中英对照布尔代数.htm

软考英语精典短文每日一练汇总 软考英语精典短文每日一练汇总

      T．Since T has five elements，there are 25 subsets of T，for we may choose 
      any individual element to be included or omitted from a subset[2]．Note 
      that these 32 subsets include T itself and the empty set，which contains no 
      elements at all[3]．If T contains all elements of concern，it is called the 
      universal set．Given a subset of T，such as S，we may define the complement 
      of S with respect to a universal set T to consist of precisely those 
      elements of T which are not included in the given subset[4]．Thus，S as 
      above defined has as its complement（with respect to T） ＝｛d，e｝．The union of 
      any two sets（subsets of a given set）consists of those elements that are in 
      one or the other or in both given sets[5]；the intersection of two sets 
      consists of those elements that are in both given sets．We use the symbol ∪ 
      to denote the union，and ∩ to denote the intersection of two sets．For 
      example，if B =｛b，d，e｝，then B∪S＝｛a，b，c，d，e｝，and B∩S＝｛b｝．<BR><BR>While other 
      set operations may be defined[6]，the operations of 
      complementation，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一that correspond to 
      the set operations of complementation，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．For any given element of a Boolean 
      algebra，there is a unique complement with the property that a＋ = 1 and a 
      ＝0．Boolean addition and multiplication are associative and commutative 
      [7]，as are ordinary addition and multiplication，but otherwise have 
      somewhat different properties．The principal properties are given in Table 
      3-1，where a，b，and c are any elements of a Boolean algebra．<BR><BR>Table 
      3-1[8]<BR><BR><BR><BR>Since 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[9]，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．Tables 3-2 
      define the Boolean operations for Boolean algebras of two 
      elements，respectively．<BR><BR>Table 3-2 Two elements<BR><BR>While 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 components．With such a 
      representation the operations of the Boolean algebra are accomplished 
      componentwise [10] by considering each component as an independent 
      two-element Boolean algebra[11]．This corresponds to representing subsets 
      of a finite set by binary vectors．For example，since the set T has five 
      elements，we may represent its subsets by five-component binary 
      vectors，each component denoting an element of the set T．A numeral 1 in the 
      ith component of the vector denotes the inclusion of the ith element of 
      that particular subset；a 0 denotes its exclusion．<BR><BR>Thus，the subset S 
      =｛a，b，c｝has the binary vector representation｛1，1，1，0，0｝．The set operations 
      become Boolean operations on the components 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．<BR><BR>布尔代数的概念最初是由英国数学家George 
      Boole于1847年提出来的。从那时起，由Boole创始的这一理论在代数学家和逻辑学家们的努力下得到了很大程度的发展和升华。由于布尔代数、集合代数、逻辑学和二进制算术之间有内在的联系，所以布尔代数的理论在电子数字计算机的开发研制中是很关键的。<BR><BR>布尔代数是集合代数的概念最自然的发展。设S= 
      {a，b，c}和T=｛a，b，c，d，e｝分别为两个含有3个或5个元素的集合。由于S中的每个元素（a，b，c）都属于T，所以我们说S是T的一个子集。由于T有5个元素，因而T共有25个子集，这是因为可以选择任何一个元素使其包含于某个子集或从该子集中删除。应该注意到这32个子集中包括T本身和空集，空集就是不含任何元素的集合。如果了包含了所讨论的所有元素，则称之为全集。给定T的一个子集，例如S，可以定义一个关于全集T的S的补集，其中正好包含那些不在子集S中而在T中的元素。于是，如上定义的集合S就有它的一个补集（相对于集合T）， 
      =｛d，e｝。任何两个集合（都是给定集合的子集）的并集包含了出现于该两个集合中某一个集合或同时出现于该两个集合中的所有元素；两个集合的交集包含了同时出现于两个集合中的元素。我们用符号“∪”来表示两个集合的“并（运算）”；用“∪”来表示两个集合的“交（运算）”。例如，如果B= 
      {b，d，e}，那么，B∪S=｛a，b，c，d，e｝，B∩S=｛b｝。<BR><BR>虽然可以定义其他一些集合运算，但求补、并和交是人们最感兴趣的3个集合运算。一个布尔代数就是一个有限集或无限集以及建立在该集上的3个运算——否定、加法和乘法。这3个运算分别对应于集合的求补、并和交运算。在布尔代数的元素中有两个突出的元素：0，对应于空集；1，对应于全集。对于一个布尔代数中任意给定元素a，都有一个惟一的补 
      ，它满足a + = 1和a = 
      0。布尔加法和布尔乘法与普通加法和乘法一样，满足结合律和交换律，但除此之外还有一些不太相同的性质。主要的性质由表3-1给出，其中a，b和c是布尔代数中的任意元素。<BR><BR>由于n个元素的有限集恰有2n个子集，而且可以看出有限布尔代数一定是有限集合代数，所以对某个整数n而言，每个有限布尔代数也恰有2n个元素。例如，上面定义的集合T的集合代数就对应于一个有32个元素的布尔代数。表3-2定义了具有两个元素的布尔代数的布尔运算。<BR><BR>虽然可以用不同的符号来表示布尔代数中的每个元素，但最常用的方法是用一个有n个分量的二元向量来表示一个有限布尔代数的2n个元素。用这样一种表示方法，布尔代数的所有运算都以分量的方式完成，而每一个分量都作为一个独立的二元素布尔代数，这种做法对应于用二元向量来表示一个有限集的子集。例如，由于集合T有5个元素，所以可以用5个分量的二元向量表示它的子集，其中每个分量表示集合T的一个元素。向量中的第i个分量中的数字1，表示第i个元素包括在该特定的子集中，数字0表示不包括。于是，子集S= 
      {a，b，c}可用二元向量表示为｛1，1，1，0，0｝。集合运算变成了向量分量上的布尔运算。集合的这种表示方法及相应的布尔或逻辑运算，对于信息检索非常有用。由于这一原因，文件的集合与查询特征可以很容易且迅速地得到匹配<BR><BR></DIV></TD></TR></TBODY></TABLE>
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