CPSC 202: Mathematical Tools for Computer Science

Lectures

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  1. September 3, 2009: Introduction to descrete math, course schedule and requirements.
  2. September 8, 2009: Logical propositions, logical operators, operator precedence, truth tables, de Morgan's laws, CNF, DNF, start on resolution proofs.
  3. September 10, 2009: Precedence order of XOR, resolution proofs, predicates, quantifiers, negation of quantified expressions, and proofs that use instantiation, generalization, simplification, and conjunction
  4. September 15, 2009: Exercise 23 in Section 1.5; Sets, subsets, power sets, Venn diagrams, union, intersection, complement, difference, symmetric difference; Proof that, for finite sets A, B, and C, if the symmetric difference of A and B equals the symmetric difference of A and C, then B = C
  5. September 17, 2009: Proof that, for arbitrary sets A, B, and C, if the symmetric difference of A and B equals the symmetric difference of A and C, then B = C; Review of proofs by induction: The size of the power set of S, where |S| = n, is 2^n; Cartesian products of sets; Introduction to functions: domain, co-domain, image, preimage, range, injections, surjections, bijections, composition of functions, sequences, geometric progressions, geometric series
  6. September 22, 2009: Arithmetic progressions, divisibility, primes and composites, greatest common divisors, least common multiples, modular arithmetic, Euclidean algorithm
  7. September 24, 2009: Inverses mod m, Chinese Remainder Theorem; Arithmetic mod p, quadratic residues and non-residues, Wilson's Theorem
  8. September 29, 2009: Working through Wilson's Theorem for p=11; using the Euclidean Algorithm to find inverses mod m; introduction to matrix arithmetic
  9. October 1, 2009: Gaussian elimination, matrix inverses, matrix transposes. Jim Aspnes' lecture notes: http://pine.cs.yale.edu/pinewiki/LinearAlgebra
  10. October 6, 2009: Proofs by induction, Harmonic numbers, Fibonacci numbers
  11. October 8, 2009: Recursive definitions of sets, functions, and structures; Using structural induction to prove properties of recursively defined objects
  12. October 13, 2009: Review for Exam 1
  13. October 15, 2009: [Exam1] | [Solutions]
  14. October 20, 2009: Using the product rule and the pigeonhole principle to solve counting problems
  15. October 22, 2009: Binomial coefficients; Introduction to probability: sample spaces, events, probabilities, conditional probabilities, independence
  16. October 27, 2009: Conditional probability, Independence, Binomial distribution, Random Variables
  17. October 29, 2009: Bayes' Theorem, Expected value, Variance
  18. November 3, 2009: Linearity of expectation, Chebyshev's inequality, Markov's inequality, Examples; Introduction to recurrence relations
  19. November 5, 2009: Calculating the expected number of comparisons in an insertion sort. Modeling with recurrence relations, divide-and-conquer algorithms, solving recurrence relations using charateristic polynomials and initial conditions.
  20. November 10, 2009: Generating functions.
  21. November 12, 2009: Using generating functions to prove combinatorial identities, model counting problems, and solve recurrence relations; Introduction to Graphs: degrees, Handshake Theorem, adjacency matrices, paths, cycles, wheels, hypercubes, cliques, and bipartite graphs
  22. November 17, 2009: Euler circuits and the bridges of Konisberg; Hamiltonian circuits and Gray codes; Shortest paths and Dijkstra's algorithm