逻辑与形式化方法复习

# 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)

## 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)

## 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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