Дистрибутивност

Дистрибутивност је алгебарска особина понашања оператора сабирања и множења над алгебарском структуром $(K,\oplus ,\cdot )$. Конкретно када се производ два елемента скупа K може представити ако збир производа једног од њих са још два елемента који у збиру дају другог, каже се да закон дистрибуције важи за дату алгебарску структуру. Множење може бити лево и десно те отуда два различита услова:

$a\cdot (b\oplus c)=a\cdot b\oplus a\cdot c$ (дистрибутивност слева)

$(b\oplus c)\cdot a=b\cdot a\oplus c\cdot a$ (дистрибутивност здесна)

Ако су задовољени само први или само други услов, каже се да се „лево односно десно множење лепо понаша према сабирању“. Уколико су оба испуњена, каже се да се „операција множења лепо понаша према сабирању“ тј. да је дистрибутивна.

Дефиниција

Given a set $S$ and two binary operators $\,*\,$ and $\,+\,$ on $S,$

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

$$x*(y+z)=(x*y)+(x*z),$$

the operation $\,*\,$ is right-distributive over $\,+\,$ if, given any elements $x,y,{\text{ and }}z$ of $S,$

$$(y+z)*x=(y*x)+(z*x),$$

and the operation $\,*\,$ is distributive over $\,+\,$ if it is left- and right-distributive.

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

Значење

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

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

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

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

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 distributivity.

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 integers) and fields (like the field of rational numbers). 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.

Примери

Реални бројеви

In the following examples, the use of the distributive law on the set of real numbers $\mathbb {R}$ 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.

First example (mental and written multiplication)

During mental arithmetic, distributivity is often used unconsciously:

$$6\cdot 16=6\cdot (10+6)=6\cdot 10+6\cdot 6=60+36=96$$

Thus, to calculate $6\cdot 16$ in one's head, one first multiplies $6\cdot 10$ and $6\cdot 6$ and add the intermediate results. Written multiplication is also based on the distributive law.

Second example (with variables)

$$3a^{2}b\cdot (4a-5b)=3a^{2}b\cdot 4a-3a^{2}b\cdot 5b=12a^{3}b-15a^{2}b^{2}$$

Third example (with two sums)

$${\begin{aligned}(a+b)\cdot (a-b)&=a\cdot (a-b)+b\cdot (a-b)=a^{2}-ab+ba-b^{2}=a^{2}-b^{2}\\&=(a+b)\cdot a-(a+b)\cdot b=a^{2}+ba-ab-b^{2}=a^{2}-b^{2}\\\end{aligned}}$$

Here the distributive law was applied twice, and it does not matter which bracket is first multiplied out.

Fourth example

Here the distributive law is applied the other way around compared to the previous examples. Consider

$$12a^{3}b^{2}-30a^{4}bc+18a^{2}b^{3}c^{2}\,.$$

Since the factor $6a^{2}b$ occurs in all summands, it can be factored out. That is, due to the distributive law one obtains

$$12a^{3}b^{2}-30a^{4}bc+18a^{2}b^{3}c^{2}=6a^{2}b\left(2ab-5a^{2}c+3b^{2}c^{2}\right).$$

Матрице

The distributive law is valid for matrix multiplication. 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.

Други примери

• Multiplication of ordinal numbers, 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 of sets is distributive over intersection, and intersection is distributive over union.
• Logical disjunction ("or") is distributive over logical conjunction ("and"), and vice versa.
• For real numbers (and for any totally ordered set), the maximum operation is distributive over the minimum operation, and vice versa:
  $$\max(a,\min(b,c))=\min(\max(a,b),\max(a,c))\quad {\text{ and }}\quad \min(a,\max(b,c))=\max(\min(a,b),\min(a,c)).$$
• For integers, the greatest common divisor is distributive over the least common multiple, and vice versa:
  $$\gcd(a,\operatorname {lcm} (b,c))=\operatorname {lcm} (\gcd(a,b),\gcd(a,c))\quad {\text{ and }}\quad \operatorname {lcm} (a,\gcd(b,c))=\gcd(\operatorname {lcm} (a,b),\operatorname {lcm} (a,c)).$$
• For real numbers, addition distributes over the maximum operation, and also over the minimum operation:
  $$a+\max(b,c)=\max(a+b,a+c)\quad {\text{ and }}\quad a+\min(b,c)=\min(a+b,a+c).$$
• For binomial multiplication, distribution is sometimes referred to as the FOIL Method (First terms $ac,$ Outer $ad,$ Inner $bc,$ and Last $bd$) such as: $(a+b)\cdot (c+d)=ac+ad+bc+bd.$
• In all semirings, including the complex numbers, the quaternions, polynomials, and matrices, multiplication distributes over addition: $u(v+w)=uv+uw,(u+v)w=uw+vw.$
• In all algebras over a field, including the octonions and other non-associative algebras, multiplication distributes over addition.

Пропозициона логика

Rule of replacement

In standard truth-functional propositional logic, distribution in logical proofs uses two valid rules of replacement to expand individual occurrences of certain logical connectives, within some formula, into separate applications of those connectives across subformulas of the given formula. The rules are

$$(P\land (Q\lor R))\Leftrightarrow ((P\land Q)\lor (P\land R))\qquad {\text{ and }}\qquad (P\lor (Q\land R))\Leftrightarrow ((P\lor Q)\land (P\lor R))$$

where "$\Leftrightarrow$", also written $\,\equiv ,\,$ is a metalogical symbol representing "can be replaced in a proof with" or "is logically equivalent to".

Truth functional connectives

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 tautologies.

$${\begin{alignedat}{13}&(P&&\;\land &&(Q\lor R))&&\;\Leftrightarrow \;&&((P\land Q)&&\;\lor (P\land R))&&\quad {\text{ Distribution of }}&&{\text{ conjunction }}&&{\text{ over }}&&{\text{ disjunction }}\\&(P&&\;\lor &&(Q\land R))&&\;\Leftrightarrow \;&&((P\lor Q)&&\;\land (P\lor R))&&\quad {\text{ Distribution of }}&&{\text{ disjunction }}&&{\text{ over }}&&{\text{ conjunction }}\\&(P&&\;\land &&(Q\land R))&&\;\Leftrightarrow \;&&((P\land Q)&&\;\land (P\land R))&&\quad {\text{ Distribution of }}&&{\text{ conjunction }}&&{\text{ over }}&&{\text{ conjunction }}\\&(P&&\;\lor &&(Q\lor R))&&\;\Leftrightarrow \;&&((P\lor Q)&&\;\lor (P\lor R))&&\quad {\text{ Distribution of }}&&{\text{ disjunction }}&&{\text{ over }}&&{\text{ disjunction }}\\&(P&&\to &&(Q\to R))&&\;\Leftrightarrow \;&&((P\to Q)&&\to (P\to R))&&\quad {\text{ Distribution of }}&&{\text{ implication }}&&{\text{ }}&&{\text{ }}\\&(P&&\to &&(Q\leftrightarrow R))&&\;\Leftrightarrow \;&&((P\to Q)&&\leftrightarrow (P\to R))&&\quad {\text{ Distribution of }}&&{\text{ implication }}&&{\text{ over }}&&{\text{ equivalence }}\\&(P&&\to &&(Q\land R))&&\;\Leftrightarrow \;&&((P\to Q)&&\;\land (P\to R))&&\quad {\text{ Distribution of }}&&{\text{ implication }}&&{\text{ over }}&&{\text{ conjunction }}\\&(P&&\;\lor &&(Q\leftrightarrow R))&&\;\Leftrightarrow \;&&((P\lor Q)&&\leftrightarrow (P\lor R))&&\quad {\text{ Distribution of }}&&{\text{ disjunction }}&&{\text{ over }}&&{\text{ equivalence }}\\\end{alignedat}}$$

Двострука дистрибуција

$${\begin{alignedat}{13}&((P\land Q)&&\;\lor (R\land S))&&\;\Leftrightarrow \;&&(((P\lor R)\land (P\lor S))&&\;\land ((Q\lor R)\land (Q\lor S)))&&\\&((P\lor Q)&&\;\land (R\lor S))&&\;\Leftrightarrow \;&&(((P\land R)\lor (P\land S))&&\;\lor ((Q\land R)\lor (Q\land S)))&&\\\end{alignedat}}$$

Види још

• Асоцијативност
• Антикомутативност
• Комутативност

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