ECS 20  Fall 2008  LecturebyLecture Summaries 


Lecture  Topic 
Lect 1  R 9/25  Introduction. Meet the TAs. What is discrete math? Sample problems: (0) summing 1..n. (1) Necessary and sufficient number of moves for solving Towers of Hanoi. (2) sqrt(2) is irrational. (3) Five shuffles aren't enough to mix a deck of cards (unfinished). Student scribe notes. 
Lect 2  T 9/30  Finish proof that five shuffles aren't enough to mix a deck of cards: rising sequences. Sentential logic. Logical operators and their truth tables. Rendering English into logic (it doesn't really work). Formal definition of a wellformed formula (of sentential logic). Student scribe notes. Also, a wiki book. 
Lect 3  R 10/2  Definition of WFFs, and how to extend truth assignments to WFFs. Designing Boolean formulas for specific tasks (majority, exactly one, binary addition). Representing Boolean formulas by circuits. Tautologies, satisfiability, logical equivalence. Student scribe notes. 
Lect 4  T 10/7  Review: tautologies, satisfiability, equivalence and truthtable algorithm. The = and = = symbols. Logical "laws". Elements of proof systems (by example). Completeness and soundness. Compactness theorem and TILING. Firstorder logic: Adding quantifiers. Prof's lecture notes. 
Lect 5  R 10/9  Practice converting English to quantified boolean formulas. Semiformal description of the syntax of a quantified Boolean formula. How to complement a QBF. Two examples: number theory, set theory. Begin treatment of set theory, discussing in, subset, emptyset, union, intersection. Prof's lecture notes. 
Lect 6  T 10/14  Quiz 1. Solutions to quiz 1. Review of sets: membership, union, intersection. Complement, difference, symmetric difference. Venn diagrams. Laws for set manipulations, such as DeMorgan's laws. Cross products. Alphabets and Strings. Prof's lecture notes. 
Lect 7  R 10/16  More on set: review of operations, power set operator, and proving elementary properties of operators. Formal languages. Alphabets, strings, languages. Concatenation. Extending operations from strings to languages. The * operator. Why study languages? Regular languages. Student scribe notes. 
Lect 8  T 10/21  Review of strings, languages, regular expressions. Examples. DFAs. Claim: the languages of regular expression = the languages of DFAs (proven in ecs120). Relations. Definitions and examples. Inverse of a relation, composition of relations. Prof's lecture notes. 
Lect 9  R 10/23  Quiz 2. Solutions to quiz 2. Review of relations. Properties of an equivalence relation:
reflexive, symmetric, transitive. Partitions of a set. Relationship between equivalence relations
and partitions. Various examples, such as x~y if 3  (xy)
x~y if x and y are regular expressions denoting the same language. Prof's lecture notes.

Lect 10  T 10/28  Plan: Why partitions and equivalence relations are equivalent. More examples of how to identify the blocks of a relation. Functions. Onetoone, onto, and bijective functions. Inverse of a function. Prof's lecture notes. 
Lect 11  R 10/30  Midterm 
Lect 12  T 11/4  Review of terminology associated to functions (image, preimage, onetoone, etc). Examples of commonly used functions. Review of exp and log functions. Ignoring constants: BigO and Theta notations. ( Wikipedia page on this.) Prof's lecture notes. 
Lect 13  R 11/6  Review of bigO and Theta definitions, with additional examples. Infinite sets. Definition for when sets are equinumerous. \N ~ \Z and \N ~ \Q. The powerset of \N is uncountable. The reals are uncountable. The continuum hypothesis. Return and discussion of midterms. Prof's lecture notes. 
Lect xx  T 11/11  Holiday. 
Lect 14  R 11/13  The Pigeonhole Principle. Examples: genders among three people; number of friends; distance of points in a square; mod 10 sum of 10 numbers; the Ramsey number R(3,3)=6. Strong form of pigeonhole principle. Induction and recursion. Buying envelopes. Prof's lecture notes. (Applications of Ramsey theory, in response to a student question.) 
Lect 15  F 11/14  (Lecture in discussion section.) Finish buyingenvelopes example. Unwinding the solution to get an algorithm. Tiling with trominos. The sum of the first n odd natural numbers is n^2. Unwinding the solution. Cakecutting. Prof's lecture notes. 
Lect 16  T 11/18  Quiz 3. Recurrence relations. Review of cakecutting problem and analyzing its running time. Review of Towers of Hanoi. Repeated substitution and induction. Another recurrence relation. The Master Theorem. Binary search. Cakecutting. Prof's lecture notes. 
Lect 17  R 11/20  Sample missteps from Quiz 3. Wrong is better than garbled. Counting. A zillion examples using permutations P(n,r), Combinations C(n,r), factorials, and just a bit of cleverness. Prof's lecture notes. 
Lect 18  T 11/25  Probability. Definitions of: probability spaces, events, independence of events. Inclusion/exclusion. Examples, like: probability of 50 heads out of 100 with a fair or unfair coin, getting a fullhouse in poker, two birthdays on the same day. Prof's lecture notes. 
Lect xx  R 11/27  Holiday. 
Lect 19  T 12/2  Probability, part two. Conditional probability. Random variables and their expectation. The guesssmaller/guesslarger problem, the Monty Hall problem, the Northbound/Southbound subways problem, etc. Prof's lecture notes. 
Lect 20  R 12/4  Students grade me. Graph theory. Basic definitions. Paths, cycles. Trees. Eulerian graphs and their characterization. Easy/hard pairs of related problems problems: Eulerian vs. Hamiltonian; shortest paths vs. longest paths; twocolorable (bipartite) vs. threecolorable. Graph isomorphism. Goodbye! Prof's lecture notes 
Lect xx  F 12/12  Final. The exam is Friday, 1–3 pm, in our usual room (202 Wellman). Is it not cool that our exam is on 12/12, a very pretty date? 