Suggested Homework


You will have at least a week from the posted date before I expect you to have worked the problems. Please, if you have any questions, don't hesitate to ask.

Week 1

Jan 14, 2003
Section 1.2) Problems 1,2,5,7,10,12,13,17,19,24,25,29
Section 1.3) Problems 1-5,11,14,15,19,20,23-29

Jan 15, 2003
Section 1.4) Problems 2,10,11,14,28,30,31


Week 2

Jan 21, 2003
Section 1.5) Problems 11, 13, 19

Jan 22, 2003
Section 1.5) Problems 22,23,26,27,36


Week 3

Jan 28, 2003
Section 1.6) Problems 1, 3, 5, 11, 15, 19, 26

Jan 29, 2003
Section 1.7) Problems 3, 5, 9, 11a), 15, 19, 21, 37 (see definitions 6 and 7 on pages 92 and 93 to work #37)

Jan 30, 2003
Section 1.8) Problems 7 a) b), 10, 11, 15, 17, 19, 21, 25, 26, 27, 30, 32

**THERE IS A TEST SCHEDULED FOR FEB 6, 2003**
All material covered up to and including Jan 30, 2003 is fair game. If you have any questions please come see me during office hours. If you can't make my office hours, send me an email and we can try to work something out.



Week 4

Feb 4, 2003
Foreshadowing about what's to come after the exam. Review for the exam.

Feb 5, 2003
Review for the exam.

Feb 6, 2003
Get psyched up for more math.



Week 5

Feb 11, 2003
Section 2.4) Problems 1-7

I. Show that if A = {a1, a2, a3,...} is a countably infinite set and x is any element not in A then the set B = {x, a1, a2, a3,...} is also countably infinite. (You must construct a function f: B --> N, where N denotes the natural numbers, and show that it is bijective.)

Extra Credit: (+2 (max) points on the next quiz) Let N denote the natural numbers. Show that the set NxNxN is countably infinite. Again, you must construct a function g: NxNxN --> N and prove that it is bijective. (Hint, we proved that NxN is countably infinite in class via the function f : NxN --> N defined by f((m,n))=[2^(m-1)][2n+1]. Think composition of functions.)

Feb 12, 2003
Section 2.4) Problems 8-13, 15-19

Feb 13, 2003
Section 2.4) Problems 29, 31, 32, 38, 40, 51, 53, 54
I. Let m be a positive integer. Prove that if a is congruent to b modulo m and c is congruent to d modulo m, then (a + c) is congruent to (b + d) modulo m and ac is congruent to bd modulo m.



Week 6

Feb 18, 2003
Section 2.5) Problems 5-10

Feb 19, 2003
Section 2.5) Problems 21,22

Feb 20, 2003
Section 2.6) Problems 1-5
Extra Credit: (+2 (max) points) Section 2.6) Problem 10



Week 7

Feb 25, 2003
Section 2.6) 6-8, 11 (See #5), 12 (See #6), 18-21,25

Extra Credit: (+2 (max) points) Section 2.6) Problem 10 (Turn in Wednesday Feb 26, 2003)

The grades on the last quiz were (in general) not good. The quizzes are intended to raise your grade since the problems come verbatim from the homework. So, I continue to emphatically suggest that you work the homework! If you have any questions about it, please ask me. I know that this stuff is difficult but if you don't ask me questions I have no idea exactly how much people understand until I grade papers.

Enough of the tirade. I will allow folks to rework the problems on Thursday's quiz (rework meaning you had to have taken the quiz Thursday) for a regrade to be turned in next Tuesday Feb 4, 2003. I expect the solutions to be written up neatly. Two of the quiz problems are in the book (Section 2.4 #6, #38) and the other problem I wrote on Feb 11, 2003 (Show that if A = {a1, a2, ...} is countably infinite, then B = {x, a1, a2, ...} is also countably infinite where x is not in A).

Feb 26, 2003
Study for the exam.

Feb 27, 2003
Study for the exam.

**THERE IS A TEST SCHEDULED FOR MARCH 4, 2003**
All material covered up to and including Feb 26, 2003 is fair game (the new stuff that makes up the majority of the exam will be sections 2.4, 2.5, and 2.6 as well as the material on countability we covered in class) . If you have any questions please send me email or come by and see me in my office. I will not be able to be around Thursday afternoon, but I will be in my office Friday morning from 10:00-12:20 if you would like to come by. In addition I will try to be around Monday morning from 10:00-12:20 as well.




Week 8

March 6, 2003
Section 3.2) 13-16
Prove using induction that 1^2 + 2^2 + 3^2 + ... +n^2 = [n(n + 1)(2n + 1)]/6, and 1^3 + 2^3 + 3^3 + ... + n^3 = [n^2(n + 1)^2]/4
Section 3.3) 1-3



Week 9

March 11, 2003
Section 3.3) 5, 12, 13, 21

March 12, 2003
Section 3.3) 22, 25, 38
Bonus : Show that given any n circles in the plane it is possible to color the regions bounded by edges of the circles with exactly two colors so that no two adjacent regions are the same color. (You may assume that if two circles intersect, then they intersect at two points. I.e., the circles don't intersect at a single point.) This is to be turned in Wednesday after spring break. Send me email if you have questions.

March 13, 2003
Section 4.1) 1, 2, 4, 7, 8, 9, 14, 19, 24, 25, 33, 37, 39



Week 10
SPRING BREAK




Week 11

March 25, 2003
Review Sections on induction and counting (3.3 and 4.1).

March 26, 2003
Continue to review induction material.

March 27, 2003
Read about pigeons in section 4.1.



Week 12

April 1, 2003
Section 4.2) 1, 3, 4, 5, 6, 7, 15, 18, 19
Section 4.3) 3, 4, 10, 11, 15, 19, 20, 21

April 2, 2003
Section 4.4) 4, 5, 12, 15, 19, 21, 33
Bonus : 36



Week 13

April 8, 2003
Review for the exam.

April 9, 2003
Continue review.

April 10, 2003
Exam 3.


Week 14

April 15, 2003
Section 2.2) 1 a-d, 3, 5, 7, 13, 15

April 16, 2003
Section 2.2) 22 a-d, 25, 27, 29, 31, 39

April 17, 2003
Section 2.3) 3, 4, 7, 9, 10
There will be a quiz next Thursday and you will have two different ways of being graded. You may choose one and only one of the following two options.
a) Take an in class open book exam over material we have covered this week.
b) Write a two page typed paper explaing what mathematical induction is and why it works. I will grade the papers on completeness, clarity, grammer, and a little style. I want the paper to be written in your own words with as little mathematical notation (none is best) as possible. Your grade on this paper will be used to replace the score for problem number two on the last exam provided you get a higher score on the paper. Otherwise the higher test problem score will count. In addition, the score on this paper will count as a quiz grade. If you have any questions please email me. I have not graded all of the problems on the exam yet. If you would like to know how you did on the second problem, drop me a line.


Week 15

April 22, 2003
Work on the induction paper and/or read about recurrence relations in section 6.1

April 23, 2003
Continuing writing and reading about recurrence relations in sections 6.1 and 6.2.

April 24, 2003
Section 6.1) 1, 5, 9
Section 6.2) 1, 3, 13, 15, 19, 21