Introduction to Discrete Mathematics

1. Introduction to Discrete Mathematics Lecture 1: Sep 1

2. Basic Information • Course homepage: http://www.cse.cuhk.edu.hk/~chi/csc2110/ • Newsgroup: cuhk.cse.csc2110. at news.cse.cuhk.edu.hk • Instructor: Lau, Lap Chi • Office hour: W7 (SHB 911) • Lectures: M7-8 (ERB LT), W6 (TYW LT) • Tutors: Chan Yuk Hei, Tom Fung Wai Shing, Isaac Yung Chun Kong, Darek Zhang Zixi, Jesse • Tutorials: H5 (ERB 404) or H6 (EGB 404)

3. Course Material • Textbook: Discrete Mathematics with Applications (DMA) Author: Susanna S. Epp Publisher: • Reference: Course notes from “mathematics for computer science” http://courses.csail.mit.edu/6.042/spring07/

4. Course Requirements • Homework, 20% • Midterm, 30% • Course project, 10% • Final Exam, 40% Midterm: Oct 27 (Monday), M7-8

5. Course Project Pick an interesting mathematical topic, write a report from 5 to 10 pages. 4 students in a group Can use any references, but cite them. Choose 3 groups to present, up to 5% bonus

6. A Project Tell an interesting story related to mathematics. More about good topic and nice presentation, than mathematical difficulty. • Interesting or curious problems, interesting history • Surprising or elegant solutions • Nice presentation, easy to understand

7. Checker x=0 Start with any configuration with all men on or below the x-axis.

8. Checker x=0 Move: jump through your adjacent neighbour, but then your neighbour will disappear.

9. Checker x=0 Move: jump through your adjacent neighbour, but then your neighbour will disappear.

10. Checker x=0 Goal: Find an initial configuration with least number of men to jump up to level k.

11. K=1 x=0 2 men.

12. K=2 x=0

13. K=2 x=0 Now we have reduced to the k=1 configuration, but one level higher. 4 men.

14. K=3 x=0 This is the configuration for k=2, so jump two level higher.

15. K=3 x=0 8 men.

16. K=4 x=0

17. K=4 x=0

18. K=4 x=0

19. K=4 x=0

20. K=4 x=0 Now we have reduced to the k=3 configuration, but one level higher 20 men!

21. K=5 • 39 or below • 40-50 men • 51-70 men • 71- 100 men • 101 – 1000 men • 1001 or above

22. Example 1 How to play Rubik Cube? Google: Rubik cube in 26 steps http://blog.sciencenews.org/mathtrek/2007/08/cracking_the_cube.html

23. Example 2 The mathematics of paper folding http://www.ushistory.org/betsy/flagstar.html http://erikdemaine.org/foldcut/

24. Example 3 3D-images http://128.100.68.6/~drorbn/papers/PDI/

25. Project Ideas • Magic tricks • More games, more paper folding, etc • Famous paradoxes • Prime numbers • Game theory http://www.cse.cuhk.edu.hk/~chi/csc2110/project.html Deadline: November 17.

26. Why Mathematics? Design efficient computer systems. • How did Google manage to build a fast search engine? • What is the foundation of internet security? algorithms, data structures, database, parallel computing, distributed systems, cryptography, computer networks… Logic, number theory, counting, graph theory…

27. Topic 1: Logic and Proofs How do computers think? Logic: propositional logic, first order logic Proof: induction, contradiction Artificial intelligence, database, circuit, algorithms

28. Topic 2: Number Theory • Number sequence • Euclidean algorithm • Prime number • Modular arithmetic Cryptography, coding theory, data structures

29. Topic 3: Counting • Sets • Combinations, Permutations, Binomial theorem • Functions • Counting by mapping, pigeonhole principle • Recursions, generating functions Probability, algorithms, data structures

30. Topic 3: Counting How many steps are needed to sort n numbers?

31. Topic 4: Graph Theory • Relations, graphs • Degree sequence, isomorphism, Eulerian graphs • Trees Computer networks, circuit design, data structures

32. What is discrete mathematics? Logic: artificial intelligence (AI), database, circuit design Number theory: cryptography, coding theory Counting: probability, analysis of algorithm Graph theory: computer network, data structures logic, sets, functions, relations, etc CSC 2100, ERG 2040, CSC 3130, CSC 3160

33. c b a Pythagorean theorem Familiar? Obvious?

34. Good Proof c b a Rearrange into: (i) a cc square, and then (ii) an aa & a bb square

35. Good Proof c b-a c c a b c

36. b-a b-a Good Proof c b a

37. Good Proof a b a a b-a b 74 proofs in http://www.cut-the-knot.org/pythagoras/index.shtml

38. Bad Proof

39. Statement (Proposition) Statement is either True or False True 2 + 2 = 4 Examples: False 3 x 3 = 8 787009911 is a prime Non-examples: Hello. How are you?

40. P P Q Q P Q P Q Logic Operators

41. Compound Statement p = “it is hot” q = “it is sunny” It is hot and sunny It is not hot but sunny It is neither hot nor sunny

42. Exclusive-Or  exclusive-or coffee “or” tea How to construct a compound statement for exclusive-or?

43. Check by Truth Table

44. Logical Equivalence Two statements have the same truth table De Morgan’s Law De Morgan’s Law

45. Simplifying Statement A tautology is a statement that is always true. A contradiction is a statement that is always false.

46. Two Important Things http://appsrv.cse.cuhk.edu.hk/~acmprog/web2008/ Class Photos! Identify your face and send us your name and nicknames