命题及常用的符号
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命题及常用的符号
由变量定义的命题,
该变量称为自由变量因为给它一个值能产生一个命题。
例如集合生成器符:{x:x具有性质P},表示为P(x)。
•其真值取决于x的值,它的真值集是x的真值集;
属性可能有多个自由变量,描述遵循笛卡尔产品的方式,
表示为P(x,y,z)
通用量化:
如果属性对所有值都为true:
•表示为∀x P(x)
普遍量化的否定:
如果无值的属性为真:真值集必须是空集;
属性并非适用于所有值;
•表示为∀x ¬ P(x)
存在量化:
如果属性对某些值为真:
(真理集不能是空的)它被认为是存在的量化
表示为∃x P(x)
存在量化的否定:
如果属性对于某些值不为真:
真值集是宇宙的严格子集,可能是空集;这又是存在的量化;
表示为 ∃x ¬ P(x)
有界量化:
将量化限制在属性;
对于通用量化:
•写入∀x∈A P(x),
•等价于∀x(x∈A⇒P(x));
•相当于∀x(A(x)⇒P(x));
(其中A(x)是描述x到的条件的谓词集合A的元素。)
对于存在量化:
•写入∃x∈A P(x),
•等价于∃x(x∈A∧P(x));
•相当于∃x(A(x)∧P(x));
独特量化:
谓词对于一个元素是真的;存在量化的一种形式:
∃!x P(x)
相当于∃x∃x (P(x) ∧ ¬ ∃y(P(y) ∧ y ≠ x)).
总结
1.∃xP(x)
The existential quantifier always means “at least one”, which
means that there may be one or more of the specified thing in
existence.
2.∃!xP(x)
Sometimes, it may be useful to say that there is only one. In these
cases, an existential quantifier is written as ∃!, which means “there
exists exactly one
3.∀xP(x)
In predicate logic, a universal quantification is a type
of quantifier, which is interpreted as “for all”. It expresses
that a propositional function can be satisfied by every
member of a domain of discourse.
例子:
将 L(x) 设定为‘x likes Brussel sprouts’
∃xL(x)-----Someone likes brussels sprouts
∃x¬L(x)-----someone doesn’t like brussels sprouts
¬∃xL(x)-----everyone dislikes brussels sprouts
¬∃x¬L(x)----everyone likes brussels sprouts
∀xL(x)---- everyone likes brussels sprouts
∀x¬L(x)----everyone dislikes brussels sprouts
¬∀xL(x)----someone doesn’t like brussels sprouts
¬∀x¬L(x)----Someone likes brussels sprouts
标签:常用,真值,符号,命题,----,likes,量化,sprouts,brussels 来源: https://blog.csdn.net/kirsten111111/article/details/112912305