目录
第 1章绪论 .......................................................................................................1
第一部分数学基础知识
第 2章命题 .......................................................................................................7
2.1定义和举例 .............................................................................................7
2.2命题联结词 .............................................................................................8
2.3重言式和矛盾式 .................................................................................... 13
2.4命题形式化 ........................................................................................... 17
2.5命题的量化 ........................................................................................... 18
第 3章集合和集合运算..................................................................................... 21
3.1集合 ..................................................................................................... 21
3.2集合相等 .............................................................................................. 23
3.3补集 ..................................................................................................... 25
3.4空集 ..................................................................................................... 26
3.5子集和超集 ........................................................................................... 27
3.6幂集和集合族 ....................................................................................... 28
3.7集合的交集、并集和补集 ....................................................................... 30
3.8笛卡儿积 .............................................................................................. 34
3.9集合运算的其他基本规律 ....................................................................... 37
第 4章数学证明............................................................................................... 39
第 5章关系 ..................................................................................................... 43
5.1定义和举例 ........................................................................................... 43
5.2关系运算 .............................................................................................. 47
5.3关系的重要性质 .................................................................................... 50
5.4等价关系与划分 .................................................................................... 52
计算机数学基础 (第 6版)
5.5等价关系的运算 .................................................................................... 57
5.6偏序关系 .............................................................................................. 61
第 6章映射与函数 ........................................................................................... 65
6.1定义及第一个例子 ................................................................................. 65
6.2满射、单射和双射 ................................................................................. 69
6.3序列和集合族 ....................................................................................... 74
6.4集合的基数 ........................................................................................... 77
6.5参考资料 .............................................................................................. 80
第二部分技术支持
第 7章数学证明方法 ........................................................................................ 85
7.1直接证明法 ........................................................................................... 85
7.2换质位法证明 ....................................................................................... 87
7.3反证法 ................................................................................................. 88
7.4等价证明 .............................................................................................. 89
7.5原子命题证明 ....................................................................................... 90
7.6个案分析证明 ....................................................................................... 92
7.7带量词的命题证明 ................................................................................. 93
7.8组合证明 .............................................................................................. 96
第 8章完全归纳法 ......................................................................................... 100
8.1完全归纳法的思路 ............................................................................... 101
8.2归纳证明举例 ..................................................................................... 101
8.3归纳证明的结构 .................................................................................. 104
8.4广义完全归纳法 .................................................................................. 106
8.5归纳定义 ............................................................................................ 107
第 9章组合计数............................................................................................. 116
9.1基本计数原则 ..................................................................................... 116
9.2排列和二项式系数 ............................................................................... 121
9.3计算二项式系数 .................................................................................. 125
第 10章离散概率论 ....................................................................................... 133
10.1随机试验和概率 ................................................................................ 133
10.2条件概率 .......................................................................................... 141
10.3随机变量 .......................................................................................... 143
目录
10.4二项分布和几何分布 .......................................................................... 149
10.5参考资料 .......................................................................................... 153
第三部分数学结构
第 11章布尔代数........................................................................................... 157
11.1布尔函数及其表达形式 ...................................................................... 157
11.2布尔代数的定义 ................................................................................ 163
11.3布尔代数示例 .................................................................................... 164
11.4布尔代数的性质 ................................................................................ 170
11.5布尔代数中的偏序 ............................................................................. 174
11.6布尔代数的原子 ................................................................................ 176
11.7布尔表达式的规范形式 ...................................................................... 180
11.8最小化布尔表达式 ............................................................................. 182
11.9同构基本定理 .................................................................................... 184
11.10电路代数 ......................................................................................... 188
第 12章图和树 .............................................................................................. 193
12.1基本概念 .......................................................................................... 194
12.2图中的通路和回路 ............................................................................. 199
12.3图和矩阵 .......................................................................................... 203
12.4图同构 .............................................................................................. 210
12.5树 .................................................................................................... 212
第 13章命题逻辑........................................................................................... 218
13.1布尔代数和命题逻辑 .......................................................................... 218
13.2范式 ................................................................................................. 223
13.3可满足性等价公式 ............................................................................. 225
13.4不可满足的子句集合 .......................................................................... 229
13.5霍恩子句的可满足性 .......................................................................... 232
13.6归结原理 .......................................................................................... 235
13.7 2KNF中的子句集 ............................................................................. 242
第 14章模算术 .............................................................................................. 245
14.1因数关系 .......................................................................................... 246
14.2模的加法和乘法 ................................................................................ 249
14.3模运算 .............................................................................................. 253
计算机数学基础 (第 6版)
14.4最大公因数和欧几里得算法 ................................................................ 257
14.5费马小定理 ....................................................................................... 261
14.6使用费马小定理的加密 ...................................................................... 265
14.7 RSA加密算法 .................................................................................. 270
14.8参考资料 .......................................................................................... 272