1、PART COMPUTER SOFTWARECHAPTER 3 MATHEMATIC FUNDAMENTALS 3.1 BOOLEAN ALGEBRA,chapter 3-3.1,3.1 BOOLEAN ALGEBRA,*Main point,Main point: Development and function of boolean algebra Concept of set algebra Set operations Boolean operations,计算机英语（第三版）,*?1,计算机英语（第三版）,The concept of a Boolean algebra was fi
2、rst proposed by the English mathematician George Boole in 1847. Since that time, Booles original conception has been extensively developed and refined by algebraists and logicians1. The relationships among Boolean algebra, set algebra, logic, and binary arithmetic have given Boolean algebras a centr
3、al role in the development of electronic digital computers.,?1 What is the fundamentals of electronic digital computers ?,计算机英语（第三版）,The most intuitive development of boolean algebras arises from the concept of a set algebra. Let S =a,b,c and T=a,b,c,d,e be two sets consisting of three and five elem
4、ents, respectively. We say that S is a subset of T, since every element of S (namely, a, b, and c) belongs to 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 subset2.Note that these 32 subsets include T itself and the
5、 empty set, which contains no elements at all3.,原因状语从句,原因状语从句,计算机英语（第三版）,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
6、 included in the given subset4. Thus, S as above defined has as its complement (with respect to T) S=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 sets5 ; the intersection of two sets consists of those elements that ar
7、e in both given sets.,宾语补足语,定语从句,定语从句,定语从句,We use the symbol to denote the union, and to denote the intersection of two sets. For example, if B=b,d,e, then BS=a,b,c,d,e, and BS=b.,*?2,计算机英语（第三版）,?2 Explain the operations of union andintersection,While other set operations may be defined6, the operat
8、ions of complementation, union, and intersection are of primary interest to us .,让步状语从句,A Boolean algebra is a finite or infinite set of elements together with three operationsnegation, addition, and multiplicationthat correspond to the set operations of complementation, union, and intersection, res
9、pectively.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+a=1 and aa=0.,同位语,定语从句,条件状语,计算机英语（第三版）,Boolean
10、 addition and multiplication are associative and commutative7, 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.,*?3,?3 List the principal properties
11、 of Booleanalgebra,计算机英语（第三版）,Table 3-1,Table3-1 8 Distributivity: a(b+c)=ab+ac a+(bc)=(a+b)(a+c)Idempotency: a+a=aaa=aAbsorption laws: a+ab=aa(a+b)=aDe Morgans laws: (a+b)= a b(ab)=a + b,Since a finite set of n elements has exactly 2n subsets,and it can be shown that the finite Boolean algebras are
12、 precisely the finite set algebras9, each finite Boolean algebra consists of exactly 2n elements for some integer n.,原因状语从句,主语从句,计算机英语（第三版）,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 a
13、lgebras of two elements, respectively.,Table 3-2 Two elements,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.,让步状语从句,Table 3-2,With such
14、 a representation the operations of the Boolean algebra are accomplished componentwise10 by considering each component as an independent twoelement Boolean algebra11 .This corresponds to representing subsets of a finite set by binary vectors. For example, since the set T has five elements, we may re
15、present 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.,计算机英语（第三版）,原因状语从句,Thus, the subset S=a,b,c has the binar
16、y 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 m
17、ay be easily and rapidly matched.,*?4,?4 What is very useful in information retrieval ?,计算机英语（第三版）,NOTES1,1refine原意为“精炼、提纯、精纯”,此处翻译为“精炼”;该句中使用了完成时被动式,从而强调从这一概念产生时起,就一直被发展(developed)与精炼(refined)。 2该句中since引导一个原因状语从句;for也引导一个原因状语从句,补充说明子集总数为什么是这么多。 3which引导一个非限制性定语从句,说明空集的含义。,NOTES,计算机英语（第三版）,4句中Given
18、为分词引导的条件状语;不定式to consist of 对complement of S起补充说明的作用。which are not.引导一个限制性定语从句,修饰elements。 5in one or the other or in both given sets该并列介词短语说明并集的元素来源于两个集合中的一个(in one)或另一个(the other)或同时出现于两个之中(in both given sets)。 6句中operation在布尔代数中译为“运算”,而不译为“操作”；while引导一个让步状语从句。,计算机英语（第三版）,NOTES2,7associative结合的; c
19、ommutative交换的。 8表31中distributivity为分配律;idempotency为同一律;Absorption laws为吸收律;Demorgans lows为德摩根定律。 9“有限布尔代数”和“有限集合代数”中的“有限”是指其中集合的元素个数有限,运算也建立在该有限集之上。 10componentwise:wise后缀,用以构成副词,表示“逐一”、“状态”、“样子”等。,计算机英语（第三版）,NOTES3,11by considering.此处by表示方法、手段等,作状语。twoelement Boolean algebra意指只含0和1的布尔代数。,计算机英语（第三版）
20、,NOTES4,KEYWORDS,Boolean algebra 布尔代数 union 并set algebra 集合代数 Intersection 交set 集（合） complement 补empty set 空集 finite set 有限集universal set 全集 infinite set 无限集subset 子集 exclusion “异”运算,KEYWORDS,计算机英语（第三版）,(1) The most intuitive development of Boolean algebras arises from the concept of a .(2) Let S=a,
21、b,c and T=a,b,c,d,e be two sets consistingof three and five elements respectively, we say that .(3) As above condition, we say S=d,e .(4) A set that contains all elements of concern is called .(5) In a Boolean algebra 0 represents .,EXERCISES1.1,c,EXERCISES,计算机英语（第三版）,1. Fill in the blanks.,i,a,j,d,
22、EXERCISES1.2,计算机英语（第三版）,For element “a” of a Boolean algebra, a+a= and aa= . (7) A finite set of n elements has exactly subsets. (8) We often represent the 2n elements of a finiteBoolean algebra by having n components. (9)Also as the condition of question (2), the corresponding binary vector of the
23、subset S=a,b,c is represented with . (10) The binary vector representation of sets is very useful in .,a,b,d,f,h,b,e,g,f,answer,计算机英语（第三版）,a. the complement of Sb. binary vectorsc. the empty setd. set algebrae. 1,1,1,0,0f. 1, 0g. information retrievalh. 2ni. the universal setj. S is a subset of T,AN
24、SWER,计算机英语（第三版）,2. Multiple choices.,2. Multiple choices.(1) If set T=a,b,c,d, there are subset of T.a. 32 b. 25c. 24 d. 14+T itself+the empty set(2) In Boolean algebra primary operations that weare interested are: .a. complement b. negationc. union d. intersection,c,d,a,b,c,d,计算机英语（第三版）,(3) Boolean
25、 algebra became a central role in thedevelopment of electronic digital computers bymeans of .a. logic b. set algebrac. binary arithmetic d. decimal arithmetic,a,b,c,计算机英语（第三版）,3. Find the corresponding items with drawing lines. De Morgans laws a. addition intersection b. negation Idempotency c. (a+b)= a b union d. a+a=a Distributivity e. multiplication complement f. a+ab=a Absorption laws g. a(b+c)=ab+ac ,3. Find the,c,e,d,a,g,b,f,
