Pre/co-requisites: Two semesters of Calculus (or permission of the instructor)

Homework:

Numerous problems will be assigned each class to be done for the following class (see the attached problems list). Some of these problems will be designated as "hand-in" problems, as discussed in the next section of this syllabus. In general, however, other homework problems will not be collected (although you may submit them if you would like me to correct them). Nevertheless, it is expected that you attempt all of the problems in a timely manner, and that you keep up with the pace set in class. This is part of your responsibility to yourself as a student and to whoever is paying for your education. THE ONLY WAY TO LEARN MATHEMATICS IS TO DO MATHEMATICS.

I suggest the following routine for this course (as well as for any other mathematics course you may take):

1. Try to read the next scheduled section of the text BEFORE it is covered in class. It is not necessary that you understand it completely, but try to get an idea of what it is about and where any difficulties you may have in understanding it lie.

2. Pay attention in class and try to take as careful notes as you can.

ASK QUESTIONS about material you do not understand.

3. AFTER class, reread the text and your notes, and try to fill in any gaps that may exist in your notes. Make a list of items you do not understand and ASK about them during the following class or during my office hours.

4. Attempt all of the assigned problems. Answers and, occasionally, partial solutions to some of the problems are contained at the back of the book; but do not rely on this as a crutch. If you need to consult the answers to guide you through some problems, that is OK; but, if you find that you have to "massage" most of your work to get the answers in the back of the book, then it is time to make an appointment to see me. No solutions or answers will be available to guide you when it comes time to take a test.

5. When studying for each test, review your notes and the text and the problems which have been assigned. If you have been keeping up with the work and doing all of the above as you go along, it should not be necessary to "cram" for the test; a brief review is all that should be needed.

Writing:

Since this is a W course, you will be expected to do a substantial amount of writing. Each week (more or less) some problems will be assigned to be handed in. These "hand-in" problems will be graded for written organization, style and grammatical correctness as well as for the mathematical content. The hypothetical audience for all written work will be the other students in the class. You must explain your reasoning to them and must convince them that your solution to the problem you are submitting is both correct and complete. While we are primarily interested in the organization and the persuasiveness of your arguments, correct grammar and spelling are not to be ignored. Over the course of the semester, you may be expected to revise and rewrite some of these assignments, to improve your writing (and your grade).

In addition, there will be a group paper on cardinality, a topic that I am not able (due to lack of time) to cover fully in class. You and the other members of your group will be asked to explain how the development of this topic depends on other topics which have been discussed in class. (See page 5 of this syllabus for more details.)

After EACH class meeting, you will be expected to prepare one index card in accordance with the instructions (see page 3). These index cards are to be submitted once each week.

Format instructions for "hand-in problems":

1. Each problem is to begin on a new page.

2. You are to write on one side of the page only, leaving ample blank space for me to write comments.

3. Pages of a problem may be stapled or clipped together. However, different problems are NOT to be stapled to each other.

4. Your name is to appear on the top of the first page of each problem, and your name or initials are to appear at the top of each subsequent page.

5. Everything you write is to be in grammatically and idiomatically correct English. This does not mean that you may not use equations and other forms of mathematical shorthand. What is does mean is that, when read aloud, everything you write, including equations etc., should read as complete and correct sentences. Transitions and paragraph formation are also important considerations.

6. Although there is no specific requirement to this effect, word processed or typed work is preferred, with work written is in blue or black pen preferable to work done is in pencil, even though you may have to cross out rather than erase mistakes. Is in any case, all work submitted must be clearly legible.

Index cards:

Index cards (see above) are to be prepared as follows: On one side of the card, PRINT your name in the upper left hand corner and the date in the upper right hand corner. On the same side of the card write one or two sentences addressing one of the following: What the class was about; what was the most important thing you learned; what was the most interesting thing we discussed (and why you found it interesting). On the other side of the card, write one or two sentences addressing one of the following: something you did not understand well; something which reminded you of something you learned in another course; a mathematically inquisitive question provoked by the class - e.g., "I wonder whether such and such a result would still be true if we removed the restriction of finiteness." During the course of the semester, vary which of the above items you choose to address, so that each is addressed approximately equally often. In addition, if, during the course of the semester, something comes up in another course you are taking which relates to something we discussed in this course, make an index card (including your name, the date and the course number or title) briefly discussing the connection.

Group work:

Each student in the class will be assigned to a group. Members of a group should sit together in class and are even encouraged to get together outside of class to discuss the course material, to study and to do the "non-hand-in" homework, with the proviso that all members of the group are to be active participants rather than just listeners or observers. As the name suggests, the "group paper" is also to be a group effort, with each member of the group responsible to assure that every other member understands the concepts involved. (These concepts will appear on the final examination.)

On the other hand, since the hand-in homework will constitute part of your grade, it is to be an individual effort and is not to be discussed with anyone other than the instructor until after it has been submitted. Breaking this rule is a serious violation of the Student Code of Conduct and will be severely punished. (I caution you, based on past experience, that I almost always can tell when two students have worked together or when one has copied from the other, so I strongly advise you not to work with anyone and not to copy anyone else's work nor allow anyone to copy yours. This warning is not intended as a challenge but, rather, as a word to the wise.) If you need help with a hand-in problem, come to me.

Some classes may begin with time set aside for the group to meet to review the previous night's homework and to select one (or possibly more, depending on the number of groups) problem which they would like me to do on the board. Portions of other classes may also be devoted to group work.

After each test, group members are encouraged to get together to review the test and to help each other master material not mastered previously.

The group paper:

Each group is to write a paper discussing the notion of cardinality. Although most of the material for this paper can be found is in Chapter 5 of our text, you might find it helpful to consult other sources (such as the supplementary text) as well. I am also available for consultation.

I would like certain changes in notation and terminology from what is found is in our text. In particular, the text speaks of sets as being "equivalent". This is a poor choice of terminology because there are many different possible equivalence relations on a set of sets. The more traditional terminology for the type of equivalence which concerns us here is "equipotence"; this is the terminology I would like you to use. Also, I would like you to denote the cardinality of a set A by A rather than by as is in the text. (#(A) is also used sometimes, but A is preferable.)

The paper should include all of the following, as well as anything else which you consider helpful and relevant:

1. A definition of what it means for two sets to be equipotent.

(The term "bijective mapping" should appear somewhere is in the definition.)

2. Proof of the fact that equipotence of sets is an equivalence relation.

3. Proof of the fact that Z, N and E are equipotent, where N is the set of natural numbers, Z is the set of integers, and E is the set of even integers.

4. A definition of what it means for a set to be finite. (It will be useful to first define the sets N_k = {1,2,3,...,k}.)

5. A definition of what it means for a set to be infinite.

6. Proof of the fact that N is infinite.

7. Proof of the fact that Z is infinite.

8. A definition of what it means for a set to be countable (denumerable).

9. Proof of the fact that Q (the rational numbers) is countable.

10. Proof of the fact that R is not countable.

11. A definition of cardinality as a mapping whose domain is the set of equivalence classes with respect to the equivalence relation "equipotence".

12. A definition of what it means for one cardinal number to be less than or equal to another.

13. Justification of the fact that is a reflexive and transitive relation on the set of cardinal numbers.

14. A discussion (without proof) of the theorem which tells us that is an antisymmetric relation on the set of cardinal numbers and hence that is an order relation.

15. Justification of the fact that, for cardinal numbers,

16. Proof of the fact that there is no largest cardinal number.

17. A brief discussion of the continuum hypothesis.

The paper should be clearly written, thoughtfully organized and presented in a manner which shows that you understand what you are writing and not simply regurgitating what is in the text. It should be typed (or computer printed) double spaced. As with other written work, the proposed audience should be other students is in the class. A first draft of the paper will be due on Monday, November 26, with the final draft due on Wednesday, December 10.

Grading:

There will be two midterm examinations, each worth approximately 20% of the final grade, and a cumulative final examination, worth approximately 25%. Grades for the written homework assignments will be worth approximately 25% and the group paper will be worth the remaining 10%. In the case of borderline grades, class participation (attendance as well as asking and answering questions) may be the deciding factor.

During the course of the semester, when I return your "hand-in" homework assignments, I will occasionally write "resubmit" or "rewrite" at the top of some problems. These problems are to be rewritten, taking any comments I have made into consideration, and resubmitted in an improved form. These will then be regraded, and any improvement in your grade will contribute to your grade for the problem. In grading resubmitted work, I will pay special attention to the style of your presentation and the technical aspects of your writing (in particular, grammar and spelling), so that you may seek help at the Writing Center if you feel you need it.

Although this may seem unduly complicated and may provide more detail than you really want to know, I would now like to explain the final grading system.

Grades on exams will be assigned in the following manner: Each problem will be worth a certain number of points (which may or may not add up to a total of 100 on any one exam). Problems will be graded based on the indicated number of points and the total of these awarded points will be computed, giving a numerical score for the exam. These numerical scores will then be converted into letter grades in accordance with the following procedure: Prior to my totaling individual scores, I will determine what I deem to be appropriate cut-off level for each letter grade. For example, I might decide, "On this exam, in order to get an A-, a student should get a total of 72 points." After each student's grade is totaled, I may make small modifications in my predetermined cut-offs. For example, if my predetermined cut-off score for an A- is 72 points, and if there is one student with 71 points and the next highest point total is 54, I may decide to move the cut-off for A- to 71.

The grade for homework will be determined similarly, except that the letter grade will be based on the total points each student accumulates for the semester rather than on having a letter assigned for each assignment. (Most problems assigned will be graded on a basis of 5 points. Although I may modify this, past experience suggests that the cut-off for an A- will be a per problem average of 4 points out of 5. Thus, if, for example, 22 problems are assigned for the semester and if you get a total of 88 points, then your homework grade will be A-. Cut-offs for B-, C- and D- respectively will probably be 3+, 3- and 2. [+ represents 1/3, and - represents 1/3 off]. )

Finally, the group paper will be assigned a letter grade directly.

After a letter grade is assigned to each component of the grade, the letters will be converted to numbers according to a scale such as the following: A+=97; A=94; A-=90; B+=87; B=84; B-=80; C+=77; C=74; C-=70; D+=67; D=64; D-=60; F+=55; F=50. The weighted average of the components will then be computed and reconverted to a letter grade using a slightly more liberal scale such as: A=92; A-=88; B+=85; B=82; B-=78; C+=75; C=72; C-=68; D+=65; D=60; D-=55.

As an example, suppose a student receives grades of A, and B- on the two midterm exams, and grades of A- on the final; A- for the homework and B+ for the group paper. Then, using the scale above, these will be converted to 94, 80, 90, 90, and 87 respectively. Using the weights indicated above, the weighted average would be (.2)94 + (.2)80 + (.25)90 + (.25)90 + (.1)87 = 88.5. This would then be converted to a letter using the second scale. The resulting grade for the course would be A-.

Attendance and lateness:

It is expected that you attend class regularly and on time. I expect this not only because I want you to share in the benefit of my "words of wisdom" but also because a group with several members absent no longer can function as a group. While I do not have any hard and fast rules such as "five absences means an automatic F", I do take attendance, and you can be sure that excessive unexcused absence or lateness will affect your grade. In particular, you can't participate if you are not here.

You are also expected to be present for each scheduled exam. If you determine in advance that you will not be able to be present on the date of a scheduled exam, I expect you to notify me immediately of that fact, so that we can discuss alternate arrangements. If a last minute emergency prevents you from being able to take a scheduled exam, I expect you to call my office AS SOON AS POSSIBLE. (If you are ill or your car won't start, I expect to hear from you on the morning of the exam. If you are in a coma, then when you emerge from the coma will be soon enough, assuming that you have a doctor's note attesting to the fact that you were in the coma!) If I am not available when you call, leave a message with the secretary stating your name, the class you are in, your reason for missing the exam and a phone number at which I can reach you later that day. If you don't follow these instructions, I am likely to treat your absence as unexcused.

If we have not made arrangements in advance for you to take a make-up exam, you should be prepared to take a make-up on the day you return to campus. In general, I find it difficult to compose two exams which are truly comparable, so I prefer not to. For an excused absence, I may decide to give a make-up on the day you return, or I may decide to disregard the exam and to increase the percentages which your other exams and/or writing grades contribute your final grade. (This is my decision, not yours.)

Exams which are missed due to unexcused absence will receive the grade of zero.

Students who miss more than one exam may receive a zero for the second exam, regardless of the reason for the absence, unless advance arrangements have been made.

The Course:

This course serves three purposes. It is a first introduction to mathematical abstraction and to mathematical proof; it introduces a number of topics which arise time and time again in more advanced math courses; it serves as an important indicator of whether mathematics is right for you and whether you are right for mathematics.

The course begins by considering the nature of mathematics as the study of axiomatic systems and the implications this has for the relationship between mathematics and the real world.

Chapter 1 introduces the tools of propositional logic, the rules of inference and methods of proof.

Chapter 2 deals with the basic notions of set theory. It also considers mathematical induction (a very important technique for proving certain types of theorems) and some techniques of counting.

Chapter 3 in concerned with the notion of mathematical relations. In particular, it considers equivalence relations and their connection with partitions. This will be supplemented by a discussion of order relations (not adequately covered in the text).

Chapter 4 examines the notion of a function, as a particular kind of relation. It also considers what it means for a function to be injective (one-to-one), surjective (onto) or bijective (a one-to-one correspondence).

Chapter 5 considers the notion of cardinality as an example of an equivalence relation. Much (but not all) of the material needed for the group paper may be found in this chapter. I will not cover this chapter in class.

Chapter 6 is an introduction to group theory. We will only consider the beginning of the chapter, discussing what is meant by an algebraic operation and what is meant by a group.

Chapter 7 deals with the axiomatic structure of the real number system. As with Chapter 6, this chapter delves too deeply into topics covered in more advanced courses. Instead of following the thread of the text, we will consider the construction of the real numbers as an illustration of the application of the concepts found in the earlier parts of the course.

Sections covered:

Chapter 1, sections 1-6

Chapter 2, sections 1-5

Chapter 3, sections 1-3, supplemented by material on order relations

Chapter 4, sections 1-4

Chapter 5, sections 1 & 2, selected topics from sections 3-5

Chapter 6, sections 1 & 2, selected topics from section 3

Chapter 7, section 1, supplemented with discussion of the construction of the real numbers starting with Peano's postulates.

Overnight assignments (homework to be done and discussed in class but not collected): Do these as soon as the relevant section of the text has been discussed in class.

Section 1.1, page 7: 1, 2a,b,d,f,g,h,j, 3a,c,e,g,i,j,k,m, 4, 7, 10a,b,d

Section 1.2, page 15: 1, 2a,c,f,i, 3a,c,e,g, 4, 5, 7, 8, 9c, 10, 11, 14c,e,f,i

Supplementary exercises page 1: #s 1

Section 1.3, page 24: 1a-k, 2, 4, 5, 6, 7, 8, 9,10

A function f(x) is said to be bounded on an interval I if there is a number M such that the value of f(x) never exceeds M whenever x is in I. Introduce appropriate propositional functions and represent the statement that f(x) is bounded on I symbolically. Then negate the symbolic definition and give an idiomatic English version of a useful denial.

Supplementary exercises, page 1: #s 2, 3, 4, 5

Section 1.4, page 35: 2, 3, 6, 7, 8a⁽²⁾

Section 1.5, page 42: 1, 2, 3a,h, 4a, 6b,d, 7a²

Section 1.6, page 50: 2, 3, 4, 5a-i, 6a, 7e,f,g ²

Secvtion 1.7, page 61: 1c,e, 5b, 8a, 10c

Section 2.1, page 71: 1, 2, 3, 4, 5, 7, 8, 13, 14, 15, 18 ²

Section 2.2, page 77: 1, 2, 3, 4, 5, 6, 7, 10, 14a,b,d, 15 ²

Supplementary exercises, page 2 # 3

Section 2.3, page 87: 1, 2, 3, 4, 5, 6a, 7, 8, 11, 12, 17a,b, 19 ²

Section 2.4, page 100: 1, 2, 4, 5, 6, 7, 8a,b,g,k,n, 9a,c, 10, 13 ²

Section 2.5, page 109: 1, 3a, 5d, 8, 12

Section 2.6, page 119: 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 14, 15, 16, 17, 18

Section 3.1, page 135: 1a,d, 2, 3b,f, 5, 6, 8a,b,d,f,h,i,j,k, 9, 10a,b,c,i,o,p, 12, 13, 17 ²

Section 3.2, page 142: 1, 2, 4a,c,d,e,f,g,h,j, 6, 7, 8, 12, 13, 15 ²

Section 3.3, page 150: 1, 2a,b, 3, 4, 5a,c,d, 7a,b, 9 ²

Section 3.4, page 158: 1a,b,c,d, 2, 3, 5, 7, 8, 13, 15, 18 ²

Supplemental material: Use quantifiers to express the definitions of "upper bound",

"least upper bound", "maximal element", "maximum element" symbolically

Supplementary exercises, pages 5, 6:#s 2, 3, 4, 5, 6, 7

Section 4.1, page 178: 1, 2, 3, 4, 6, 7, 8, 9a,c, 10, 11, 14, 16a,b,c

Section 4.2, page 187: 1a,b,c,e,f,g, 3a-g, 4, 7a,b, 16, 18 ²

Section 4.3, page 197: 1, 2, 3, 5, 7, 8, 9, 10, 11, 14, 15, 16 ²

Supplementary exercises, page 7 # 1

Section 4.4, page 204: 1, 2, 3, 4, 5, 8, 9, 10, 14, 16, 17, 18

Section 6.1: page 250: 1, 2, 3, 4, 5, 6, 7, 8, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24 ²

Supplementary exercises, pages 8-11 #s: 1, 2, 3, 4, 5, 6,

Section 6.2:

Supplementary exercises, pages 8-11 #s: 7, 8, 10, 11, 12, 13, 15, 16, 19, 20, 21, 22, 23, 24

Written work to be handed in and graded: (Due dates may be adjusted)

1. (Due 1/24/02)

Student information sheet

Pretest

2. (Due 1/29/02)

Section 1.1, page 8: #5

3. (Due 2/5/02)

Section 1.2, page 17: #12

Section 1.3: A function f(x) is said to be continuous on an interval I if

(>0)(xI)(>0)(yI)[ (x-y<) (f(x)-f(y)<) ];

f(x) is said to be uniformly continuous on I if

(>0)(>0)(xI)(yI)[ (x-y<) (f(x)-f(y)<) ].

a) Discuss the difference between continuity and uniform continuity. Are they logically equivalent? Does one of these imply the other? Explain.

b) Give an idiomatic English version of a useful denial of the statement that f(x) is continuous on I.

4. (Due 2/12/02)

Section 1.4, page 35: #4, 10b ⁽³⁾

Section 1.5, page 42: #3c,d,f,g, 9, 12a,c,d ³

5. (Due 2/19/02)

Section 1.6, page 50 #1b,d,e,f,g, 8c,e,f,g,h,i ³

Section 1.7, page 61 #1a, 3a, 5d, 11a,b,d,e,f ³

6. (Due 2/26/02)

Section 2.1, page 71: #10, 11 12, 19a,b,d,f,g ³

Section 2.2, page 77: #8a,g,k,n,o,r, 9, 11, 13, 17b,c,f,g,h ³

Supplementary exercises, page 2 # 2

7. (Due 3/5/02)

Section 2.3, page 87: #6b, 18, 20c,d,e ³

Supplementary exercises, page 2 # 1, 5

8. (Due on 3/19/02)

Section 2.4, page 100: #8d,e,j,o, 9b, 11, 15b,c,f

Section 2.5, page 109: #2, 6b,d, 15a,c,d ³

9. (Due 3/26/02)

Section 2.6, page 119: #19 ³

Section 3.1, page 125: #3d, 20c,e,f,g³

Supplementary exercises, page 2 #4

10. (Due 4/2/02)

Section 3.2, page 135: #9, 15, 16a,b,c³

Supplementary exercises, pages 3 & 4, #1, as assigned

11. (Due 4/9/02)

Section 3.3, page 140: #14a³

Section 3.4, page 148: #6, 23, 25a,c³

12. (Due 4/16/02)

Section 4.1, page 170: #17, 18³

Section 4.2, page 177: #11,20a,b,d³

13. (Due 4/23/02)

Section 4.3, page 185: #4, 6 12, 17a,b,d,f,g,h³

Section 4.4, page 194: #19, 20b,c³

Supplementary exercises, page 7 #2

14. First draft of group paper (See syllabus page 4) - Due 4/25/02

15. (Due 4/30/02)

Section 6.1, page 239: #10, 11, 17, 20, 25a,b³

Supplementary exercises, page 10 # 14, 17, 18

16. Final draft of group paper - Due 5/9/02

Tentative examination dates: 2/28/02 and 4/16/02

Final examination: 5/14/02

1. Primarily for those who feel they need additional assistance. I will not be using this text in the course, but, although the material is arranged in a different order, the explanations are clear and there are many worked examples.

2. See footnote 3 on page 10.

3. On all "proofs to grade" problems, use the following grading scale: A if the proof is correct and nicely presented; A- if the proof is correct but not well presented; B if the proof is essentially correct, but something is missing; C if part of the result is correctly proven but the proof is not complete; D if the approach is correct but the details are not; and F if the result is not correct, if the proof contains a serious flaw in reasoning or if the proof presented makes no significant progress toward proving the stated result. If you assign any grade other than A, you must explain why - i.e, what do you find objectionable about the given proof. DO NOT tell me how the result should have been proven, simply tell me what is wrong with the proof which has been presented.