Topics Covered

Remark: topics covered in each lecture are subject to change.

  1. Wed, Jan 18, 06: Setion 1.1: sets, sequences, multisets, empty set, substes, equal sets, belongs.

  2. Fri, Jan 20, 06: Section 1.1 contin: proper sets, sets of integers, rationals, natural numbers, irrationals, and real numbers, intervals, positive part and negative part of a set A (i.e. A+ and A-); begin Section 1.2: statements (propositions), formulas, simple statements, compound statements, logical connectives, truth tables, and, or, not, if then (implies).

  3. Mon, Jan 23, 05: Section 1.2 contin: implies contin, converse, contrapositive, inverse, negation of "if then", biconditional (if and only if - iff), negation of iff, examples.

  4. Tuesday, Jan 24, 05: Section 1.2 contin: converse, inverse, more examples; begin Section 1.3 (logical equivalence): definition of logical equivalence, tautology, contradiction, theorems, examples.

  5. Wed, Jan 25, 05: Section 1.3 contin, begin Section 1.4 (existential and universal quantifiers: definition of "there exists" and "for all", other expressions used for them, their negations, examples.

  6. Fri, Jan 27, 05: Section 1.4 contin.

  7. Mon, Jan 30, 06: Section 1.5 and related material (Operations on Sets): union, intersection, difference set (relative complement), symmetric difference, Cartesian product, universal set, complement, power set, examples involving discrete and continuous sets.

  8. Tues, Jan 30, 06: Section 1.5 Contin: a+S, a-S, aS, where S is a set, disjoint, notation for union/intersection/Cartesian prodcut of finite sets, pairwise disjoint, partition, cardinality, cardinality of product of finite sets, cardinality of the union of pairwise disjoint sets, related theorems, examples.

  9. Wed, Feb 1, 06: Section 1.6 and related material (Proof Techniques): Direct proofs os general statements and of p -> q, proof by counterexample to show that statements of the form "forall x in S, t(x)" are false (as a special case "forall x in S, p(x) -> q(x))", examples.

  10. Fri, Feb 3, 06: Section 1.6 contin: proof by contrapositive of statements of the form p -> q, proof by contradication.

  11. Mon, Feb 6, 06: Quiz, proof by cases, proof of "if and only if", proof of "the following are equivalent", examples and exercises.

  12. Tues, Feb 7, 06: more exercises, begin Section 1.7 (Proof by the weak/strong form of Mathematical Induction).

  13. Wed, Feb 8, 06: Section 1.7 Contin.

  14. Fri, Feb 10, 06: Finish 1.7, strong form of Mathematical Induction.

  15. Mon, Feb 13, 06: Begin Chapter 2 of Straight: Sections 1 and 2 and related material (Principle of Well-Ordering and the Division Algorithm): quotient and remainder and how to find them, floor and ceil. functions.

  16. Tues, Feb 14, 06: Section 3 of Chapter 2 (The Euclidean Algorithm): the greatest common divisor, the least common multiple, how to find them, and how to write the greatest common divisor as a linear combination of the two numbers, relatively prime.

  17. Wed, Feb 15, 06: Section 3 contin., theorems, begin Section 2.4 (prime numbers).

  18. Fri, Feb 17, 06: Section 2.4 (prime numbers), prime numbers, composite numbers, Fundamental Theorem of Arithmetic, Canonical Factorization, theorems; begin Section 2.5 (Integers modulo n).

  19. Mon, Feb 20, 06: Go over homework, Section 2.5 continued, addition and multiplication in Zn, negatives in Zn, inverses in Zn, when the inverse exists and how to find it, theorems.

  20. Tues, Feb 21, 06: Quiz 2, solution of the quiz, Section 2.5 continued, examples.

  21. Wed, Feb 22, 06: Group exercises on 2.5; begin Chapter 3 of Straight (Matrices), definition of matrices, vectors, square matrices, zero matrix.

  22. Fri, Feb 24, 06: Operations on vectors (addition, scalar multiplication, dot/inner product, addition of matrices, scalar multiplication of matrices, theorems.

  23. Mon, Feb 27, 06: Multiplication of matrices, theorems.

  24. Tuesday, Feb 28, 06: Exam I.

  25. Wed, Mar 1, 06: Special matrices (lower-triangular, upper-triangular, diagonal, identity matrix), definition of inverse and theorems about it.

  26. Fri, Mar 3, 06: transpose, symmetric matrices, skew-symmetric matrices, writing a matrix as a sum of a symmetric matrix and a skew-symmetric matrix, theorems, powers of matrices, theorems.

  27. Mon, Mar 6, 06: Determinants, singular and non-singular matrices.

  28. Tues, Mar 7, 06: How to find the inverse.

  29. Wed, Mar 8, 06: Inverse continued, using the inverse to solve linear systems.

  30. Fri, Mar 10, 06: Section 2.3 of Goodaire - Binary Relations, reflexive, symmetric, antisymmetric, transitive, equivalence, partial orders.

  31. Mon, Mar 13, 06: Examples, equivalence classes, partitions.

  32. Wed, Mar 15, 06:

  33. Fri, Mar 17, 06: Exercises, representation of a binary relation as a matrix and as a digraph, compirable and imcomapribale elements, total orders.

  34. Mon, Mar 20, 06: Quiz, total orders continued, inverses and composition of binary relations.

  35. Tues, Mar 21, 06: Exercises.

  36. Wed, Mar 22, 06: Begin functions (Chapter 3 of Goodaire); domain, target, image, preimage, image of a set, preimage of a set.

  37. Mon, Apr 3, 06: One-to-one (injective) functions.

  38. Tues, Apr 4, 06: Onto (surjective) functions and bijections (one-to-one correspondence).

  39. Wed, Apr 5, 06: The identity function, function inverse.

  40. Fri, Apr 7, 06: More examples, composition of functions.

  41. Mon, Apr 10, 06: Composition of functions continued.

  42. Tuesday, Apr 11, 06: Cardinality, countable and uncountable sets, countably infinite sets.

  43. Wednesday, Apr 12, 06: Continue.

  44. Friday, Apr 14, 06: Continue.

  45. Tues, Apr 18, 06: Sections 6.1 (Principle of Inclusion-Exclusion)

  46. Wed, Apr 19, 06: Section 6.2 (Addition and Multiplication Principles/Rules)

  47. Fri, Apr 21, 06: Section 7.1 (Permutations).

  48. Mon, Apr 24, 06: Section 7.2 (Combinations).

  49. Tues, Apr 25, 06: Exam 2 (Material: Matrices, binary relations, functions)

  50. Wed, Apr 26, 06: Finish combinations; repetitions (Section 7.3).

  51. Fri, Apr 28, 06: Repeitions contin.

  52. Mon, Apr 1, 06: Pigenonhole principle.

  53. Wed, Apr 3, 06: Derangements.

  54. Fri, Apr 5, 06: Derangements contin, Binomial Theorem. Then linear transformations.