1、2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,1,基础部分:逻辑(Logic)集合(Sets)算法(Algorithms)数论(Number Theory),2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,2,1.1 逻辑Logic,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,3,逻辑学：研究推理的一门学科 数理逻辑：用数学方法研究推理的一门数学学科,- 一套符号体系 + 一组规则,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,
2、4,数理逻辑的内容：古典数理逻辑：命题逻辑、谓词逻辑现代数理逻辑：公理化集合论、递归论、模型论、证明论,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,5,命题 Proposition:一个有确定真或假意义的语句.,命题逻辑 Proposition Logic,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,6,EXAMPLE 1,All the following statements are propositions.1. Washington, D.C., is the capital of the Unit
3、ed States of America.2. Toronto is the capital of Canada.3. 1+1=2.4. 2+2=3.,Propositions 1 and 3 are true, whereas 2 and 4 are false.,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,7,EXAMPLE 2,Consider the following sentences.1. What time is it?2. Read this carefully.3. x+1 =2.4. x+y = z.,Sentences 1 a
4、nd 2 are not propositions because they are not statements. Sentences 3 and 4 are not propositions because they are neither true nor false, since the variables in these sentences have not been assigned values. Various ways to form propositions from sentences of this type will be discussed in Section
5、1.3.,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,8,命题的语句形式陈述句 非命题语句：疑问句命令句感态句非命题陈述句：悖论语句,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,9,命题的符号表示：大小写英文字母：P、Q、R、p 、q 、r、命题真值（Truth Values）的表示：真：T、1假：F、0,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,10,命题语句真值确定的几点说明：1、时间性2、区域性3、标准性命题真值间的关系表示：真值表（Truth Table
6、),2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,11,DEFINITION 1.,Let p be a proposition. The statement“It is not the case that p.“ is another proposition, called the negation of p. The negation of p is denoted by p. The proposition p is read “not p.“,p的否定,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,12
7、,EXAMPLE 3,Find the negation of the proposition“Today is Friday“and express this in simple English.,The negation is“It is not the case that today is Friday.“This negation can be more simply expressed byToday is not Friday.“,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,13,Table 1,2/8/2019 7:54 AM,Dere
8、n Chen, ZheJiang Univ.,14,DEFINITION 2.,Let p and q be propositions. The proposition “p and q,“ denoted by pq, is the proposition that is true when both p and q are true and is false otherwise. The proposition pq is called the conjunction of p and q. The truth table for pq is shown in Table 2.,p和q的合
9、取,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,15,Table 2,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,16,EXAMPLE 4,Find the conjunction of the propositions p and q where p is the proposition “Today is Friday“ and q is the proposition “It is raining today.“,Solution: The conjunction of these propositi
10、ons, pq, is the proposition “Today is Friday and it is raining today.“ This proposition is true on rainy Fridays and is false on any day that is not a Friday and on Fridays when it does not rain.,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,17,DEFINITION 3.,Let p and q be propositions.The proposition
11、 “p or q,“ denoted by pq, is the proposition that is false when p and q are both false and true otherwise. The proposition pq is called the disjunction of p and q. The truth table for pq is shown in Table 3.,p和q的析取,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,18,Table 3,2/8/2019 7:54 AM,Deren Chen, Z
12、heJiang Univ.,19,EXAMPLE 5,What is the disjunction of the propositions p and q where p and q are the same propositions as in Example 4?,Solution: The disjunction ofp and q, pq, is the proposition“Today is Friday or it is raining today.“This proposition is true on any day that is either a Friday or a
13、 rainy day (including rainy Fridays). It is only false on days that are not Fridays when it also does not rain.,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,20,DEFINITION 4.,Let p and q be propositions. The exclusive or of p and q, denoted by p q, is the proposition that is true when exactly one of p
14、 and q is true and is false otherwise. The truth table for the exclusive or of two propositions is displayed in Table 4.,p和q的对称差,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,21,Table 4,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,22,DEFINITION 5.,Let p and q be propositions.The implication pq is the p
15、roposition that is false when p is true and q is false and true otherwise. In this implication p is called the hypothesis (or antecedent or premise) and q is called the conclusion (or consequence).,如果p,则q,单条件, 蕴涵 P:前提 Q:结论,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,23,Table 5,2/8/2019 7:54 AM,Deren
16、 Chen, ZheJiang Univ.,24,EXAMPLE 6,What is the value of the variable x after the statementif 2+2=4 then x := x+ 1if x = 0 before this statement is encountered? (The symbol: = stands for assignment. The statement x: = x + 1 means the assignment of the value of x + 1 to x.),Solution: Since 2 + 2 = 4 i
17、s true, the assignment statement x: = x + 1 is executed. Hence, x has the value 0+1=1 after this statement is encountered.,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,25,implication pq The converse of pq : qpThe contrapositive of pq :q p,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,26,EXAMPLE 7,Find
18、the converse and the contrapositive of the implication“If today is Thursday, then I have a test today.“,Solution: The converse is“If I have a test today, then today is Thursday.“ And the contrapositive of this implication is“If I do not have a test today, then today is not Thursday.“,2/8/2019 7:54 A
19、M,Deren Chen, ZheJiang Univ.,27,DEFINITION 6.,Let p and q be propositions, The biconditional p q is the proposition that is true when p and q have the same truth values and is false otherwise. The truth table for p q is shown in Table 6.,P当且仅当q,双条件，等价,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,28,T
20、able 6,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,29,EXAMPLE 8,How can the following English sentence be translated into a logical expression?“You can access the Internet from campus only if you are a computer science major or you are not a freshman,“,Solution:a (c f ).,2/8/2019 7:54 AM,Deren Chen,
21、 ZheJiang Univ.,30,EXAMPLE 9,How can the following English sentence be translated into a logical expression?“You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old.“,Solution:(r s) q.,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,31,EXAMPLE 10,说离散数学是枯燥无味
22、的或毫无价值的，那是不对的。P：离散数学是有味道的； Q：离散数学是有价值的；,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,32,APPLICATION 1,Web Page Searching. Most Web search engines support Boolean searching techniques, which usually can help find Web pages about particular subjects. For instance, using Boolean searching to find Web pa
23、ges about universities in New Mexico, we can look for pages matching NEW AND MEXICO AND UNIVERSITIES. The results of this search will include those pages that contain the three words NEW, MEXICO, and UNIVERSITIES.,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,33,APPLICATION 2,A bit string is a sequenc
24、e of zero or more bits. The length of this string is the number of bits in the string.,The bitwise of two strings of the same length:Bitwise ORBitwise ANDBitwise XOR,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,34,Table 7,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,35,EXAMPLE 12,Find the bitwise OR,
25、bitwise AND, and bitwise XOR of the bit strings 01 1011 0110 and 11 0001 1101. (Here, and throughout this book, bit strings will be split into blocks of four bits to make them easier to read.),Solution:The bitwise OR, bitwise AND, and bitwise XOR of these strings are obtained by taking the OR, AND,
26、and XOR of the corresponding bits, respectively. This gives us 01 1011 0110 11 0001 1101 11 1011 1111 bitwise OR 01 0001 0100 bitwise AND 10 1010 1011 bitwise XOR,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,36,P、Q、R 称为原子命题（Atomic Proposition）。 原子命题或加上逻辑联结词组成的表达式成为复合命题（Compositional Proposition）。 从命题
27、常量 到 命题变量(Propositional Variable),命题公式： 1、原子命题是命题公式； 2、设P是命题公式，则P也是命题公式； 3、设P、Q是命题公式，则（P Q）、（P Q）、（P Q）、（P Q）也是命题公式； 4、有限次地使用1、2、3所得到的也是命题公式。Proposition Formulas, Well-Formed Formulas(wff),2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,37,命题公式的运算规则：逻辑联接词的优先级： 、 、 、 、命题公式的表达式的运算规律：同代数表达式命题公式的运算方法：所有公式中的
28、命题变量用指定命题（真值）代入（或指派），得到一个公式对应的真值。,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,38,性质1： 如果一个命题公式有N个互异的命题变量，则命题公式对应的真值有2的N次幂种可能分布。,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,39,永真命题公式（Tautology） 公式中的命题变量无论怎样代入，公式对应的真值恒为T。 永假命题公式（Contradiction) 公式中的命题变量无论怎样代入，公式对应的真值恒为F。可满足命题公式（Satisfaction） 公式中的命题变量无论
29、怎样代入，公式对应的真值总有一种情况为T。 一般命题公式（Contingency） 既不是永真公式也不是永假公式。,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,40,性质2：（1）设P是永真命题公式，则P的否定公式是永假命题公式；（2）设P是永假命题公式，则P的否定公式是永真命题公式；（3）设P、Q是永真命题公式，则（P Q）、（P Q）、（P Q）、（P Q）也是永真命题公式,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,41,小 结1、命题的概念：定义、逻辑值、 符号化表示 2、从简单命题到复合命题：逻辑联接词：运算方法、运算优先级 3、从命题常量到命题变量，从复合命题到命题公式：命题公式的真值描述：真值表 4、命题公式的分类：永真公式、永假公式、可满足公式 、一般公式,2/8/2019 7:54 AM,Deren Chen, ZheJiang Univ.,42,进一步的思考：1、从二值逻辑到多值逻辑2、从确定值到模糊值模糊逻辑（Fuzzy Logic),练习题：1、2、7、9、25（d）、30（b）,
