离散数学Nested Quantifiers 嵌套量词.ppt

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,The Foundations:Logic and Proof Guo Jian,*,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,The Foundations:Logic and Proof Guo Jian,*,1.4 Nested Quantifiers,（嵌套量词）,Introduction,nested quantifiers,-occur within the scope of other,quantifiers,x,y(x+y,=0),2026/5/18 周一,1,The Foundations:Logic and Proof Guo Jian,(1)Example 1(page 51),the domains for x,，,y and z,-all real numbers,x y(,x+y,=,y+x,)-commutative law,-true,x,y,(,x+y,=0),-true,x y z(,x+(y+z,)=(,x+y)+z,),-true -associative law,2026/5/18 周一,2,The Foundations:Logic and Proof Guo Jian,(2)Example 2(page 45),x y(x0)/(y0),(xy0y0)(x+y)0),(,b)The,domain-positive integer,xy(x+y,0),2026/5/18 周一,9,The Foundations:Logic and Proof Guo Jian,Example7,“Every real number except 0 has a multiplicative inverse”,Solution:,The domain-real number,x(x0)y(xy=1),2026/5/18 周一,10,The Foundations:Logic and Proof Guo Jian,4.Translating from Nested quantifiers into English,(1)Example 9(page 55),x(,C(x,)/,y(,C(y,)/,F(x,y,),C(x)-”x,has a computer”,F(x,y)-”x,y,are friends”,the domain for both x and y,-all students in the school,?What does the formula mean?,2026/5/18 周一,11,The Foundations:Logic and Proof Guo Jian,(2)Example 10(page 55),x,y z(,(,F(x,y,)/,F(x,z,)/(,y,z,),F(y,z,),),F(a,b,)-a and b are friends,the domains-for x,y and z,-all students in your school,What does this formula mean?,2026/5/18 周一,12,The Foundations:Logic and Proof Guo Jian,5.Translating English Sentences Into Logical Expression,Example 11(page 56),“,If a person is female and is parent,then this person is someones mother,.”,the domain,-all people,2026/5/18 周一,13,The Foundations:Logic and Proof Guo Jian,Solution:,“,For every person,if person x is a female and person x is a parent,then there exists a person y such that person x is the mother of person y,.”,F(x)-x,is female;,P(x)-x,is a parent,M(x,y)-x,is the mother of y,Then,the formula is:,x(,F(x,)/,P(x,),y,M(x,y,),)or,x,y,(,F(x,)/,P(x,),M(x,y,),),2026/5/18 周一,14,The Foundations:Logic and Proof Guo Jian,(2)Example 12(page 56),“,Everyone has exactly one best friend,”,The domain,-all people,Solution:,“,For every person x,person x has exactly one best friend,”,B(x,y,)-y is the best friend of x,y(,B(x,y,)/,z(,z,y,),B(x,z,),),),x(y,(,B(x,y,)/z(,z,y,),B(x,z,),),2026/5/18 周一,15,The Foundations:Logic and Proof Guo Jian,(3)Example 13(page 56,57),Please learn by yourself.,2026/5/18 周一,16,The Foundations:Logic and Proof Guo Jian,6.,Negating Nested Quantifiers,(1)Example 11(page 48),Express the negation of ,xy,(,xy,=1),so that no negation precedes a quantifier.,Solution:,xy,(,xy,=1),x,y(,xy,=1),x y,(,xy,=1),x y (xy,1),2026/5/18 周一,17,The Foundations:Logic and Proof Guo Jian,