Suary for lectures: MCS 361 Discrete Matheatics, Fall 2016 36568 LCD - undergrad MWF 1:00-1:50, TH 309 Instructor: Shuel Friedland Office: 715 SEO, phone: 413-2150, e:ail: friedlan@uic.edu web: //www.ath.uic.edu/ friedlan Last update Deceber 1, 2016 All the references are fro the book [1] 1 Week 1: 8/22-8/29, 2016 1.1 August 22, 2016 Did : 1.1, 1.3, 1.4. 1.2 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, 2016 2.1 August 29 Finished 2.1 and 2.2. 2.2 August 31 Finished 2.3. Gave quiz 1. 2.3 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
3 Week 3: 9/7-9/9, 2016 3.1 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 4. 3.3: Ex1., Ex.2 (explained Algo. 2 in 3.1), Ex. 4, Matrix Multiplication, Strassen algo https://en.wikipedia.org/wiki/strassen 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, 2016 4.1 Septeber 12 Finished 4.1. 4.2: p 257-260. 4.2 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, 2016 5.1 Septeber 19 Stated and proved Ferat s little theore by proving Newton s theore (x + 1) = x + x 1 +... + x k +... + x + 1. 1 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 12. 5.2 Septeber 21 4.5: Hashing functions, Exaple 2; Check Digits: Exaple 4, Exaple 6. Started 4.6 5.3 Septeber 23 Finished 4.6. Started 5.1. Did Exaple 1. Discussed Why Matheatical Induction is Valid, on page 314. Gave quiz 4. 2
6 Week 6: 9/26-9/30, 2016 6.1 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 5.2. 6.2 Septeber 28 5.3: Exaples 1, 4, Theore 1, Definition 1, Definition 3. 6.3 Septeber 30 6.1: 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 555. 7 Week 7: 10/3-10/7, 2016 7.1 October 3 Finished 6.1. 6.2: The pigeonhole principle:theore 1, Corollary 1. 7.2 October 5 Continued 6.2: Exaple 1, 3, 4; Theore 2; Exaple 6; Exaple 10; Exaple 11. Started Exaple 13. 7.3 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, 2016 8.1 October 10 Fisihed 6.4. Started 6.6: Exaple 1, Exaple 2. Explained Algorith 1. 8.2 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 1 7. 9 Week 9: 10/17-10/21, 2016 9.1 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
9.2 October 19 Continued 7.2: Exaples 5, 6, 8, 9, 10, DEFINITION 5, 6, Theore 8. 9.3 October 21 Discussed Exaple 13 and 14. Proved the forula n 1.1774 as stated in the first paragraph on page 463. See Product forula on y website: http://hoepages.ath.uic.edu/ friedlan/productestoct16.pdf Started to disucss Monte Carlo Algoriths. Did Exaple 15. 10 Week 10: 10/24-10/28, 2016 10.1 October 24 Finished 7.2: Did Exaple 16. Explained The Probabilistic Method. Stated Theore 4 and discussed its proof. Gave Quiz 6. 10.2 October 26 8.1: Exaples 1-5. 10.3 October 28 8.2: Definition 1, Theore 1, Exaple 4, Theore 3, Theore 2, Exaple 5. 11 Week 11: 10/31-11/4, 2016 11.1 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 10. 11.2 Noveber 2 Finished 8.2: Did Exaples 10, 11, 13. Started 10.1: Definition 1, Explained Figure 1. Gave quiz 7. 11.3 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 4. 12 Week 12: 11/7-11/11, 2016 12.1 Noveber 7 Continued 10.2: Definition 5, Theore 3, Exaple 8, Definition 6, Exaple 9, Exaple 11, Theore 4, Exaple 12, Matching, Exaple 14, 15. 12.2 Noveber 9 Continued 10.2: Proved Theore 5, Definition 7, 8, Exaple 18. 4
12.3 Noveber 11 Finished 10.2. Started 10.3: Representing graphs, adjacency atrices, incidence atrices. Gave quiz quiz 8. 13 Week 13: 11/14-11/18, 2016 13.1 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, 11. 13.2 Noveber 16 Started 10.4. Did Definitions 1, 2, 3, Exaples: 1, 2, 3, 4, 5, 7, 8. 13.3 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, 11. 14 Week 14: 11/21-11/23, 2016 14.1 Noveber 21 Finished 10.4 Gave quiz 9. 14.2 Noveber 23 10.5: Euler Path and Circuits, Theore 1, 2. Hailtonian paths and Circuits, Definition 1, Gray code. 15 Week 15: 11/28-12/2, 2016 15.1 Noveber 28 11.1: Did pages 745-753, up to included Theore 4. 15.2 Noveber 30 Continued 11.1: Finished 11.1. Started 11.2: Exaple 1. References [1] Kenneth H. Rosen, Discrete Matheatics and Its Applications, 7th Edition, McGraw-Hill, 2012. 5