1
课程详述
COURSE SPECIFICATION
以下课程信息可能根据实际授课需要或在课程检讨之后产生变动。如对课程有任何疑问，请
联系授课教师。
The course information as follows may be subject to change, either during the session because of unforeseen
circumstances, or following review of the course at the end of the session. Queries about the course should be
directed to the course instructor.
1.
课程名称 Course Title
离散数学（H） Discrete Mathematics (H)
2.
授课院系
Originating Department
计算机科学与工程系 Department of Computer Science and Engineering
3.
课程编号
Course Code
CS215
4.
课程学分 Credit Value
3
5.
课程类别
Course Type
专业基础课 Major Foundational Courses
6.
授课学期
Semester
秋季 Fall
7.
授课语言
Teaching Language
中英双语 English & Chinese
8.
授课教师、所属学系、联系方
式（如属团队授课，请列明其
他授课教师）
Instructor(s), Affiliation&
Contact
（For team teaching, please list
all instructors）
王琦，副教授， 计算机科学与工程系， wangqi@sustech.edu.cn
Qi Wang, Associate Professor, Department of Computer Science and Engineering,
wangqi@sustech.edu.cn
9.
实验员/助教、所属学系、联系
方式
Tutor/TA(s), Contact
待公布 To be announced
10.
选课人数限额(可不填)
Maximum Enrolment
（Optional）
50
2
11.
授课方式
Delivery Method
讲授
Lectures
习题/辅导/讨论
Tutorials
实验/实习
Lab/Practical
其它(请具体注明)
Other（Please specify）
总学时
Total
学时数
Credit Hours
48
48
12.
先修课程、其它学习要求
Pre-requisites or Other
Academic Requirements
MA102B 高等数学（下）A Calculus II A
M107A 线性代数 I-A Linear Algebra I-A
13.
后续课程、其它学习规划
Courses for which this course
is a pre-requisite
CS403 密码学与网络安全 Cryptography and Network Security
14.
其它要求修读本课程的学系
Cross-listing Dept.
无 Not applicable for other departments beside CSE.
教学大纲及教学日历 SYLLABUS
15.
教学目标 Course Objectives
本课程旨在理解和应用在计算机科学与工程中广泛存在的一系列抽象的离散结构。具体地说，本课程将介绍逻辑、集合与
函数、数学证明、计算复杂度、数论及其应用、密码学、群环域代数结构及应用、数学归纳法、计数、递归、关系、图论
及算法等内容，特别是这些内容在计算机中的实际应用。
The objective of this course is to understand and use (abstract) discrete structures that are backbones of computer
sciences. In particular, this course is meant to introduce logic, sets and functions, mathematical proofs, complexity,
number theory, cryptography, algebraic structures and applications, induction, counting, recurrences, relations, graph
theory and related algorithms, with an emphasis on applications in computer science.
16.
预达学习成果 Learning Outcomes
本课程预期达到以下学习效果：
• 能够较好地阅读、理解、完成数学证明
• 理解离散数学中各部分问题的形式化表述，包括计数、数论、密码学、逻辑和证明、递归、图论等
• 学习一系列的离散数学工具并学会应用这些工具和方法解决计算机科学中的实际问题
On completion of this course, the students are expected to:
• be able to read, understand, and construct mathematical arguments and proofs
• understand the formulation of common problems in several areas of discrete mathematics, including counting,
number theory, cryptography, logic and proof, recursions, graph theory, etc.
• learn a number of discrete mathematical tools and apply discrete mathematical tools to solve certain problems in
computer science
17.
课程内容及教学日历 （如授课语言以英文为主，则课程内容介绍可以用英文；如团队教学或模块教学，教学日历须注明
主讲人）
Course Contents (in Parts/Chapters/Sections/Weeks. Please notify name of instructor for course section(s), if
this is a team teaching or module course.)
3
第一课：离散数学概论、命题逻辑
离散数学介绍、典型问题
命题逻辑、逻辑连接符、真值表
第二课：逻辑等价性、谓语逻辑
命题逻辑的应用
逻辑等价性及证明
命题逻辑的限制
谓语逻辑及量词
第三课：逻辑推导、数学证明思想
逻辑推导规则及应用
五种数学证明思路及举例证明
第四课：集合与函数
集合及运算、逻辑表示
定义函数、单射、满射函数
第五课：复合函数、序列、可数集
复合函数、函数逆定义
序列、序列求和、公式推导
可数集定义、证明及举例
第六课：计算复杂度 I
大 O 符号、复杂度计算举例
NP 理论介绍、决定和优化问题
第七课：计算复杂度 II、初等数论 I
NP 完全问题
整除、模运算、b 进制表示及相关算法
第八课：初等数论 II
素数、最大公约数
欧几里得算法、Bezout 等式
线性同余方程、模 n 逆及求解
第九课：初等数论应用
中国剩余定理、向后置换法
线性同余法生成伪随机数
费马小定理、欧拉定理
本原根定义
第十课：群环域抽象代数结构及应用
群、环、域定义及举例
有限域介绍、性质
有限域在随机序列、纠错编码的应用及相关举例
4
第十一课：密码学
密码学历史
对称密码学介绍、公钥密码学、RSA 加密机制
离散对数问题介绍、El Gamal 加密机制
第十二课：数学归纳法、递归 I
数学归纳法介绍及证明
数学归纳法弱准则、强准则
汉诺塔举例
递归式求解
第十三课：递归 II
递归式求解
主定理
第十四课：计数 I
排列组合计数、加法乘法规则
容斥原理及证明
鸽巢原理
一一对应原理
第十五课：计数 II
二项式系数及性质
Pascal 恒等式
组合证明
第十六课：高级计数方法
欧几里得算法复杂度分析
求解线性递归式
生成函数
第十七课：关系 I
二元关系
关系的性质及计数
关系复合
第十八课：关系 II
传递关系性质及证明
关系闭包
关系数据库
第十九课：关系 III
连通关系与传递闭包的关系
Roy-Warshall 算法
等价关系、等价类
偏序、Hasse 图
第二十课：图论 I
5
图论基本概念
无向图、有向图、二分图等
匹配、Hall 定理及证明
第二十一课：图论 II
图表示、邻接矩阵、关联矩阵
图同构
路径、连通性、欧拉图
第二十二课：图论 III
Hamilton 图
最短路径问题、Dijkstra 算法
平面图、欧拉公式
图染色问题
第二十三课：树 I
树基本概念
平衡树、相关数型数据结构介绍及算法
先序、中序、后序遍历
第二十四课：树 II、复习课
最小生成树及算法
深度、广度优先搜索
复习课
Overview of Discrete Math, Propositional Logic
Introduction to Discrete Math, typical problems
Propositional logic, logical connectives, truth tables
Logical Equivalence, Predicate Logic
Application of propositional logic
Logical equivalence and proof
Limitations of propositional logic
Predicate logic, quantifiers
Logical Inference, Proof Methods
Rules of logical inferences and applications
Five methods of proof, proof exercises
Set and Functions
Set, set operations, and representations using logic
Definition of functions, one-to-one functions, onto functions
Composite Functions, Sequences, Countable Sets
6
Composite function, inverse function
Sequences, sum of sequences, closed-form formula
Countable sets, proofs and examples
Computational Complexity I
Big-O notation, examples of complexity
NP theory
Decision problem, optimization problem
Computational Complexity II, Number Theory I
NP-Completeness
Divisibility, modular operation, base-b representation, related algorithms
Number Theory II
Primes, greatest common divisor
Euclidean algorithm, Bezout identity
Linear congruential equation, inverse modulo n
Applications of Number Theory
Chinese remainder theory, back substitution
Pseudorandom numbers using linear congruential method
Fermat’s little theorem, Euler’s theorem
Primitive root
Abstract Algebra and applications
Definitions of group, ring and field
Introduction to finite fields
Applications in pseudorandom sequences, error-correcting codes
Cryptography
History of cryptography
Symmetric encryption, public-key cryptography, RSA scheme
Discrete logarithm problem, El Gamal encryption scheme
Mathematical Induction, Recurrence I
Introduction to induction, typical proofs
Weak principle, strong principle of mathematical induction
Hanoi tower and recurrence
Solving recurrences with initial conditions
Recurrence II
Solving recurrences, more examples
7
The master theorem
Counting I
Permutations, combinatorial numbers, the sum/product rule
Inclusion-Exclusion principle and its proof
Pigeonhole principle
Bijection principle
Counting II
Binomial coefficient and properties
Pascal identity
Combinatorial proofs
Advanced Counting Techniques
Complexity of Euclidean algorithm
Solving linear recurrence relations with initial conditions
Generating functions
Relation I
Binary relation
Properties of relation, and counting
Composite relations
Relation II
Transitive relations, properties and proofs
Transitive closure
Relational database
Relation III
Connectivity relation and transitive closure
Roy-Warshall algorithm
Equivalence relations, equivalence class
Partial ordering, Hasse diagram
Graph Theory I
Basic concepts of graph theory
Undirected graphs, directed graphs, Bipartite graphs
Matching, Hall’s marriage theorem, proof
Graph Theory II
Representations of graphs, adjacency matrix, incidence matrix
Isomorphism of graphs
8
Path, connectivity, Euler graph
Graph Theory III
Hamilton graph
Shortest path, Dijkstra algorithm
Planar graphs, Euler formula
Graph coloring
Tree I
Basic concepts of tree
Balanced tree and counting
More related data structures and algorithms
Preorder, inorder, postorder traversal
Tree II, Review Lecture
Minimum spanning tree and algorithms
Depth-first search and breadth-first search
review
18.
教材及其它参考资料 Textbook and Supplementary Readings
Textbook：Kenneth Rosen, Discrete Mathematics and Its Applications, 7
th
Edition, Mc Graw Hill Education, 2012.
Reference books: Ronald Graham, Donald Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for
Computer Science, 2
nd
Edition, Addison-Wesley Professional, 1994.
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein, Introduction to Algorithms, 3
rd
Edition,
MIT Press, 2009.
课程评估 ASSESSMENT
19.
评估形式
Type of
Assessment
评估时间
Time
占考试总成绩百分比
% of final
score
违纪处罚
Penalty
备注
Notes
出勤 Attendance
课堂表现
Class
Performance
小测验
Quiz
10%
2-3 次 2-3 times
课程项目 Projects
5% optional
额外加分 optional
平时作业
Assignments
20%
5-7 次 5-7 times
期中考试
Mid-Term Test
30%
覆盖本课程前半部分 Convers the
first part of the course
期末考试
Final Exam
40%
覆盖本课程全部内容 Covers the
whole course
9
期末报告
Final
Presentation
其它（可根据需要
改写以上评估方
式）
Others (The
above may be
modified as
necessary)
20.
记分方式 GRADING SYSTEM
A. 十三级等级制 Letter Grading
B. 二级记分制（通过/不通过） Pass/Fail Grading
课程审批 REVIEW AND APPROVAL
21.
本课程设置已经过以下责任人/委员会审议通过
This Course has been approved by the following person or committee of authority
