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2020-05-26 15:02:28
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# 第2章 数学基础 $$ 对应学时(指第几学时) \begin{cases} 2 \ 集合论\\ 3 \ 命题逻辑的语法和语义\\ 4 \ 命题逻辑的形式证明\\ 5 \ 一阶逻辑的语法和语义\\ 6 \ 一阶逻辑的形式证明\\ \end{cases} $$ [TOC] 注:笔记为截取ppt重点以及记录个人理解为主,另对于课堂练习进行归纳汇总。 ## 2 集合论 $$ 集合论 \begin{cases} 集合的表示\\ 集合谓词\\ 特殊集合\\ 集合运算\\ 序偶与笛卡尔积\\ \end{cases} $$ ### 集合的表示 ![集合的表示](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414213043.png) ### 集合谓词 ![集合谓词](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414213103.png) ### 特殊集合 #### 特殊的集合 ![特殊集合](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414213130.png) #### 关于$\emptyset$ , $\subseteq$ , $=$ , $\subset$ , $\in$ , $\mathbb{P}$ , $\mathbb{P}_1$ 的性质 ![关于$\emptyset$ , $\subseteq$ , $=$ , $\subset$ , $\in$ , $\mathbb{P}$ , $\mathbb{P}_1$ 的性质](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414214056.png) ### 集合运算 #### 集合的运算 ![集合运算](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414214220.png) #### 关于集合并、交、差运算的性质 ![关于集合并、交、差运算的性质](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414214401.png) #### 集合的运算(续) ![集合的运算(续)](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414214727.png) ### 序偶和笛卡尔积 ![序偶和笛卡尔积1](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414214813.png) ![序偶和笛卡尔积2](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414214827.png) ![序偶和笛卡尔积3](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414214840.png) ### 课堂练习2 ![课堂练习2](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414215527.png) <img src="https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414221201.png" style="zoom:35%" alt="解答2-1" /> ## 3 命题逻辑的语法和语义 命题是具有确定真假意义的陈述句,称无法再分解的命题为原子命题。 ### 命题逻辑的语法 #### 符号 ![命题逻辑的语法 - 符号](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414223109.png) #### 公式 ![命题逻辑的语法 - 公式](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414223302.png) ### 命题逻辑的语义 #### 解释与真值表 ![命题逻辑的语义1](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414223507.png) ![命题逻辑的语义2](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414223536.png) #### 永真式、矛盾式、可满足式 ![永真式、矛盾式、可满足式](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414223720.png) #### 逻辑推论 ![逻辑推论](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414223807.png) #### 逻辑等价 ![逻辑等价](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414223856.png) ### 课堂练习3 ![课堂练习3](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414224554.png) <img src="https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414230404.png" style="zoom:35%" alt="解答 3-1" /> <img src="https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414230457.png" style="zoom:35%" alt="解答 3-2" /> ## 4 命题逻辑的形式证明 ### 自然演绎系统 ![自然演绎系统](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414230841.png) ### 推理规则 ![推理规则1](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414230952.png) ![推理规则2](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414231033.png) ![推理规则3](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414231525.png) ### 形式证明 ![形式证明](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414231901.png) ### 推理规则 $\in$ ![推理规则 $\in$](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414231947.png) ### 推理的传递性 ![推理的传递性](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414232023.png) ### 例子 ![例1](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414232057.png) ![例2](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414232115.png) ### 完全性定理 ![完全性定理](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414232148.png) ### 课堂练习4 ![课堂练习4](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200414230723.png) ![解答 4-1](https://picture-bed-bucket.oss-cn-beijing.aliyuncs.com/img/20200415001727.png)
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