Distributive law - Distributive law

It is usually a law relating to two algorithms, and is also called the distributive law. For example, in the case of multiplication × and addition +, 2×(3+4) and (2×3)+(2×4) are equal, which is 14, and so the law states that for two algorithms and elements a, b, and c, a゜(bc) and (a゜b)(a゜c) are equal. If an algorithm satisfies the commutative law, then the property that (bc)゜a and (b゜a)(c゜a) are equal can be derived, but in general it is not always the case that ゜ satisfies the commutative law, so when it is simply called the distributive law, it usually means that both of these hold true.

Examples of cases where the distributive law holds include the multiplication × and addition + of numbers, the multiplication and addition of matrices, the sum and intersection of sets, and the intersection and sum. It also holds for propositions in the sense that A and (B or C) and (A and B) or (A and C) are equivalent. For other cases, for example, when writing powers as a ^b , (a×b)c and (ac)×(bc) are equal, but there are also cases where only one form of the distributive law holds, such as 2(3×2)=64 and (23)×(22)=32.

[Namba Kanji]

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

通常二つの算法に関する法則で、分配律ともよばれている。たとえば数の乗法×と加法＋について2×(3+4)と(2×3)+(2×4)が等しく14であるというように、二つの算法゜とおよび元a、b、cについてa゜(bc)と(a゜b)(a゜c)が等しいという法則。もし算法゜が交換法則を満足すれば(bc)゜aと(b゜a)(c゜a)が等しいという性質が導かれるが、一般には゜が交換法則を満足するとは限らないので、通常、単に分配法則とよべば、この両方が成立することをいう。

　分配法則が成立する例としては、数の乗法×と加法＋、行列の積と加法、集合の和と共通部分、共通部分と和などがある。また、たとえば命題についてもA and(B or C)と(A and B) or (A and C)が同値という意味で成立する。その他、たとえばべきについてa^bをabと記すと(a×b)cと(ac)×(bc)は等しいが2(3×2)=64,(23)×(22)=32のように一方の形の分配法則しか成立しない場合もある。

［難波完爾］

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