Predicates and Quantiflers 调词与量词 则对任意整数i, 1=(-1)(i+1) 恒成立。也就是说,该命题函数的真值无论用什么个体代 入总是对应为T。此类命题函数的真值描述通过一个称为全称 量词的特殊符号来量化 2/24/202111:37PM Deren Chen, ZheJiang Univ 11
P redica t es a nd Qua nt if iers 谓词与量词 2/24/2021 11:37 PM Deren Chen, ZheJiang Univ. 11 则对任意整数i， i 2–1=(i–1)(i+1) 恒成立。也就是说，该命题函数的真值无论用什么个体代 入总是对应为T。此类命题函数的真值描述通过一个称为全称 量词的特殊符号来量化
DEFINITIONZ Predicates and Quantiflers 调词与量词 定义2命题函数P(x)的全称量化( universal qualification) 是一个按如下规则确定真值的命题:如果对每一个个体a代入 得到的P(a均为T。则该命题为T。记为vkP(x)。这里V是全称 量词( universal quantifier),表示为“对任意的”、“所有 的”、“对每一个”等等。 2/24/202111:37PM Deren Chen, ZheJiang Univ
P redica t es a nd Qua nt if iers 谓词与量词 2/24/2021 11:37 PM Deren Chen, ZheJiang Univ. 12 定义2 命题函数P(x)的全称量化（universal quanification） 是一个按如下规则确定真值的命题：如果对每一个个体a代入 得到的P(a)均为T。则该命题为T。记为VxP(x)。这里V是全称 量词（universal quantifier），表示为“对任意的” 、 “所有 的” 、 “对每一个”等等。 DEFINITION 2
DEFINITIONZ Predicates and Quantiflers 调词与量词 The universal quantification of P(x)is the proposition P(x)is true for all values of x in the universe of discourse.” 2/24/202111:37PM Deren Chen, ZheJiang Univ 13
P redica t es a nd Qua nt if iers 谓词与量词 2/24/2021 11:37 PM Deren Chen, ZheJiang Univ. 13 DEFINITION 2. The universal quantification of P(x) is the proposition “P(x) is true for all values of x in the universe of discourse
EXAMPLE 5 Predicates and Quantiflers 调词与量词 Express the statement Every student in this class has studied calculus as a universal quantification It can be written as yxP(x)oryx(S(X)→P(×) P(x= has studied calculus S(x)=x is in this class 2/24/202111:37PM Deren Chen, ZheJiang Univ 14
P redica t es a nd Qua nt if iers 谓词与量词 2/24/2021 11:37 PM Deren Chen, ZheJiang Univ. 14 EXAMPLE 5 Express the statement "Every student in this class has studied calculus" as a universal quantification. It can be written as xP(x) or x (S(x)→P(x)) P(x) = "x has studied calculus." S(x) = "x is in this class
EXAMPLE 8 Predicates and Quantiflers 调词与量词 What is the truth value of yX P(x), where P(x)is the statement x2<10'and the universe of discourse consists of the positive integers not exceeding 4? yxP(x)=P(1)∧P(2)∧P(3)∧P(4)=F. 2/24/202111:37PM Deren Chen, ZheJiang Univ
P redica t es a nd Qua nt if iers 谓词与量词 2/24/2021 11:37 PM Deren Chen, ZheJiang Univ. 15 EXAMPLE 8 What is the truth value of V x P(x), where P(x) is the statement "x2 < 10" and the universe of discourse consists of the positive integers not exceeding 4? x P(x)=P(1) ∧P(2) ∧P(3) ∧P(4) = .F