离散数学及应用

定　　价：49.8 元

• 作者：黄玮
• 出版时间：2024/3/1
• ISBN：9787121476396
• 出 版 社：电子工业出版社

• 中图法分类：O158 
• 页码：188
• 版次：01
• 开本：16开

内容简介

本书内容分为数理逻辑、集合论、代数结构、图论等四个主要部分，包含命题逻辑、一阶谓词逻辑、集合、二元关系、函数、代数结构、特殊代数系统、图、特殊图及图的应用、树等10个章节。本书以应用型人才培养为目标，突出离散数学作为计算机及相关本科的专业基础课这一定位，本书可作为计算机科学与技术、软件工程、智能科学与技术、物联网工程、网络工程等专业离散数学课程的教材。

作者介绍

黄玮，男，1978年生人，副教授，在读博士，现任教于福州理工学院，具有知名企业的从业经历，曾获"微软金牌讲师”等头衔，长期从事软件工程、虚拟现实研发与教学工作，主持和完成过各类省级、市级科研项目5项，获软件著作权14项，发表科研论文9篇。

目 录

目 录
第 1 章 命题逻辑 ····································································································.1
1.1 命题与联结词 ·····························································································.1
1.2 命题公式及其赋值 ······················································································.5
1.3 等值式 ·······································································································.8
1.4 析取范式与合取范式 ·················································································.10
1.4.1 范式的基本概念 ··············································································.10
1.4.2 主范式及其表示 ··············································································.12
1.4.3 主范式的求解 ··················································································.13
1.4.4 主范式的数量 ··················································································.15
1.5 联结词的完备集 ························································································.16
1.5.1 真值函数 ·························································································.16
1.5.2 联结词完备集 ··················································································.17
1.6 命题逻辑的推理理论 ·················································································.18
1.6.1 推理的基本概念 ··············································································.18
1.6.2 推理定律和推理规则 ·······································································.19
1.6.3 命题逻辑的推理方法 ·······································································.19
1.7 命题逻辑的应用 ························································································.23
1.7.1 命题公式在计算机中的表示 ·····························································.23
1.7.2 程序优化 ·························································································.24
1.7.3 资源分配 ·························································································.24
1.7.4 计算机硬件电路设计 ·······································································.25
1.8 命题逻辑实验 ····························································································.25
习题 1 ···············································································································.27
第 2 章 一阶逻辑 ···································································································.30
2.1 一阶逻辑的基本概念 ·················································································.30
2.2 一阶逻辑公式及其解释 ··············································································.33
2.2.1 一阶逻辑公式的符号 ·······································································.33
2.2.2 一阶逻辑公式的结构 ·······································································.34
2.2.3 量词的辖域 ·····················································································.34
2.2.4 公式的解释 ·····················································································.35
2.3 一阶逻辑等值式及演算 ··············································································.37
2.4 一阶逻辑前束范式 ·····················································································.38
2.5 一阶逻辑的推理理论 ·················································································.39
2.6 一阶逻辑实验 ····························································································.43
习题 2 ···············································································································.43
第 3 章 集合论 ·······································································································.47
3.1 集合的基本概念 ························································································.47
3.2 集合的基本运算 ························································································.50
3.3 有限集的计数 ····························································································.54
3.4 集合论实验 ·······························································································.55
习题 3 ···············································································································.57
第 4 章 二元关系 ···································································································.60
4.1 有序对与笛卡儿积 ·····················································································.60
4.2 二元关系的定义 ························································································.62
4.3 关系的运算 ·······························································································.63
4.4 关系的性质 ·······························································································.67
4.4.1 性质的定义 ·····················································································.67
4.4.2 性质的判定 ·····················································································.70
4.5 关系的闭包 ·······························································································.72
4.5.1 闭包的基本概念 ··············································································.72
4.5.2 闭包的性质 ·····················································································.74
4.5.3 用 WarShall 算法求传递闭包 ····························································.74
4.6 等价关系 ···································································································.76
4.6.1 等价关系的概念 ··············································································.76
4.6.2 等价类 ····························································································.76
4.6.3 商集与划分 ·····················································································.77
4.7 偏序关系 ···································································································.79
4.7.1 偏序关系的概念 ··············································································.79
4.7.2 哈斯图 ····························································································.79
4.7.3 偏序集中的特殊元素 ·······································································.80
4.7.4 偏序关系的应用 ··············································································.81
4.8 二元关系实验 ····························································································.83
习题 4 ···············································································································.84
第 5 章 函数 ··········································································································.86
5.1 函数的定义与性质 ·····················································································.86
5.1.1 函数的定义 ·····················································································.86
5.1.2 函数的性质 ·····················································································.88
5.2 函数的复合与反函数 ·················································································.90
5.2.1 函数的复合 ·····················································································.90
5.2.2 反函数 ····························································································.91
5.3 函数实验 ···································································································.92
习题 5 ···············································································································.92
第 6 章 代数结构 ···································································································.94
6.1 运算及其性质 ····························································································.94
6.1.1 代数的构成 ·····················································································.94
6.1.2 二元运算的性质 ··············································································.97
6.2 代数系统 ···································································································.99
6.3 半群与群 ·································································································.102
6.4 子群 ········································································································.104
6.5 代数结构实验 ··························································································.107
习题 6 ·············································································································.108
第 7 章 特殊代数系统 ··························································································.111
7.1 环与域 ····································································································.111
7.2 格的定义与性质 ······················································································.113
7.2.1 格的定义 ·······················································································.113
7.2.2 格的性质 ·······················································································.114
7.2.3 特殊的格 ·······················································································.115
7.3 布尔代数 ·································································································.116
7.4 布尔代数的应用 ······················································································.118
7.4.1 逻辑门电路 ···················································································.118
7.4.2 电路的极小化 ················································································.118
7.5 特殊代数系统实验 ···················································································.119
习题 7 ·············································································································.120
第 8 章 图 ············································································································.123
8.1 图的基本概念 ··························································································.123
8.1.1 图的概念 ·······················································································.123
8.1.2 握手定理 ·······················································································.125
8.1.3 图的同构 ·······················································································.127
8.1.4 图的运算 ·······················································································.128
8.1.5 子图与补图 ···················································································.128
8.2 通路与回路 ·····························································································.129
8.3 图的连通性 ·····························································································.130
8.3.1 无向图的连通性 ············································································.130
8.3.2 有向图的连通性 ············································································.132
8.4 图的矩阵表示 ··························································································.133
8.5 图的实验 ·································································································.135
习题 8 ·············································································································.137
第 9 章 特殊图及其应用 ·······················································································.139
9.1 欧拉图 ····································································································.139
9.1.1 欧拉图的概念 ················································································.139
9.1.2 欧拉图的判别法 ············································································.140
9.2 汉密尔顿图 ·····························································································.141
9.3 平面图 ····································································································.143
9.3.1 平面图的基本定义 ·········································································.143
9.3.2 欧拉公式 ·······················································································.145
9.3.3 欧拉公式的推广 ············································································.146
9.3.4 平面图的判断 ················································································.147
9.4 图的应用 ·································································································.148
9.4.1 最短路径问题 ················································································.148
9.4.2 中国邮递员问题 ············································································.150
9.5 特殊图实验 ·····························································································.152
习题 9 ·············································································································.153
第 10 章 树 ··········································································································.155
10.1 无向树 ··································································································.155
10.2 生成树及应用 ························································································.157
10.2.1 生成树的基本概念 ·······································································.157
10.2.2 最小生成树 ··················································································.158
10.3 根树及应用 ···························································································.161
10.3.1 根树的相关概念 ················································································.161
10.3.2 最优二叉树 ······················································································.162
10.3.3 树的遍历 ·························································································.165
10.4 树的实验 ·······························································································.166
习题 10 ···········································································································.167
模拟卷 A ················································································································.169
模拟卷 B ················································································································.172
参考文献 ···············································································································.176