CS 2233: Discrete Mathematical Structures

Instructor: Jessica Sherette

Office Hours in FLN 3.01.04 #7

Tuesdays 5:30pm to 6:30pm, Wednesdays 5:00pm to 6:00pm, and by appointment

Handouts:

Homework:

Homework 1; due **1/23/14** before class

Homework 2; due **1/30/14** before class

Homework 3; due **2/6/14** before class (solutions: 1,2,3,4,5)

Homework 4; due **2/27/14** before class

Homework 5; due **3/6/14** before class

Homework 6; due **3/20/14** before class (solutions: 1,2,3,4,5)

Homework 7; due **4/10/14** before class

Homework 8; due **4/17/14** before class

Homework 9; due **4/29/14** before class (solutions:
1,
2,
3,
4,
5,
6,
7,
8)

Class Notes:

01/14 - 1 propositions, 2 negation, 3 conjunction, 4 disjunction, 5 xor, 6 conditional statement, 7 biconditional statement, 8 precedence of operations, 9 translating english, 10 propositional equivalences, 11 truth table limitations

01/16 - 1 contrived example proof, 2 sequence of equivalences, 3 predicates, 4 quantifiers, 5 existential quantifiers, 6 negating quantifiers

01/21 - 1 negating quantifiers , 2 negating quantifiers examples, 3 translating english statements, 4 nested quantifiers, 5 nested quantifiers examples, 6 nested quantifiers table, 7 additional nested quantifiers , quiz 1 solutions

01/23 - 1 negating nested quantifiers, 2 introduction to proofs, 3 direct proof, 4 indirect proof, 5 indirect proof motivation, 6 proof of equivalence, 7 proof by contradiction, 8 proof by contradiction (cont.)

01/28 - 1 proof by cases, 2 existance proof (constructive), 3 existance proof (non-constructive), 4 mistakes in proofs, 5 mistakes in proofs (cont.), 6 introduction to sets, 7 commonly used sets, quiz 2 solutions

01/30 - 1 equality of sets, cardinality of sets, and the empty set, 2 power set, 3 Cartesian product, 4 compliment and union, 5 intersection and difference, 6 de Morgan's law for sets

02/04 - 1 introduction to functions, 2 example of a mapping that is not a function, 3 definition of image and range, 4 definition of one-to-one, onto, and one-to-one correspondence, 5 examples of one-to-one and onto, 6 additional example of one-to-one and onto, 7 using graohs to determine onto and one-to-one, quiz 3 solutions

02/06 - 1 inverse and composition, 2 inverse and composition examples, 3 floor and ceiling, 4 proof of lemma about floor and ceiling, 5 sequences definition, 6 geometric progression and arithmetic progression, 7 summations, 8 geometric series proof, 9 geometric series proof cont.

02/11 - 1 midterm notes, 2 introduction to algorithms, 3 compute max algorithm, 4 runtime of compute max algorithm, 5 big-oh definition and motivation, 6 big-oh examples

02/18 - 1 common runtime functions, 2 big-oh, omega, and theta definitions, 3 omega examples, 4 theta example, 5 code snippets, quiz 4 solutions

02/20 - 1 introduction to induction, 2 induction example 1, 3 induction intuition, 4 induction example 3, 5 induction example 4, 6 induction example 4 cont., 7 induction example 5

02/25 - quiz 5 solutions

02/27 - 1 introduction to strong induction, 2 strong induction example 1, 3 strong induction example 2, 4 introduction to loop invariants and example 1, 5 example 1 (cont.)

03/04 - 1 multiply example, 2 multiply example (cont.), 3 introduction to recursion, 4 introduction to recursion examples, 5 introduction to recursion correctness proof, 6 recursively defined sequences, quiz 6 solutions 1, quiz 6 solutions 2

03/06 - 1 towers of Hanoi, 2 number of moves to solve the towers of Hanoi, 3 estimating an explicit formula using the expansion method, 4 proof of correctness for the estimate, 5 recursive definition example, 6 structural induction example 1, 7 structural induction example 1 (cont.), 8 structural induction example 2, 9 structural induction example 2 (cont.)

03/18 - 1 divide and conquer algorithms, 2 new recurrence for pow, 3 algorithm based on the recurrence, 4 runtime recurrence and expansion method, 5 inductive proof of runtime, 6 inductive proof of runtime (cont.), quiz 7 solutions

03/20 - 1 recursion tree method, 2 divide and conquer, Master Theorem, 3 master theorem examples 1, 4 master theorem examples 2

03/25 - 1 introduction to solving linear recurrences relations, 2 characteristic equation and solving for constants, 3 example 1, 4 example 1 (cont.), 5 example 2, 6 example 2 (cont.), quiz 8, quiz 8 solutions

04/01 - 1 introduction to relations, 2 relations vs functions, 3 example relation, 4 properties of relations, 5 examples of properties of relations, 6 introduction to n-ary relations, 7 operations on n-ary relations, 8 example 1-join operation

04/03 - 1 definition of mod, 2 equivalence relation example, 3 equivalence relation example (cont.), 4 example graphs, Notes on Graphs

04/08 - 1 undirected and directed graphs, 2 maximum number of edges in a graph, 3 degree in an undirected graph, 4 degree in a directed graph, 5 paths in a graph, 6 connected and strongly connected graphs, 7 connected components and introduction to trees, quiz 9 solutions

04/10 - 1 theorem about the number of edges in a tree, 2 notation for trees, 3 binary and k-ary trees, 4 full, balanced, and complete trees, 5 theorem about the number of leaves in a tree, 6 full k-ary tree lemma

04/15 - 1 introduction to languages, 2 introduction to grammars, 3 directly derivable and derivable, 4 example grammar 1, 5 example grammar 1 (cont.), 6 example grammar 2, 7 example language 1, quiz 10 solutions

04/17 - 1 example language 2, 2 example language 2 and simpler grammar, 3 example language 3, 4 context free grammar and Backus-Naur form, 5 ALGOL 60 specification

04/22 - 1 introduction to finite state machines, 2 example FSM, 3 definition of accepted and example FSM 1, 4 example FSM 2 and example FSM 3, 5 example FSM 4 and example FSM 5, 6 example FSM 6, 7 example FSM with output (check book for clearer image), 8 example language which is not recognized by any FSM, quiz 11 solutions

04/22 - 1 hierarchy of languages, solutions to the problems in the notes 1, solutions to the problems in the notes 2