1 CS233601: Discrete Mathematics Department of Computer Science National Tsing Hua University.
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1 CS233601: Discrete Mathematics Department of Computer Science National Tsing Hua University
2 Instructor –SHIH YU CHANG ( 張適宇 ), firstname.lastname@example.org@cs.nthu.edu.tw –Office Number: R743, EECS Building Time and Location –Wednesday 10:10 ~ 12:00, Friday 11:10 ~ 12:00 –EECS 236 Office Hours –Friday 15:00~1700 Textbook –"Elements of Discrete Mathematics" (McGraw-Hill), by Prof. C. L. Liu Reference –"Discrete Mathematics and Its Applications" (McGraw- Hill), by Kenneth H. Rosen
4 What is Discrete Mathematics? Discrete mathematics is the part of mathematics devoted to the study of discrete objects. Here discrete means consisting of distinct or unconnected elements. The kind of problems solved using discrete mathematics include: –How many ways are there to choose a valid password on a computer system?
5 What is Discrete Mathematics? (Cont.) –What is the probability of winning a lottery? –Is there a link between two computers in a network? –What is the shortest path between two cities using a transportation system? –How can a list of integers be sorted so that the integers are in increasing order? –How many steps are required to do such a sorting? –How can it be proved that a sorting algorithm correctly sorts a list?
6 What is Discrete Mathematics? (Cont.) –How can a circuit that adds two integers be designed? –How many valid Internet addresses are there? You will learn the discrete structures and techniques needed to solve such problems. More generally, discrete mathematics is used whenever objects are counted, when relationships between finite (or countable) sets are studied, and when processes involving a finite number of steps are analyzed.
7 Why Study Discrete Mathematics? Through this course you can develop your mathematical maturity, that is, your ability to understand and create mathematical arguments. Discrete mathematics provides the mathematical foundations for many computer science courses, including data structures, algorithms, database theory, automata theory, formal languages, compiler theory, computer security, and operating systems.
8 Five Themes in Discrete Mathematics? Mathematical Reasoning –Understanding mathematical reasoning in order to read, comprehend, and construct mathematical arguments –Mathematical logic, methods of proof, mathematical induction Combinatorial Analysis –The ability to count or enumerate objects –The basic techniques of counting, permutations, combinations Discrete Structures –The abstract mathematical structures used to represent discrete objects and relationships between these objects –Sets, permutations, relations, graphs, trees, and finite-state machines
9 Five Themes in Discrete Mathematics? Algorithmic Thinking –The specification of an algorithm that solves certain classes of problems –The specification of the algorithm, the verification of the algorithm, the analysis of the space and time complexities of the algorithm Applications and Modeling –Computer science, data networking, chemistry, business, etc. –Modeling with discrete mathematics is an extremely important problem-solving skill
10 Course Outline Sets and Propositions Computability and Formal Languages Permutations, Combinations, and Discrete Probability Relations and Functions Graphs and Planar Graphs Trees and Cut-Sets Finite State Machines
11 Course Outline (Cont.) Analysis of Algorithms Discrete Numeric Functions and Generating Functions Recurrence Relations and Recursive Algorithms Groups and Rings Boolean Algebras