離散數學

臺北科技大學 115-1 學期課程，課號 362410。

課程大綱

Course Introduction Discrete mathematics is a branch of mathematics devoted to the study of discrete objects (i.e., objects that are distinct and unconnected). This course will introduce and apply various discrete structures and techniques to solve real-world problems. Main Topics 1. The Foundations: Logic and Proofs o Understanding the fundamentals of logical reasoning and constructing mathematical proofs. 2. Basic Structures: Sets, Functions, Sequences, Sums, and Matrices o Exploring fundamental mathematical structures used in computer science and mathematics. 3. Algorithms o Designing and analyzing algorithms for various computational problems. 4. Number Theory and Cryptography o Applying number theory in cryptographic algorithms and understanding their mathematical underpinnings. 5. Counting o Learning basic counting principles and techniques, including permutations and combinations. 6. Discrete Probability o Introducing probability theory with discrete random variables and their applications. 7. Advanced Counting Techniques o Delving deeper into combinatorial analysis and advanced counting methods. 8. Relations o Examining relations and their properties, including equivalence relations and partial orderings. Learning Objectives • Development of Mathematical Thinking and Logical Reasoning: Cultivate the ability to understand and create mathematical arguments, laying the foundation for advanced mathematical studies. • Foundations in Computer Science: Provide the necessary mathematical background for advanced courses in computer science, such as data structures, algorithms, compiler theory, and more. • Operations Research and Applications: Equip students with the mathematical background required to solve problems in operations research, chemical engineering, biology, and related fields. Course Features • Emphasis on Problem-Solving: The course focuses on teaching mathematical reasoning and problem-solving rather than merely imparting skills. • Fostering Creativity and Critical Thinking: Through diverse topics and problems, students are encouraged to apply their knowledge to new challenges. • Practice in Mathematical Proofs: Students will learn how to read, write, and validate mathematical proofs, which will be valuable for advanced courses. This course aims to equip students with the necessary tools and skills to tackle challenges in future courses and professional work.

課程進度

Week 1: The Foundations: Logic and Proofs • Introduction to Propositional Logic • Logical Equivalences and Quantifiers Week 2: The Foundations: Logic and Proofs • Methods of Proof • Proof Strategies and Techniques Week 3: Basic Structures: Sets, Functions, Sequences, Sums, and Matrices • Introduction to Sets and Operations • Functions and Their Properties Week 4: Basic Structures: Sets, Functions, Sequences, Sums, and Matrices • Sequences and Summations • Introduction to Matrices Week 5: Algorithms • Definition and Complexity of Algorithms • Algorithmic Problem-Solving Week 6: Number Theory and Cryptography • Basic Concepts in Number Theory • Introduction to Cryptography Week 7: Counting • Basic Counting Principles • Permutations and Combinations Week 8: Counting • Advanced Counting Methods • Inclusion-Exclusion Principle Week 9: Midterm Exam Week 10: Discrete Probability • Introduction to Discrete Probability • Probability Rules and Bayes’ Theorem Week 11: Discrete Probability • Expected Value and Variance • Applications of Discrete Probability Week 12: Advanced Counting Techniques • Generating Functions • Recurrence Relations Week 13: Advanced Counting Techniques • Pigeonhole Principle • Applications of Advanced Counting Techniques Week 14: Relations • Introduction to Relations • Properties of Relations Week 15: Relations • Equivalence Relations • Partial Orders Week 16: Final Exam

評量方式

Pop Quizzes + Homework 40% Midterm Exam 30% Final Exam 25% Flexible Learning Activities (Weeks 17–18) 5%

教材

【遵守智慧財產權觀念，請使用正版教科書，不得使用非法影印教科書】 使用外文原文書：是 Discrete Mathematics and Its Applications 作者：Kenneth Rosen ISBN：9781260091991 年份：2019 出版商：McGraw-Hill 參考網頁：Discrete Mathematics and Its Applications 8/e