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Discrete Mathematics 離散數學 顏嗣鈞 Instructor: 顏嗣鈞 Class: Thursday 10:20--12:10 Office: BL- 525 Phone: 3366-3581 E-mail: [email protected] URL: http://www.ee.ntu.edu.tw/~yen Office Hours: By appointment Textbook Discrete Mathematics and Its Applications 5th Edition Kenneth H. Rosen McGraw-Hill Class Website http://www.ee.ntu.edu.tw/~yen/courses/d m2006.html Course Requirements There will be homework, one midterm exam, and a final exam. The weightings within the semester grade will be: Homework Assignments 20 % Midterm 40 % Final exam 40 % Only homework turned in by the due date is guaranteed to be graded. Any special circumstances that cause difficulty in meeting the deadlines should be brought to the attention of the instructor in advance. What is Mathematics, really? • It’s not just about numbers! • Mathematics is much more than that: Mathematics is, most generally, the study of any and all absolutely certain truths about any and all perfectly well-defined concepts. • But, these concepts can be about numbers, symbols, objects, images, sounds, anything! Physics from Mathematics • Starting from simple structures of logic & set theory, – Mathematics builds up structures that include all the complexity of our physical universe… • Except for a few “loose ends.” • One theory of philosophy: – Perhaps our universe is nothing other than just a complex mathematical structure! • It’s just one that happens to include us! So, what’s this class about? What are “discrete structures” anyway? • “Discrete” ( “discreet”!) - Composed of distinct, separable parts. (Opposite of continuous.) discrete:continuous :: digital:analog • “Structures” - Objects built up from simpler objects according to some definite pattern. • “Discrete Mathematics” - The study of discrete, mathematical objects and structures. Discrete Structures We’ll Study • • • • • • • • • Propositions Predicates Proofs Sets Functions Orders of Growth Algorithms Integers Summations • • • • • • • • • Sequences Strings Permutations Combinations Relations Graphs Trees Logic Circuits Automata Relationships Between Structures • “→” :≝ “Can be defined in terms of” Programs Proofs Groups Trees Operators Complex Propositions numbers Graphs Real numbers Strings Functions Integers Matrices Relations Natural numbers Sequences Infinite Bits n-tuples Vectors ordinals Sets Not all possibilities are shown here. Some Notations We’ll Learn p x P( x) f : A B pq pq {a1 , , an } Z, N, R S T f 1 ( x) |S| f g pq A B pq {x | P( x)} x P( x) xS n A x a S O, , min, max a /| b gcd, lcm mod (ak a0 )b [aij ] AT AΟ B A[ n ] C (n; n1 , , nm ) p( E | F ) R [ a ]R A i i 1 n a i i 1 a b (mod m) n r deg (v) Why Study Discrete Math? • The basis of all of digital information processing is: Discrete manipulations of discrete structures represented in memory. • It’s the basic language and conceptual foundation for all of computer science. • Discrete math concepts are also widely used throughout math, science, engineering, economics, biology, etc., … • A generally useful tool for rational thought! Uses for Discrete Math in Computer Science • Advanced algorithms & data structures • Programming language compilers & interpreters. • Computer networks • Operating systems • Computer architecture • Database management systems • Cryptography • Error correction codes • Graphics & animation algorithms, game engines, etc.… • I.e., the whole field! Instructors: customize topic content & order for your own course Course Outline (as per Rosen) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Logic (§1.1-4) Proof methods (§1.5) Set theory (§1.6-7) Functions (§1.8) Algorithms (§2.1) Orders of Growth (§2.2) Complexity (§2.3) Number theory (§2.4-5) Number theory apps. (§2.6) Matrices (§2.7) Proof strategy (§3.1) Sequences (§3.2) 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. Summations (§3.2) Countability (§3.2) Inductive Proofs (§3.3) Recursion (§3.4-5) Program verification (§3.6) Combinatorics (ch. 4) Probability (ch. 5) Recurrences (§6.1-3) Relations (ch. 7) Graph Theory (chs. 8+9) Boolean Algebra (ch. 10) Computing Theory (ch.11) Why Is It Computer Science? • The is basically a mathematics course: – No programming – Lots of theorems to prove So why is it computer science? • Discrete mathematics is the mathematics underlying almost all of computer science: – – – – – Designing high-speed networks Finding good algorithms for sorting Doing good web searches Analysis of algorithms Proving program correct Course Objectives • Upon completion of this course, the student should be able to: – Check validity of simple logical arguments (proofs). – Check the correctness of simple algorithms. – Creatively construct simple instances of valid logical arguments and correct algorithms. – Describe the definitions and properties of a variety of specific types of discrete structures. – Correctly read, represent and analyze various types of discrete structures using standard notations.