Distributive property

 ( Properties Of Binary Operations ) 

In mathematics, the distributive property of is a generalization of the distributive law, which asserts that the equality $$x\; \backslash cdot\; (y\; +\; z)\; =\; x\; \backslash cdot\; y\; +\; x\; \backslash cdot\; z$$ is always true in elementary algebra. For example, in elementary arithmetic, one has $$2\; \backslash cdot\; (1\; +\; 3)\; =\; (2\; \backslash cdot\; 1)\; +\; (2\; \backslash cdot\; 3).$$ Therefore, one would say that multiplication distributes over addition.

This basic property of numbers is part of the definition of most algebraic structures that have two operations called addition and multiplication, such as , , matrices, rings, and fields. It is also encountered in Boolean algebra and mathematical logic, where each of the logical and (denoted $\backslash ,\backslash land\backslash ,$) and the logical or (denoted $\backslash ,\backslash lor\backslash ,$) distributes over the other.

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Definition Given a set $S$ and two $\backslash ,*\backslash ,$ and $\backslash ,+\backslash ,$ on $S,$

• the operation $\backslash ,*\backslash ,$ is over (or with respect to) $\backslash ,+\backslash ,$ if, given any elements $x,\; y,\; \backslash text\{\; and\; \}\; z$ of $S,$

$$x\; *\; (y\; +\; z)\; =\; (x\; *\; y)\; +\; (x\; *\; z);$$

• the operation $\backslash ,*\backslash ,$ is over $\backslash ,+\backslash ,$ if, given any elements $x,\; y,\; \backslash text\{\; and\; \}\; z$ of $S,$

$$(y\; +\; z)\; *\; x\; =\; (y\; *\; x)\; +\; (z\; *\; x);$$

• and the operation $\backslash ,*\backslash ,$ is over $\backslash ,+\backslash ,$ if it is left- and right-distributive. Distributivity of Binary Operations from Mathonline

When $\backslash ,*\backslash ,$ is commutative, the three conditions above are logically equivalent.

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Meaning The operators used for examples in this section are those of the usual addition $\backslash ,+\backslash ,$ and multiplication $\backslash ,\backslash cdot.\backslash ,$

If the operation denoted $\backslash cdot$ is not commutative, there is a distinction between left-distributivity and right-distributivity:

$$a\; \backslash cdot\; \backslash left(\; b\; \backslash pm\; c\; \backslash right)\; =\; a\; \backslash cdot\; b\; \backslash pm\; a\; \backslash cdot\; c\; \backslash qquad\; \backslash text\{\; (left-distributive)\; \}$$ $$(a\; \backslash pm\; b)\; \backslash cdot\; c\; =\; a\; \backslash cdot\; c\; \backslash pm\; b\; \backslash cdot\; c\; \backslash qquad\; \backslash text\{\; (right-distributive)\; \}.$$

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

To multiply a Summation (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: $$(a\; \backslash pm\; b)\; \backslash div\; c\; =\; a\; \backslash div\; c\; \backslash pm\; b\; \backslash div\; c.$$ In this case, left-distributivity does not apply: $$a\; \backslash div(b\; \backslash pm\; c)\; \backslash neq\; a\; \backslash div\; b\; \backslash pm\; a\; \backslash div\; c$$

The distributive laws are among the axioms for rings (like the ring of ) and fields (like the field of ). 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.

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Examples

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Real numbers In the following examples, the use of the distributive law on the set of real numbers $\backslash R$ 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.

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Matrices The distributive law is valid for matrix multiplication. More precisely, $$(A\; +\; B)\; \backslash cdot\; C\; =\; A\; \backslash cdot\; C\; +\; B\; \backslash cdot\; C$$ for all $l\; \backslash times\; m$-matrices $A,\; B$ and $m\; \backslash times\; n$-matrices $C,$ as well as $$A\; \backslash cdot\; (B\; +\; C)\; =\; A\; \backslash cdot\; B\; +\; A\; \backslash cdot\; C$$ for all $l\; \backslash times\; m$-matrices $A$ and $m\; \backslash times\; n$-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.

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Other examples

• Multiplication of , in contrast, is only left-distributive, not right-distributive.
• The cross product is left- and right-distributive over vector addition, though not commutative.
• For sets, the union is distributive over intersection, and intersection is distributive over union.
• Logical disjunction ("or") is distributive over logical conjunction ("and"), and vice versa.
• For (and for any totally ordered set), the maximum operation is distributive over the minimum operation, and vice versa: $$\backslash max(a,\; \backslash min(b,\; c))\; =\; \backslash min(\backslash max(a,\; b),\; \backslash max(a,\; c))\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\; \backslash min(a,\; \backslash max(b,\; c))\; =\; \backslash max(\backslash min(a,\; b),\; \backslash min(a,\; c)).$$
• For , the greatest common divisor is distributive over the least common multiple, and vice versa: $$\backslash gcd(a,\; \backslash operatorname\{lcm\}(b,\; c))\; =\; \backslash operatorname\{lcm\}(\backslash gcd(a,\; b),\; \backslash gcd(a,\; c))\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\; \backslash operatorname\{lcm\}(a,\; \backslash gcd(b,\; c))\; =\; \backslash gcd(\backslash operatorname\{lcm\}(a,\; b),\; \backslash operatorname\{lcm\}(a,\; c)).$$
• For real numbers, addition distributes over the maximum operation, and also over the minimum operation: $$a\; +\; \backslash max(b,\; c)\; =\; \backslash max(a\; +\; b,\; a\; +\; c)\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\; a\; +\; \backslash min(b,\; c)\; =\; \backslash min(a\; +\; b,\; a\; +\; c).$$
• For binomial multiplication, distribution is sometimes referred to as the FOIL MethodKim Steward (2011) Multiplying Polynomials from Virtual Math Lab at West Texas A&M University (First terms $a\; c,$ Outer $a\; d,$ Inner $b\; c,$ and Last $b\; d$) such as: $(a\; +\; b)\; \backslash cdot\; (c\; +\; d)\; =\; a\; c\; +\; a\; d\; +\; b\; c\; +\; b\; d.$
• In all semirings, including the , the , , and matrices, multiplication distributes over addition: $u\; (v\; +\; w)\; =\; u\; v\; +\; u\; w,\; (u\; +\; v)w\; =\; u\; w\; +\; v\; w.$
• In all algebras over a field, including the and other non-associative algebras, multiplication distributes over addition.

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Propositional logic

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Rule of replacement In standard truth-functional propositional logic, Elliott Mendelson (1964) Introduction to Mathematical Logic, page 21, D. Van Nostrand CompanyAlfred Tarski (1941) Introduction to Logic, page 52, Oxford University Press in logical proofs uses two valid rules of replacement to expand individual occurrences of certain logical connectives, within some Logical formula, into separate applications of those connectives across subformulas of the given formula. The rules are $$(P\; \backslash land\; (Q\; \backslash lor\; R))\; \backslash Leftrightarrow\; ((P\; \backslash land\; Q)\; \backslash lor\; (P\; \backslash land\; R))\; \backslash qquad\; \backslash text\{\; and\; \}\; \backslash qquad\; (P\; \backslash lor\; (Q\; \backslash land\; R))\; \backslash Leftrightarrow\; ((P\; \backslash lor\; Q)\; \backslash land\; (P\; \backslash lor\; R))$$ where "$\backslash Leftrightarrow$", also written $\backslash ,\backslash equiv,\backslash ,$ is a symbol representing "can be replaced in a proof with" or "is logically equivalent to".

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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

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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 $1/3\; +\; 1/3\; +\; 1/3\; =\; (1\; +\; 1\; +\; 1)\; /\; 3$ 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.

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In rings and other structures Distributivity is most commonly found in , notably the particular cases of rings and distributive lattices.

A semiring has two binary operations, commonly denoted $\backslash ,+\backslash ,$ and $\backslash ,*,$ and requires that $\backslash ,*\backslash ,$ must distribute over $\backslash ,+.$

A ring is a semiring with additive inverses.

A lattice is another kind of algebraic structure with two binary operations, $\backslash ,\backslash land\; \backslash text\{\; and\; \}\; \backslash lor.$ If either of these operations distributes over the other (say $\backslash ,\backslash land\backslash ,$ distributes over $\backslash ,\backslash lor$), then the reverse also holds ($\backslash ,\backslash lor\backslash ,$ distributes over $\backslash ,\backslash land\backslash ,$), and the lattice is called distributive. See also .

A 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 and near-fields instead of rings and . The operations are usually defined to be distributive on the right but not on the left.

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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 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 $\backslash ,=\backslash ,$ by either $\backslash ,\backslash leq\backslash ,$ or $\backslash ,\backslash geq.$ 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, if $(S,\; \backslash mu,\; \backslash nu)$ and $\backslash left(S^\{\backslash prime\},\; \backslash mu^\{\backslash prime\},\; \backslash nu^\{\backslash prime\}\backslash right)$ are monads on a category $C,$ a distributive law $S\; .\; S^\{\backslash prime\}\; \backslash to\; S^\{\backslash prime\}\; .\; S$ is a natural transformation $\backslash lambda\; :\; S\; .\; S^\{\backslash prime\}\; \backslash to\; S^\{\backslash prime\}\; .\; S$ such that $\backslash left(S^\{\backslash prime\},\; \backslash lambda\backslash right)$ is a lax map of monads $S\; \backslash to\; S$ and $(S,\; \backslash lambda)$ is a colax map of monads $S^\{\backslash prime\}\; \backslash to\; S^\{\backslash prime\}.$ This is exactly the data needed to define a monad structure on $S^\{\backslash prime\}\; .\; S$: the multiplication map is $S^\{\backslash prime\}\; \backslash mu\; .\; \backslash mu^\{\backslash prime\}\; S^2\; .\; S^\{\backslash prime\}\; \backslash lambda\; S$ and the unit map is $\backslash eta^\{\backslash prime\}\; S\; .\; \backslash eta.$ See: distributive law between monads.

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

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Antidistributivity The ubiquitous identity that relates inverses to the binary operation in any group, namely $(x\; y)^\{-1\}\; =\; y^\{-1\}\; x^\{-1\},$ 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).

(1997). 9783211829714, Springer. . ISBN 9783211829714

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 $a$ reverses the order of addition when multiplied to the right: $(x\; +\; y)\; a\; =\; y\; a\; +\; x\; a.$

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:

$$(a\; \backslash lor\; b)\; \backslash Rightarrow\; c\; \backslash equiv\; (a\; \backslash Rightarrow\; c)\; \backslash land\; (b\; \backslash Rightarrow\; c)$$ $$(a\; \backslash land\; b)\; \backslash Rightarrow\; c\; \backslash equiv\; (a\; \backslash Rightarrow\; c)\; \backslash lor\; (b\; \backslash Rightarrow\; c).$$

These two tautologies are a direct consequence of the duality in De Morgan's laws.

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Notes

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External links

• A demonstration of the Distributive Law for integer arithmetic (from cut-the-knot)

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