The lecture notes are fairly close, but do not exactly reflect all material covered. They do also have some extra examples not done in class:
Since I often do not follow the book closely,
they are posted as helpful aids,
ECS 20 - Spring 2009 - Lecture-by-Lecture Summaries
|
| Lecture |
Topic |
| Lect 1 - T 3/31/09 |
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).
Lecture 1 notes pdf
|
| Lect 2 - R 4/2/09 |
Finish proof that five shuffles aren't enough
to mix a deck of cards: rising sequences.
Sentential logic: Chapter 4.
Logical operators and their truth tables. Rendering English into
logic (it doesn't really work). Formal definition of a well-formed formula (of sentential logic).
Lecture 2 notes html - Notes from lecture 2, a b
it rough.
|
| Lect 3 - T 4/7/09 |
More Logic, Disjunctive Normal Form (forming WFF's from a tru
th table), Majority, Logic gates and circuits (15.10), Adding 2 4-bit numbers
Lecture 3 notes pdf - Notes from lecture 3,
|
| Lect 4 - R 4/9/09 |
Logic: Basic laws (4.7), Derivations (4.9), Quantifiers (4.10),
Negation of quantified expressions (4.11)
Lecture 4 notes text - Notes from lecture 4
|
| Lect 5 - T 4/14/09 |
Sets (chapter 1), examples, definitions, applications of sets.
Operators on sets (1.4): union, intersection, disjoint sets, Venn Diagrams, Alge
bra of sets (1.5)
Lecture 5 notes part 1: pdf - Notes from lecture 5,
Lecture 5 notes part 2: text (not as pretty) - Notes
from lecture 5, part 2.,
|
| Lect 6 - R 4/16/09 |
More on sets: review of operations, (1.7) power set operator, p
artitions and proving elementary properties of operators. (1.8)Induction
(12) Formal languages. Alphabets, strings, languages. Concatenation.
Extending operations from strings to languages. The * operator.
Lecture 6 notes pdf - Notes from lecture 6
|
| Lect 7 - T 4/21/09 |
(12),Why study languages? Operations on languages (as sets) (12.4) Regular languages.
Lecture 7 notes pdf - Notes from lecture 7
|
| Lect 8 4/23/2009 - |
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.
lecture notes.
|
| Lect 9 - T 4/28/09 |
Review of relations. Properties of an equivalence relation (2.8):
reflexive, symmetric, transitive. Partitions of a set. Relationship between equivalence relations
and partitions. Various examples, such as x~y if 3 | (x-y)
x~y if x and y are regular expressions denoting the same language.
Closures of relations (2.7)
Prof's lecture notes.
|
| Lect 10 - R 4/30 |
Functions: Chap.3 One-to-one, onto, and bijective functions. Inverse of a function, function composition, intro to growth rates.
Prof's lecture notes.
|
| Lect - T 5/5 |
Midterm
|
| Lect 11 - T 5/7 |
Recursive functions: factorial, fibonacci (3.4), analysis
Ignoring constants: Big-O and Theta notations.
( Wikipedia page on this.)
Prof's lecture notes.
|
| Lect 12 - T 5/12 |
. More on big O/Theta. Program examples: searching a list in O(n) time.
Infinite sets. Integers and rationals are countable, equinumerous sets, P(N) is uncountable, the Reals and even Reals in the range (0,1) are uncountable.
Prof's lecture notes.
|
| Lect 13 - R 5/14 |
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, Binary Search
Prof's lecture notes.
(Applications of Ramsey theory, in response
to a student question.)
|
| Lect 14 - T 5/19 |
Binary Search Analysis, T(n) = Theta(logn), correctness proof using
computational induction. Recurrence relations: Towers of Hanoi using induction,
Recursion Tree method, Master Theorem. Introduction to Counting (chapter 5)
Prof's lecture notes.
|
| Lect 15 - R 5/21/18 |
Counting (chapter 5 and 6): Basics (5.2), Functions (5.3), Factorial, binomial coefficients; LOTS of counting examples.
Prof's lecture notes.
|
| Lect 16 - T 5/26 |
Probability. Chap. 7: Definitions of: probability spaces, events, independence of events.
Inclusion/exclusion. Examples, like: probability of 50 heads out of 100 with a fair coin. Conditional probability, random variables.
Getting a full-house/straight in poker,
Prof's lecture notes.
|
| Lect 17 - T 5/28 |
More probability: flipping a biased coin, die rolles, white and black balls from an urn (conditional probability), medical testing, monty hall problem, birthday collisions.
Prof's lecture notes.
|
| Lect 18 - T 6/2 |
Graph theory. Basic definitions *8.2). 8.4:Paths, cycles. connectivity, Trees.
8.5 Eulerian graphs and their characterization. Easy/hard pairs of related problems problems:
Eulerian vs. Hamiltonian; shortest paths vs. longest paths;
two-colorable (bipartite) vs. three-colorable.
Students grade me.
Prof's lecture notes
|
| Lect 19 - R 6/4 |
More Graph theory.
Graph coloring and bipartite graphs. Proof of 8.8 on trees. 8.9: planar graphs,8.3 Graph isomorphism.
Edge disjoint and Vertex Disjoint paths.
Prof's lecture notes
|