Class Log
- 1. Wednesday, 8/24:
Handed out handouts and basic info about the class.
Introduction, sets (Section 1.6). Set operations (Section 1.7).
- 2. Friday, 8/26:
Quantifiers (Section 1.6).
Set operations, proofs of set identities (Section 1.7).
Started functions (Section 1.8).
- 3. Monday, 8/29:
Functions (Section 1.8). The notion of algorithm, an example.
Greedy algorithms (Section 2.1).
- 4. Wednesday, 8/31:
Sorting algorithms (Section 2.1), Merge sort (from Section 3.5)
and Tournament sort (Page 657).
- 5. Friday, 9/2:
Quiz 1. Big-Oh notation and growth of functions (Section 2.2).
- 6. Wednesday, 9/7:
Sequences and summation.
Formulas and examples for sums of arithmetic and geometric
sequences. (Section 3.2). Started mathematical induction.
- 7. Friday, 9/9:
Quiz 2. Mathematical induction (Section 3.3).
- 8. Monday, 9/12:
More on mathematical induction (Section 3.3).
Started counting: Sum Rule, Product Rule, the number of functions
(Section 4.1).
- 9. Wednesday, 9/14:
More on counting functions. Counting complements. The simplest form
of Inclusion-Exclusion Principle with examples(Section 4.1).
Pigeonhole Principle (Section 4.2).
- 10. Friday, 9/16:
Quiz 3. Pigeonhole Principle (Section 4.2).
- 11. Monday, 9/19:
Ramsey's Theorem (Section 4.2). r-Permutations and r-combinations,
binomial coefficients (Section 4.3).
The Binomial Theorem and some corollaries (Section 4.4).
- 12. Wednesday, 9/21:
Quiz 4. Pascal Identity and
Triangle. Vandermonde's identity (Section 4.4).
Permutations with repetitions (Section 4.5).
- 13. Friday, 9/23:
Test 1.
- 14. Monday, 9/26:
Discussion of test problems.
Combinations with repetitions (Section 4.5).
- 15. Wednesday, 9/28:
More on combinations with repetitions (Section 4.5).
Permutaions of groups of indistinguishable objects; distributing
objects into boxes (Section 4.5).
- 16. Friday, 9/30: Quiz 5.
Probability for finite sample spaces: definitions and
examples (Sections 5.2 and 5.1).
- 17. Monday, 10/3:
On Problem 6 from HW5.
Properties of probabilities. Conditional probabilities, independent
events (Section 5.2).
- 18. Wednesday, 10/5:
More on independent events. Bernully trials.
random variables, expected values
of random variables (Sections 5.2 and 5.3).
The example with Fibonacci rabbits (Section 6.1).
- 19. Friday, 10/7:
Quiz 6.
Linearity of expectations. Examples (Section 5.3).
Modelling with recurrence relations, Fibonacci numbers (Section 6.1).
- 20. Monday, 10/10:
Solving linear homogeneous recurrence relations with
constant coefficients of the second order (Section 6.2).
General theorem on linear homogeneous recurrence relations with
constant coefficients. Started nonhomogeneous relations (Section 6.2).
- 21. Wednesday, 10/12:
Solving nonhomogeneous recurrence relations with
constant coefficients (Section 6.2). Examples.
- 22. Friday, 10/14:
Quiz 7. Inclusion-Exclusion Principle. Applications of the
Inclusion-Exclusion Principle:
the number of onto functions and derangements (Section 6.5).
- 23. Monday, 10/17:
More on the
Inclusion-Exclusion Principle: two examples (Section 6.5).
Relations: definitions and examples (Section 7.1).
- 24. Wednesday, 10/19:
Test 2.
- 25. Friday, 10/21:
Quiz 8. Discussion of Test 2 problems.
- 26. Monday, 10/24:
Relations and their properties. Representing relations.
Equivalence relations (Sections 7.1, 7.3, 7.5).
- 27. Wednesday, 10/26:
More examples of solutions of recurrence relations (Section 6.2).
Calculating
the number of reflexive and antisymmetric relations on a set of
given cardinality (Sections 7.1, 7.3). Started graphs (Section 8.1).
- 28. Friday, 10/28:
Quiz 9.
Graphs: basic definitions, types of graphs, graph models,
degrees, Handshake Lemma, representing graphs (Sections 8.1-8.3).
- 29. Monday, 10/31:
More on graphs: connectivity, isomorphism.
Euler circuits, Euler's Theorem. Hamiltonian cycles (Sections 8.3,8.5).
- 30. Wednesday, 11/2:
More on Hamiltonian cycles.
Traveling Salesman Problem (Section 8.3). The Shortest
Path Problem - Dijkstra Algorithm (Section 8.6).
- 31. Friday, 11/4:
Finished Dijkstra Algorithm (Section 8.6). Floyd-Warshall
algorithm for finding a shortest path (Pages 602-603 and a handout).
- 32. Monday, 11/7:
Finished the Shortest Path Problem (Section 8.6).
Trees, characterizations of trees (Section 9.1).
Planar and plane graphs: stated Euler's Formula (Section 8.7).
- 33. Wednesday, 11/9:
Proof of Euler's Formula. Applications. Proved that K_5 and
K(3,3) are not planar. Stated
Kuratowski's Theorem (Section 8.7).
- 34. Friday, 11/11:
Quiz 11. More on planar graphs.
Graph colorings and
their applications.
- 35. Monday, 11/14:
Greedy colorings. Four Color Theorem.
Proof of 5-Color Theorem.
- 36. Tuesday, 11/15:
Test 3 at 5pm.
- 37. Wednesday, 11/16:
Discussion of Test 3. Rooted trees, m-ary trees and their properties
and applications. Balanced trees and their properties.
- 38. Monday, 11/28:
Decision trees. Trees and prefix codes. Started Huffman codes.
- 39. Wednesday, 11/30:
Huffman codes. Spanning trees. Minimum spanning trees and
their properties.
Prim's and Kruskal's algorithms for finding minimum
spanning trees.
- 40. Friday, 12/2:
Make-up Test 3.
- 41. Monday, 12/05:
A discussion of Test 3 (make-up) and HW12. Quiz 12.
Stable matchings. The Gale-Shapley algorithm for finding
stable matchings.
- 42. Wednesday, 12/07:
Test 4.
Last changed on December 7, 2005.