111 命题逻辑

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1、,单击此处编辑母版标题样式,单击此处编辑母版文本样式,2/22/2009 8:44 PM,Deren Chen,ZheJiang Univ.,#,Logic,命题逻辑,11/30/2024 3:40 PM,Deren Chen,ZheJiang Univ.,1,基础部分,:,逻辑,(Logic),集合,(Sets),算法,(Algorithms),数论,(Number Theory),11/30/2024 3:40 PM,Deren Chen,ZheJiang Univ.,2,1.1.1,命题逻辑,Proposition Logic,11/30/2024 3:40 PM,Deren Chen,

2、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,11/30/2024 3:40 PM,Deren Chen,ZheJiang Univ.,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,11/30/2024 3:40 PM,Deren Chen,ZheJiang Univ.,5,Proposition:,一个有确定真或假意义的语句,.,命题逻辑,Proposition Logic,11/30/2024 3:40 PM,Deren Chen

3、,ZheJiang Univ.,6,EXAMPLE,1,All the following statements are propositions.,1.Washington,D.C.,is the capital of the United 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.,11/30/2024 3:40 PM,Deren Chen,ZheJiang Univ.,7,E

4、XAMPLE 2,Consider the following sentences.,1.What time is it?,2.Read this carefully.,3.x+1=2.,4.x+y=z.,Sentences 1 and 2 are not propositions because they are not statements.Sentences 3 and 4 are not propositions because they are neither tree nor false,since the variables in these sentences have not

5、 been assigned values.Various ways to form propositions from sentences of this type will be discussed in Section 1.3.,11/30/2024 3:40 PM,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,11/30/2024 3:40 PM,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

6、真值（,Truth Values,）的表示：,真：,T,、,1,假：,F,、,0,11/30/2024 3:40 PM,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),11/30/2024 3:40 PM,Deren Chen,ZheJiang Univ.,11,DEFINITION 1.,Let p be a proposition.The statement,It is not the case that p.,is another proposit

7、ion,called the negation of p.,The negation of p is denoted by p.The proposition p is read not p.,p,的否定,11/30/2024 3:40 PM,Deren Chen,ZheJiang Univ.,12,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 F

8、riday.,This negation can be more simply expressed by,Today is not Friday.,11/30/2024 3:40 PM,Deren Chen,ZheJiang Univ.,13,Table 1,11/30/2024 3:40 PM,Deren 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

9、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,的合取,11/30/2024 3:40 PM,Deren Chen,ZheJiang Univ.,15,Table 2,11/30/2024 3:40 PM,Deren Chen,ZheJiang Univ.,16,EXAMPLE 4,Find the conjunction of the proposi

10、tions 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 propositions,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

11、and on Fridays when it does not rain.,11/30/2024 3:40 PM,Deren Chen,ZheJiang Univ.,17,DEFINITION 3.,Let p and q be propositions.The proposition 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

12、.,The truth table for pq is shown in Table 3.,p,和,q,的析取,11/30/2024 3:40 PM,Deren Chen,ZheJiang Univ.,18,Table 3,11/30/2024 3:40 PM,Deren Chen,ZheJiang 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 disjunc

13、tion 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 rainy day(including rainy Fridays).It is only false on days that are not Fridays when it also does not rain.,11/30/2024 3:40 PM,Deren Chen,ZheJiang Univ.,20,D

14、EFINITION 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 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,11/30/2024 3:40 PM,Deren Chen,Z

15、heJiang Univ.,21,Table 4,11/30/2024 3:40 PM,Deren Chen,ZheJiang Univ.,22,DEFINITION 5.,Let p and q be propositions.The implication pq is the proposition 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 cal

16、led the conclusion(or consequence).,如果,p,则,q,单条件,蕴涵,P:,前提,Q:,结论,11/30/2024 3:40 PM,Deren Chen,ZheJiang Univ.,23,Table 5,11/30/2024 3:40 PM,Deren Chen,ZheJiang Univ.,24,EXAMPLE 6,What is the value of the variable x after the statement,if 2+2=4 then x:=x+1,if 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 is true,the assignment statement x:=x+1 is executed.Hence,x has the value 0