Summary for lectures: MCS 361

Transcription

1 Suary for lectures: MCS 361 Discrete Matheatics, Fall LCD - undergrad MWF 1:00-1:50, TH 309 Instructor: Shuel Friedland Office: 715 SEO, phone: , e:ail: friedlan@uic.edu web: // friedlan Last update Deceber 1, 2016 All the references are fro the book [1] 1 Week 1: 8/22-8/29, August 22, 2016 Did : 1.1, 1.3, August 24, 2016 Did 1.5. Started 1.6. explained Rsolution Principle. 1.3 August 26, 2016 Did Methods of Proving in 1.7: Ex. 1; proved by contradiction that there is an infinite nuber of pries; Ex. 7; Ex. 13; Ex. 15. Started 2.1: did Definition 1; Ex. 1 2 Week 2: 8/29-9/2, August 29 Finished 2.1 and August 31 Finished 2.3. Gave quiz Septeber 2 3.1: Definition 1, Exaple 1, Algorith 1; Buble Sort, Algorith 4; Greedy Algoriths, Algorith 6, Lea 1, Theore 1; The Halting Proble - explained it and told about the exsitence of unsolvable probles. 1

2 3 Week 3: 9/7-9/9, Septeber 7 3.2: Def.1, Ex. 1, Ex. 3, Th 1, Ex. 5, Ex. 6, Ex. 7, Th 2, Cor. 1, Th. 3, Def. 2, Def. 3, Ex. 12, Th : Ex1., Ex.2 (explained Algo. 2 in 3.1), Ex. 4, Matrix Multiplication, Strassen algo algorith 3.2 Septeber 9 3.3: Exaple 10, Coplexity of algoriths. Started 4.1. Showed Th 1 and Corol. 1. Gave quiz 2. 4 Week 4: 9/12-9/16, Septeber 12 Finished : p Septeber 14 Finished 4.2 and 4.3. For an integer > 1 let φ() be the Euler function: nuber of integers in [] = {1,..., } that are coprie with. Assue that = p a1 1 pa k k, where 1 < p 1 < < p k are pries and a 1,..., a k are positive integers. Then 4.3 Septeber 16 φ() = p a1 1 1 (p 1 1) p a k 1 k (p k 1). 4.4: Theore 1, Exaple 2, Theore 2. Gave quiz 3. 5 Week 5: 9/19-9/23, Septeber 19 Stated and proved Ferat s little theore by proving Newton s theore (x + 1) = x + x x k x k 1 Here l = ( 1) ( l+1) l!. If prie and l is an integer satisying 1 l < then l is divisible by. Thus for an integer y 1 and p prie we have y p = ((y 1)+1) p (y 1) p +1 od (p). Therefore for any positive integer y we have that y p y od (p). That is, y(y p 1 1) = y p y divisible by p if y is not divisible by p it follows that y p 1 1 is divisible by p, which is Ferat s little theore. Did Exaple 9. Discussed pseudopries. Defined priitive root. Did Exaple Septeber : Hashing functions, Exaple 2; Check Digits: Exaple 4, Exaple 6. Started Septeber 23 Finished 4.6. Started 5.1. Did Exaple 1. Discussed Why Matheatical Induction is Valid, on page 314. Gave quiz 4. 2

3 6 Week 6: 9/26-9/30, Septeber 26 Did Exaple 9 in 5.1. Explained the strong induction and well-ordering in 5.2. Proved Theore 1 and Lea 1 ib Septeber : Exaples 1, 4, Theore 1, Definition 1, Definition Septeber : Product rule, Exaples 4, 5,6,7, 10, 11; The su rule, Ex. 16, Inclusion-exclusion forula, Theore 1, page 556, for two sets page 393, for three sets page Week 7: 10/3-10/7, October 3 Finished : The pigeonhole principle:theore 1, Corollary October 5 Continued 6.2: Exaple 1, 3, 4; Theore 2; Exaple 6; Exaple 10; Exaple 11. Started Exaple October 7 Finished the discussion about Rasey nubers in 6.2. Finished 6.3. Started 6.4: Proved Theore 1. 8 Week 8: 10/10-10/14, October 10 Fisihed 6.4. Started 6.6: Exaple 1, Exaple 2. Explained Algorith October 12 Gave the idter. 8.3 October 14 Finished 6.6: Generating Cobinations; Exaple 4, Algorith 2, Exaple 5. Started 7.1: Definition 1, Exaples Week 9: 10/17-10/21, October 17 Finished 7.1: Theore 1, Theore 2, Exaple 10. Started 7.2: Exaple 1, Definitios 1, 2, Exaple 2, Theore 1, Definition 3, Definition 4. 3

4 9.2 October 19 Continued 7.2: Exaples 5, 6, 8, 9, 10, DEFINITION 5, 6, Theore October 21 Discussed Exaple 13 and 14. Proved the forula n as stated in the first paragraph on page 463. See Product forula on y website: friedlan/productestoct16.pdf Started to disucss Monte Carlo Algoriths. Did Exaple Week 10: 10/24-10/28, October 24 Finished 7.2: Did Exaple 16. Explained The Probabilistic Method. Stated Theore 4 and discussed its proof. Gave Quiz October : Exaples October : Definition 1, Theore 1, Exaple 4, Theore 3, Theore 2, Exaple Week 11: 10/31-11/4, October 31 Gave back quiz 6 and solved it in the class. Continued 8.2: Stated Theore 4. Did Exaple 8. Stated Theore 5. Started Exaple Noveber 2 Finished 8.2: Did Exaples 10, 11, 13. Started 10.1: Definition 1, Explained Figure 1. Gave quiz Noveber 4 Finished 10.1: Discussed ultigraphs, pseudographs, digraphs, ultigraphs. Graph odels: Exaple 1, Exaple 3, Exaple 5. Started 10.2: Definition 1, Definition 2, Theore 1, Exaple 3, Theore 2, Definition Week 12: 11/7-11/11, Noveber 7 Continued 10.2: Definition 5, Theore 3, Exaple 8, Definition 6, Exaple 9, Exaple 11, Theore 4, Exaple 12, Matching, Exaple 14, Noveber 9 Continued 10.2: Proved Theore 5, Definition 7, 8, Exaple 18. 4

5 12.3 Noveber 11 Finished Started 10.3: Representing graphs, adjacency atrices, incidence atrices. Gave quiz quiz Week 13: 11/14-11/18, Noveber 14 Continued 10.3: Discussed isoorphis of graphs. Graph invariants: nuber of vertices, edges; the degree sequence; the length of cycles; in particular the length of the shortest cycle; the eigenvalues of the graph: the eigenvalues of the adjacency atrix (independent of renaing the vertices). Did Exaples 9, 10, Noveber 16 Started Did Definitions 1, 2, 3, Exaples: 1, 2, 3, 4, 5, 7, Noveber 18 Continued 10.4: Exaple 9, Proved the inequality κ(g) λ(g) in v V deg(v). Discussed connectedness in directed graphs: Definition 5, Exaples 10, Week 14: 11/21-11/23, Noveber 21 Finished 10.4 Gave quiz Noveber : Euler Path and Circuits, Theore 1, 2. Hailtonian paths and Circuits, Definition 1, Gray code. 15 Week 15: 11/28-12/2, Noveber : Did pages , up to included Theore Noveber 30 Continued 11.1: Finished Started 11.2: Exaple 1. References [1] Kenneth H. Rosen, Discrete Matheatics and Its Applications, 7th Edition, McGraw-Hill,