Lattice Introduction Discrete Mathematics.ppt
- 1. Discrete Structures for Computer Science Ruoming Jin MW 2:15 – 3:00pm Spring 2010 rm MSB115
- 2. Course Material Textbook: Discrete Mathematics and Its Applications Kenneth H. Rosen, McGraw Hill
- 3. Course Requirements Homework, 20% Quiz, 20% Three Intermediate Exams: 10% Final Exam, 30% Bonus Questions 5-10%
- 4. Why Discrete Math? 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, sets/functions, counting, graph theory…
- 5. What is discrete mathematics? logic, sets, functions, relations, etc Logic: artificial intelligence (AI), database, circuit design Counting: probability, analysis of algorithm Graph theory: computer network, data structures Number theory: cryptography, coding theory
- 6. Topic 1: Logic and Proofs Logic: propositional logic, first order logic Proof: induction, contradiction How do computers think? Artificial intelligence, database, circuit, algorithms
- 7. Topic 2: Counting • Sets • Combinations, Permutations, Binomial theorem • Functions • Counting by mapping, pigeonhole principle • Recursions, generating functions Probability, algorithms, data structures
- 8. Topic 2: Counting How many steps are needed to sort n numbers?
- 9. Topic 3: Graph Theory • Relations, graphs • Degree sequence, isomorphism, Eulerian graphs • Trees Computer networks, circuit design, data structures
- 10. Topic 4: Number Theory • Number sequence • Euclidean algorithm • Prime number • Modular arithmetic Cryptography, coding theory, data structures
- 11. 2 2 2 a b c Familiar? Obvious? c b a Pythagorean theorem
- 12. c b a Rearrange into: (i) a cc square, and then (ii) an aa & a bb square Good Proof
- 13. c c c a b c b - a Good Proof 81 proofs in http://www.cut-the-knot.org/pythagoras/index.shtml