【正文】 mentation union and intersection are of primary interest to us. A Boolean algebra is a finite or infinite set of elements together with three operations— negation, addition, and multiplication— that correspond to the set operations of plementation, union, and intersection, respectively. Among the elements of a Boolean algebra are two distinguished elements: 0, corresponding to the empty set。 and 1, corresponding to the universal set. 雖然我們可以定義其他一些集合運算，但求補、并和交運算是我們最感興趣的三個集合運算。一個布爾代數(shù)就是一個有限集或無限集，以及建立在該有限集或無限集上的三種運算 —— 否定、加或乘，這三個運算分別對應于集合的求補、并和交運算。在布爾代數(shù)的元素中有兩個特殊的元素： 0，對應于空集； 1，對應于全集。 Boolean AlgebraFor any given element of a Boolean algebra, there is a unique plement a39。 with the property that a+a39。=1 and aa39。=0. Boolean addition and multiplication are associative and mutative, as are ordinary addition and multiplication, but otherwise have somewhat different properties. The principal properties are given in Table 32, where a, b, and c are any elements of a Boolean algebra.對于一個布爾代數(shù)中任意給定元素 a，都有一個唯一的補 a?，它滿足 a+a?=1和 aa?=0。布爾加和布爾乘與普通的加和乘一樣，滿足結合律和交換律，但除此之外含有一些不太相同的特性。其主要特性由表 32給出，其中 a, b和 c是一個布爾代數(shù)中的任意元素。 Boolean AlgebraTable 32Distributivity 分配律 a(b+c)=ab+aca+(bc)=(a+b)(a+c)Idempotency 同一律 a+a=aaa=aAbsorption laws 吸收律 a+ab=aa(a+b)=aDeMan’s laws德 ?摩根定理 (a+b)39。=a39。b39。(ab)39。=a39。+b39。 Boolean AlgebraSince a finite set of n elements has exactly 2n subsets, and it can be shown that the finite Boolean algebras are precisely the finite set algebras, each finite Boolean algebra consists of exactly 2n elements for some integer n. For example, the set algebra for the set T defined above corresponds to a Boolean algebra of 32 elements. 由于 n個元素的有限集有且只有 2n個子集，而且很顯然有限布爾代數(shù)一定是有限集合代數(shù)，所以對某個整數(shù) n而言，每個有限布爾代數(shù)也有且只有 2n個元素。例如，上文定義的集合 T的集合代數(shù)就對應一個有 32個元素的布爾代數(shù)。 Boolean AlgebraWhile it is possible to use a different symbol to denote each element of a Boolean algebra, it is often more useful to represent the 2n elements of a finite Boolean algebra by binary vectors having n ponents. With such a representation the operations of the Boolean algebra are acplished ponentwise by considering each ponent as an independent twoelement Boolean algebra. This corresponds to representing subsets of a finite set by binary vectors. 雖然我們可以用不同的符號來表示布爾代數(shù)中的每一個元素，但最常用的方法是用一個有 n個分量的二進制向量來表示一個有限布爾代數(shù)的 2n個元素。用這樣一種表示方法，布爾代數(shù)的所有運算都以分量的形式完成，而每一個分量被認為是一個獨立的二值布爾代數(shù)。這種做法對應于用二進制向量來表示一個有限集的子集。 Boolean AlgebraFor example, since the set T has five elements, we may represent its subsets by fiveponent binary vectors, each ponent denoting an element of the set T. A numeral l in the ith ponent of the vector denotes the inclusion of the ith element of that particular subset。 a 0 denotes its exclusion. Thus, the subset S={a, b, c} has the binary vector representation {1,1,1,0,0}. The set operations bee Boolean operations on the ponents of the vectors. This representation of sets, and the correspondence to Boolean or logical operations, is very useful in information retrieval. Because of it, sets of document and query characteristics may be easily and rapidly matched.例如，由于集合 T有 5個元素，所以我們可以用 5個分量的二進制向量表示它的子集，其中每一個分量表示集合 T的一個元素。向量中的第 i個分量為數(shù)字 1表示集合 T的第 i個元素在某一特定子集中，用數(shù)字 0表示不在某一特定子集中。于是，子集S={a,b,c}可用二進制向量表示為 {1， 1， 1， 0， 0}。集合運算變成了向量分量上的布爾運算。集合的這種表示方法以及相應的布爾或邏輯運算，對于信息檢索是非常有用的。由于這一原因，文件的集合和查詢特性可以很容易而迅速地得到匹配。 Boolean Algebr