CS243: Discrete Structures (Fall 2009)

Time & Loc.: TR 9:30am -- 10:50am, McGlothlin-Street Hall 20
Instructor: Qun Li
Office: McGlothlin-Street Hall Room 118
Office Hours: TR 1:00pm -- 3:00pm, or by appointment
Phone: 757-221-3478
Email: liqun@cs
Syllabus: PDF
TA:  Nan Zheng (nzheng@cs.wm.edu) McGlothlin-Street Hall Room 107A, Office hours:

Course Description

We will cover logic and proof, basic number theory, induction and recursion, basic counting techniques, probability, graphs, and trees. Students should learn how to write rigorous mathematical proofs and build basic mathematical foundations for more advanced courses.

Text Book

Discrete Mathematics and its Applications, by Kenneth H. Rosen, 5th edition, McGraw Hill, 2003.

Be sure to get the 5th edition. Please DO NOT get the latest 6th edition. You can find the 5th edition through online used book stores; some students purchased used books for a few dollars.  

Homework

We will have homework every week and you need to turn in your homework every Tuesday. You are encouraged to work together to do homework problems. What is important is a student's eventual understanding of homework problems, and not how that is achieved. Students may consult any source, except for another student's final draft, in learning how to do homework problems. Students must state what sources they have consulted, with whom they have collaborated, and from whom they have received help.

Exams

There will be two mid-term exams. They are tentatively scheduled on 10/1 and 11/17. The time for the final exam will be Dec. 11 (Monday) 9:00--12:00. They are all closed-book exams.

Academic Calendar for Fall 2009

Grade

Homework 20%, Mid-term I 20%, Mid-term II 20%, Final 40%. Extra 5% for class attendance.

Schedule and Homeworks

This schedule will be updated frequently. Please remember to reload to get the latest schedule.

Date

chapter

Homework

Class Topic

HW solution
8/27(Th)
 Chap 2.4
Proof Handout
 HW1:
Problem 1, 4, 5, in the Handout
Overview, Proof, divides, even/odd
 HW1 solution
9/1(Tu)
 Chap 1.6, 1.7
Sets Handout
 HW2-I:
Problem 2 and 3 in the Handout
Ex1.7 (pp. 95), 4. (a) (b)
 Proof continued, divides, even/odd, irrational number, a little bit about set

9/3
 pp. 77-108 (Chapter 1.6, 1.7, 1.8)
 HW2-II:
Ex1.7 (pp. 95) 19, 21, 37, 38
Ex1.8 (pp. 109) 16, 25, 28, 30, 31
HW2
 Sets and functions
 HW2 solution
9/8(Tu) pp. 105-108 (Chapter 1.8)
pp. 119-129 (Chapter 2.1)
Chap2.1 Handout
 HW3-I:
Ex1.8 (pp. 111) 66, 68 (hint)
Ex2.1 (pp.130) 18, 20, 24, 32
Functions (floor and ceiling), algorithms (function, linear search, binary search)
9/10
 pp. 119-129 (Chapter 2.1)
pp. 131-142 (Chapter 2.2)
Chap2.2 Handout
 HW3-II:
Ex1.6 (pp. 85) 8, 9, 17
EX2.2 (pp. 142) 2, 6, 8, 12,
 Growth of function, big-O, HW3 solution
9/15(Tu) pp. 153-179 (Chaptter 2.4)
Chap2 Handout
 HW4-I:
Ex 2.2 (pp.142) 20
Ex 2.4 (pp.166) 12 (d) (e), 36
Use Euclidean Algorithm to compute gcd(1529,14039). 		Division, Euclidean Algorithm
9/17
 Ext. Euclidean Alg. pp. 182,
Modulo inverse, linear congruence. pp. 184-5
 HW4-II: 
(1) Express gcd(1529,14039) as a linear combination of 1529 and 14039 (i.e., find integers s and t such that gcd(1529,14039)=s*1529+t*14039). 
(2) Write down the pseudo-code for the extended Euclidean algorithm. Input: integer a and b; Output: gcd(a,b) and integers s and t
such that gcd(a,b)=sa+tb see pp.196 for some hint).
(3) pp.168(Ex.2.4) 42, 43, 44
(4) Find an integer x such that 98x is congruent to 21 modulo 35.
 Extended Euclidean Algorithm, congruent. modulo, modulo inverse
Extended Euclidean Algorithm Example
 HW4 solution
9/22(Tu) Fermat's little theorem (pp.194, 17)
 HW5-I:
Ex2.6 (pp. 194) 17 (For step (c), you don't have to use Wilson's theorem. )
Ex1.8 (pp. 110) 48
Suppl. Ex. (pp. 116) 52
 HW3 review, Fermat's little theorem
9/24
 RSA (pp.192-194)
Modular exponentiation (pp. 175-177)
 HW5-II:
Ex2.6 (pp.196) 46, 47
Ex2.5 (pp.180) 20
 RSA, Exponentiation
9/29(Tu)

HW4 and HW5 review, Exam review
 HW5 solution
10/1
  
Midterm I

10/6(Tu)
HW6-I:
Here 
pp.204 (Ex.2.7) 4, 30
Matrix, exam review
10/8
 Chap. 3.2 (pp.225-231)
Chap. 3.3 (pp.238-249)
Handout
 
HW6-II:

(1) pp. 237 (Ex. 3.2) 16 (Use the formula we learned in class, i.e., the sum for arithmetic and geometric progression)

(2) A list goes like:
-13-2, -16-(2/3), -19-(2/9), -22-(2/27), ......, -109-b.
Determine how many terms in this list, the value of b, and the sum of
this list. You only need to provide the formula for the sum rather than the exact value.

pp.253 (Ex. 3.3) 8, 42, 44, 50, 52
 sequence, arithmetic and geometric progression, Mathematical induction
 HW6-I solution

HW6-II solution
Oct 13 (Tu) No class. Fall break (10/10 (Sat.) -- 10/13 (Tue.))
10/15
 Strong induction (pp.249-251)
 HW7:
Ex3.3 (pp. 253) 18, 30, 34, 36, 40
Suppl. Ex. (pp. 292) 12, 18

For Problem 40 (pp. 254), it means to compute the sum of the products computed at all steps (from the first split to the last one)  equals n(n-1)/2. For example, if we have three stones. The fist split will result in two piles of 2 and 1 stones in them (we get a product 2*1). The second split with be three piles of 1 stone (we get another product 1*1). The sum would be 2*1+1*1=3.  Use strong induction here. Consider the first time you split, you will get two smaller piles with x stones and k+1-x stones. You can use strong induction on the two smaller piles.
Mathematical induction review, strong induction
 HW7 solution
10/20(Tu) Counting
(pp.301-306)
Handout
 HW8-I:
Ex4.1 (pp. 310) 10, 26, 28, 32, 38
 Counting, product rule, sum rule,

10/22
 Counting
Chap 4
(pp.308-310, pp.320-324) 		HW8-II:
Ex4.1 (pp. 310) 36, 42,
Ex4.3 (pp. 325) 20, 22, 24, 26, 30, 32
 principle of inclusion and exclusion, permutation and combination

 HW8 solution
10/27(Tu) Counting
Chap 4
(pp. 335-341)
 HW9-I:
Ex. 4.5 (pp. 343) 12, 20, 41, 42, Supplement Exercise (pp. 352) 38, 41, 42
Card problems:
1. How many ways to choose a five-card poker hand that contains cards of five different kinds and does not contain a flush or a straight? [A flush is a five cards of the same suit. A straight is five cards that have consecutive kinds (both A2345 and 10-JQKA are straight, but JQKA2 is not).]
2. How many ways to choose a five-card poker hand that contains one pair (that is, two of one kind and three cards of different kinds)?
more counting examples, fruit problem (combination with repetition), card problem

10/29
 Counting
Chap 4 (pp. 327-333)
 HW9-II:
Ex. 4.4 (pp. 333) 8, 10, 22
For problem 22, consider different ways to select two committees from n people. One committee has k people and another one has r-k people.
 binomial theorem, combinatorial argument, HW8 review
 HW9 solution
11/3(Tu) Probability
Chap 5 (pp.355-380)
 HW10-I:
Ex 5.1 (pp. 361) 16, 18, 32, 34, 36
Ex 5.2 (pp. 377) 8, 10,
Ex 5.3 (pp. 392) 2, 6
 Probability, expectation
11/5
 Probability
Chap 5 (pp.355-392) 		HW10-II:
Ex. 5.2 (pp. 377) 24, 26, 34
Ex. 5.3 (pp. 392) 4, 8
conditional probability, Bernoulli trial, expectation HW10 solution
11/10(Tu) Probability
Chap 5 (pp. 379-392)
 HW11-I:
Ex. 5.3 (pp. 392) 16, 19, 24
Supplementary Exercises (pp. 396): 6, 10
variance, independent random variables
11/12
review sample
 Midterm II review
 sample
11/17(Tu)
Midterm II (tentative) Midterm II (tentative)
11/19
 Graph
Chap 8 (pp. 537-575)
 HW11-II:
Ex. 8.2 (pp. 554) 6, 34
Ex. 8.3 (pp. 564) 20
pp. 575 (Ex. 8.4) 40, 42 		graph, adjacency matrix, number of paths with length n
11/24(Tu)

Euler's circuit, Euler's path
 HW11 solution
Nov 26 No class. Thanksgiving break (11/25 (W) -- 11/29 (Sun.))
12/1(Tu)



12/3



12/4 (F) Last day of Fall classes

Reading Period
12/11 (Friday) Final exam (9:00--12:00)