynomial]]s, and [[matrix (mathematics)|matrices]],  multiplication distributes over addition: <math>u (v + w) = u v + u w, (u + v)w = u w + v w.</math>
* In all [[Algebra over a field|algebras over a field]], including the [[octonion]]s and other [[non-associative algebra]]s, multiplication distributes over addition.

== Propositional logic ==
{{Transformation rules}}

=== Rule of replacement ===

In standard truth-functional propositional logic, {{em|distribution}}<ref>[[Elliott Mendelson]] (1964) ''Introduction to Mathematical Logic'', page 21, D. Van Nostrand Company</ref><ref>[[Alfred Tarski]] (1941) ''Introduction to Logic'', page 52, [[Oxford University Press]]</ref> in logical proofs uses two valid [[Rule of replacement|rules of replacement]] to expand individual occurrences of certain [[logical connective]]s, within some [[Logical formula|formula]], into separate applications of those connectives across subformulas of the given formula. The rules are
<math display="block">(P \land (Q \lor R)) \Leftrightarrow ((P \land Q) \lor (P \land R)) \qquad \text{ and } \qquad (P \lor (Q \land R)) \Leftrightarrow ((P \lor Q) \land (P \lor R))</math>
where "<math>\Leftrightarrow</math>", also written <math>\,\equiv,\,</math> is a [[metalogic]]al [[Symbol (formal)|symbol]] representing "can be replaced in a proof with" or "is [[Logical equivalence|logically equivalent]] to".

=== Truth functional connectives ===

{{em|Distributivity}} 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 [[Tautology (logic)|tautologies]].
<math display="block">\begin{alignat}{13}
&(P &&\;\land &&(Q \lor R)) &&\;\Leftrightarrow\;&& ((P \land Q) &&\;\lor (P \land R)) && \quad\text{ Distribution of } && \text{ conjunction } && \text{ over } && \text{ disjunction } \\
&(P &&\;\lor &&(Q \land R)) &&\;\Leftrightarrow\;&& ((P \lor Q) &&\;\land (P \lor R)) && \quad\text{ Distribution of } && \text{ disjunction } && \text{ over } && \text{ conjunction } \\
&(P &&\;\land &&(Q \land R)) &&\;\Leftrightarrow\;&& ((P \land Q) &&\;\land (P \land R)) && \quad\text{ Distribution of } && \text{ conjunction } && \text{ over } && \text{ conjunction } \\
&(P &&\;\lor &&(Q \lor R)) &&\;\Leftrightarrow\;&& ((P \lor Q) &&\;\lor (P \lor R)) && \quad\text{ Distribution of } && \text{ disjunction } && \text{ over } && \text{ disjunction } \\
&(P &&\to &&(Q \to R)) &&\;\Leftrightarrow\;&& ((P \to Q) &&\to (P \to R)) && \quad\text{ Distribution of } && \text{ implication } && \text{  } && \text{  } \\
&(P &&\to &&(Q \leftrightarrow R)) &&\;\Leftrightarrow\;&& ((P \to Q) &&\leftrightarrow (P \to R)) && \quad\text{ Distribution of } && \text{ implication } && \text{ over } && \text{ equivalence } \\
&(P &&\to &&(Q \land R)) &&\;\Leftrightarrow\;&& ((P \to Q) &&\;\land (P \to R)) && \quad\text{ Distribution of } && \text{ implication } && \text{ over } && \text{ conjunction } \\
&(P &&\;\lor &&(Q \leftrightarrow R)) &&\;\Leftrightarrow\;&& ((P \lor Q) &&\leftrightarrow (P \lor R)) && \quad\text{ Distribution of } && \text{ disjunction } && \text{ over } && \text{ equivalence } \\
\end{alignat}</math>

;Double distribution: 
<math display="block">\begin{alignat}{13}
&((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{alignat}</math>

== Distributivity and rounding ==

In approximate arithmetic, such as [[floating-point arithmetic]], the distributive property of multiplication (and division) over addition may fail because of the limitations of [[arithmetic precision]].  For example, the identity <math>1/3 + 1/3 + 1/3 = (1 + 1 + 1) / 3</math> fails in [[decimal arithmetic]], regardless of the number of [[significant digit]]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 [[semiring]]s, notably the particular cases of [[Ring (algebra)|ring]]s and [[distributive lattice]]s.

A semiring has two binary operations, commonly denoted <math>\,+\,</math> and <math>\,*,</math> and requires that <math>\,*\,</math> must distribute over <math>\,+.</math>

A ring is a semiring with additive inverses.

A [[Lattice (order)|lattice]] is another kind of [[algebraic structure]] with two binary operations, <math>\,\land \text{ and } \lor.</math> 
If either of these operations distributes over the other (say <math>\,\land\,</math> distributes over <math>\,\lor</math>), then the reverse also holds (<math>\,\lor\,</math> distributes over <math>\,\land\,</math>), and the lattice is called distributive. See also {{em|[[Distributivity (order theory)]]}}.

A [[Boolean algebra (structure)|Boolean algebra]] can be interpreted either as a special kind of ring (a [[Boolean ring]]) 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]]s and [[Near-field (mathematics)|near-field]]s instead of rings and [[division ring]]s. The operations are usually defined to be distributive on the right but not on the left.

== Generalizations ==
{{anchor|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]] 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 {{em|one}} 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 <math>\,=\,</math> by either <math>\,\leq\,</math> or <math>\,\geq.</math> 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]], if <math>(S, \mu, \nu)</math> and <math>\left(S^{\prime}, \mu^{\prime}, \nu^{\prime}\right)</math> are [[Monad (category theory)|monad]]s on a [[Category (mathematics)|category]] <math>C,</math> a '''distributive law''' <math>S . S^{\prime} \to S^{\prime} . S</math> is a [[natural transformation]] <math>\lambda : S . S^{\prime} \to S^{\prime} . S</math> such that <math>\left(S^{\prime}, \lambda\right)</math> is a [[lax map of monads]] <math>S \to S</math> and <math>(S, \lambda)</math> is a [[colax map of monads]] <math>S^{\prime} \to S^{\prime}.</math> This is exactly the data needed to define a monad structure on <math>S^{\prime} . S</math>: the multiplication map is <math>S^{\prime} \mu . \mu^{\prime} S^2 . S^{\prime} \lambda S</math> and the unit map is <math>\eta^{\prime} S . \eta.</math> See: [[distributive law between monads]].

A [[generalized distributive law]] has also been proposed in the area of [[information theory]].

=== Antidistributivity ===

The ubiquitous [[Identity (mathematics)|identity]] that relates inverses to the binary operation in any [[Group (mathematics)|group]], namely <math>(x y)^{-1} = y^{-1} x^{-1},</math> 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]]).<ref name="BrinkKahl1997">{{cite book|author1=Chris Brink|author2=Wolfram Kahl|author3=Gunther Schmidt|title=Relational Methods in Computer Science|url=https://archive.org/details/relationalmethod00jips|url-access=limited|date=1997|publisher=Springer|isbn=978-3-211-82971-4|page=[https://archive.org/details/relationalmethod00jips/page/n16 4]}}</ref>

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 <math>a</math> reverses the order of addition when multiplied to the right: <math>(x + y) a = y a + x a.</math><ref>{{cite book|author1=Celestina Cotti Ferrero|author2=Giovanni Ferrero|title=Nearrings: Some Developments Linked to Semigroups and Groups|year=2002|publisher=Kluwer Academic Publishers|isbn=978-1-4613-0267-4|pages=62 and 67}}</ref>

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:<ref name="Hehner1993">{{cite book|author=Eric C.R. Hehner|author-link=Eric Hehner|title=A Practical Theory of Programming|year=1993|publisher=Springer Science & Business Media|isbn=978-1-4419-8596-5|page=230}}</ref>
<math display="block">(a \lor b) \Rightarrow c \equiv (a \Rightarrow c) \land (b \Rightarrow c)</math>
<math display="block">(a \land b) \Rightarrow c \equiv (a \Rightarrow c) \lor (b \Rightarrow c).</math>

These two [[Tautology (logic)|tautologies]] are a direct consequence of the duality in [[De Morgan's laws]].

== Notes ==

{{reflist|group=note}}
{{reflist}}

== External links ==

{{Wiktionary|distributivity}}
* [http://www.cut-the-knot.org/Curriculum/Arithmetic/DistributiveLaw.shtml A demonstration of the Distributive Law] for integer arithmetic (from [[cut-the-knot]])

[[Category:Properties of binary operations]]
[[Category:Elementary algebra]]
[[Category:Rules of inference]]
[[Category:Theorems in propositional logic]]