Note: lecture notes (click on Li) are my personal lecture notes, distributed as a courtesey for those who find them useful. They are not polished, nor do they necessarilly correspond precisely to what I say.

### ECS 20 — Fall 2013 — Lecture-by-Lecture Summaries

Lecture Topic
L1 R 9/26 Brief introduction. Example probs: a simple sum; sqrt(2) is irrational; moves for Towers of Hanoi; 5 shuffles won’t randomize a deck. Scribe notes.
L2 T 10/01 Sean Davis lectures on logic. Truth tables. Logical equivalence. De Morgan’s law. Circuits. Conditionals, biconditionals. Inference. Handout.
L3 R 10/03 Designing an addition circuit. Disjunctive normal form (DNF). Formal defs for WFFs. Truth asignments. Satisfiablity and tautology.
D2 M 10/07 Mock quiz (order of precedence, truth tables, De Morgan’s laws, sentential logic formulas. PS2 notes.
L4 T 10/08 Quiz 1. Axioms and formal proofs. Completeness, soundness, and compactness. A result on tiling.
L5 R 10/10 First-order logic: syntax, examples, English-translation. Completeness and soundness. Treatment of number theory and set theory.
D3 M 10/14 CNF. Quantifiers. Truth tables. Writing and negating quantified formulas. NAND is logically complete
L6 T 10/15 Axioms of arithemetic. The principle of induction, and examples: a summation; buying envelopes; trominos; cake-cutting. Sets.
L7 R 10/17 Writing sets. Russell’s paradox. Member, subset. Union, intersection, difference, xor. Groups. R, N, Z; BITS, BYTES, WORDS32, FLOAT64.
D4 M 10/21 Q2 + PS4 notes. Induction examples. Strong induction. Envelope substitution.
L8 T 10/22 Quiz 2. Cartesian product, unordered product. Power set. Representing sets on a computer: dictionaries (with a list) and UNION/FIND.
L9 R 10/24 Alphabets, strings, languages. Concatenation, Kleene closure (star). Regular expressions & languages. Relations, equivalence relations, functions.
L10 T 10/29 Blocks (equivalence classes), and modding out by an equivalence relation. Relation to partitions. Injective and surjective functions.
L11 R 10/31 Midterm. The photo was of Andrew Wiles, the force behind the proof of Fermat’s Last ‘Theorem’.
D5 R 11/04 Prof. Rogaway went over the midterm, explain the solution to each problem.
L12 T 11/05 Review of function vocabulary and notation. Common functions for CS. Comparing the size of infinite sets. Cardinal numbers.
L13 R 11/07 Comparing |A|, |B|. There are uncountably many languages. Some langauges can’t be decided by computers. Review: log, exp, n! Big-O notation.
D6 M 11/11 “Virtual discussion section” because of holiday. Review of one-to-one and onto functions. Hints on homework.
L14 T 11/12 Definition of big-O and Theta notation. Proper and informal use. Eg: searching a list, binary search, bignum multiplication.
L15 R 11/14 An odd way to multiply: Karatsuba multiplication. Solving the recurrence relation underlying it. Pigeonhole principle and applications.
D7 M 11/18 Solving recurrence relations with repeatd substitutions and recursion trees. Big-O and Theta: ranking by order of growth.
L16 T 11/19 Quiz 3. Strong form of the Pigeonhole Principle. Graph theory: formal definitions and vocabulary. Isomorphism. Representation of graphs.
L17 R 11/22 Quiz discussion: importance of precise English. Review of graph terminology. Bipartite graphs, DFS. Paths, cycles, connectivity. Euler’s theorem.
D8 M 11/25 Discussion of HW. Finish graph theory: Connectivity. Hamiltonian cycles. Bondy-Chvatal Thm. longest and shortest paths. 2- and 3-colorability.
L18 T 11/26 Counting. Lots of examples, mostly using n!, P(n,r) and C(n,r). Principle of inclusion/exclusion.
Lxx R 11/29 Holiday. You can come, but you’ll be pretty lonely in that big room.
D9 M 12/02 Counting examples: factorials, permutations, combinations, blackjack.
L19 T 12/3 Probablity. Probability of different poker hands. Formal definition of a probablity spaces. Events, sum rule, independence. Examples.
L20 R 12/5 Finishing probablity: random variables and expected values. The funky subway. Monty Hall. Practice exam. Closing comments.
Lxx R 12/12 Final 10:30 am - 12:30 pm in our usual room