CMSC-203 Discrete Math Lectures (spring 2001--Section 0101, Alan Sherman)

Lectures: CMSC-203 Discrete Math (Spring 2001)

30 lectures + final exam (Mon/Wed 2:00-3:30pm in SS 209, Jan 29-May 16). (*) denotes primary example of lecture. Most effective ways to learn through lectures are (1) to ask thoughtful questions based on home problem-solving, and (2) to thoroughly master primary example to extent that student can reproduce (and preferably also adapt, modify, and extend) complete example in technical detail without referring to any notes. Each class begins with student questions and, if requested, discussion of current homework and readings. Readings from text follow lecture topics.

Course is organized in three major parts as follows:
Proofs: 10 lectures (including 5 on induction), including proving correctness of programs
Calculation: 9 lectures, including calculating running times of iterative and recursive programs
Counting: 8 lectures, including counting number of objects (functions, data structures)
Tests: 3 + final exam

• 1. Introduction. Course goals: proving theorems, calculating, counting, problem-solving, effective communication. Motivating examples: counting Boolean functions and essentially different Boolean functions; running time of looping and recursive programs; proofs for understanding, verification of truth, research, communication of understanding (contrast with zero-knowledge proofs). Basic undefined terms. Construction of the natural numbers (*). Introduction to first-order predicate logic. Universal and existential quantifiers. Sound and unsound arguments: all cars are red (*). Administrivia. Introduce instructor and students.

• 2. Proof by counterexample: power set of union of sets (*). Construction of numbers: integers, rational numbers, real numbers, complex numbers, extended real numbers (*). Fundamental notations from set theory and logic (conjunction, disjunction, power set, complement, Cartesian product. set difference). Ordinal, cardinal, and transfinite numbers.

• 3. Direct proof: apply definitions, case analysis, subset, set equality, biconditional, circle of implications. Examples: power set of intersection, Divisibility: divisibility of sum (*). Numerical and logical mod operators. Modus ponens, modus tolens.

• 4. Indirect proof and contraposition. Examples: n^2 even implies n even; sqrt(2) is irrational (*). Logic of conditionals. Converse, contrapositive, inverse, negation. Example: the sum of any two even integers is even (*). Demorgan's Law. [HW1 due]

• 5. Induction (strong and weak forms). Well Ordering Principle. Examples: formulae for arithmetic series and Fibonacci numbers (*).

• 6. Practice quiz in small groups: an inequality by Induction (*). Divisibility. [HW 2 due]

• 7. Induction and loop invariants. Example: WOP => Induction (*). Example: correctness of an iterative version of the repeated squaring algorithm (*).

• 8. Practice quiz in small groups: Strong induction and proving correctness of recursive programs. Example: correctness of a divide-an-conquer maximum program (*). [HW 3 due]

• 9. Epsilon-delta proofs and alternating quantifiers. Definitions of limit of sequence and Big-Oh notation. Formula for geometric summation (*).

• 10. Practice quiz in small groups: correctness of an iterative max program (*). [HW 4 due]

• 11. Begin calculation. Summation notation. Summations as recurrences, and recurrences as summations. Solving first-order linear recurrences by iteration and characteristic equations (*). What is calculation? (equation solving, simplification, canonical forms, approximations, symbolic vs. numerical).

• 12. Exam I on proofs.

• 13. Running time of a looping program (*). Five common summations: constant, arithmetic, geometric, harmonic, telescoping.

• 14. Running time of a recursive program (*). [HW 5 due] Solve recurrence four ways: iteration, characteristic equations, reasoning from recursion tree, guess and check by induction.

• spring break

• 15. Introduction to Maple. (schedule earlier if possible) Computer demonstration.

• 16. Characteristic equations: multiple roots and complex roots (*). Transformation from Cartesian to polar coordinates. Expressing periodic solutions using the elementary functions sin and cos. [HW 6 due]

• 17. Functions and relations. "Latin terms" injection, surjection, bijection, permutation. Example: state transition function of Turing machine. Visual graphs vs. combinatorial graphs. Examples of interesting functions: RSA, hash functions, popcorn function, random functions, binary functions, Boolean functions. Interesting things to do with functions: composition, inversion, lifting, variable length arguments.

• 18. Equivalence and order relations. Equivalency classes and their representatives. Transitive closure. Reachability in graphs. Coloring the vertices of a triangle. [HW 7 due]

• 19. More on recurrences. The method of transformation. Systems of recurrences.

• 20. Solving linear Diophantine equations. [HW 8 due]

• 21. Begin counting: fundamental principles of counting-addition and product rules. Urn model (first three cases). Rolling dice (*). Probability.

• 22. Exam II on calculation

• 23. Urn model (D'Alembert's counting method). Counting solutions to Diophantine equations (*). Binomial theorem and Pascal's Triangle.

• 24. Inclusion/exclusion. Examples: derangements (*). Four-step method for applying inclusion/exclusion: identify good and bad items, choose which ones to count, express items as union of sets, apply ghastly formula and try to simplify. [HW 9 due]

• 25. Practice quiz in small groups on inclusion/exclusion (*).

• 26. Diagonalization. Countable and uncountable sets. Transfinite numbers. Example: Cantor's proof (*). Other counting arguments including Pigeonhole Principle. [HW 10 due]

• 27. Practice quiz in small groups on diagonalization (*).

• 28. Counting by recurrence relations. Examples: derangements, binary search trees (*). [HW 11 due]

• 29. More examples: Counting Hagelin keys, Lotto and Big Game, Monty Hall problem.

• 30. Comprehensive test of fundamentals.

• 31. Comprehensive final exam