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离散数学笔记:1.4 断言与量词

断言(Predicates)

像这样:

$x+y=z$

我一天吃 $n$ 顿饭

包含变量的语句称为断言(有人翻译为谓词,我认为不妥。) 当给变量带入数值之后,断言就变成了命题。

我们可以用命题函数 $P(x,y,\cdots)$ 来代表断言。比如用 $P(n)$ 代表 我一天吃 $n$ 顿饭

量化符(Quantifiers)

$All$ 的首字母倒过来,“$\forall x$ ”表示“对于所有$x$”。

$Exist$ 首字母倒过来,“$\exists x$” 表示“至少存在一个$x$”。

这两个符号就是量化符号。分别是全称量化和存在量化。

对断言使用量化符号便可实现语句的量化(Quantification)。


【例子】Let C(x) be the statement “x has a cat,” let D(x) be the statement “x has a dog,” and let F(x)be the statement “x has a ferret.” Express each of these statements in terms of C(x), D(x), F(x), quantifiers, and logical connectives. Let the domain consist of all students in your class.

a) A student in your class has a cat, a dog, and a ferret.

$\exists x(C(x)\and D(x) \and F(x))$

b) All students in your class have a cat, a dog, or a ferret.

$\forall x(C(x)\or D(x) \or F(x))$

c) Some student in your class has a cat and a ferret, but not a dog.

$\exists x(C(x)\and F(x) \and \neg D(x))$

d) No student in your class has a cat, a dog, and a ferret.

$\neg(\exists x(C(x)\and D(x) \and F(x)))$

e) For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has this animal as a pet.

$\exists x(C(x)\oplus D(x) \oplus F(x))$

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