Spring, 2000
Syllabus
Instructor: Dr. Orin Chein
Office: 612 Computer Building
Office hours: MWF 9:45-11:00, T 11:15-12:00 or by appointment
Phone: 204-7846
e-mail: orin@math.temple.edu
Text: A Transition to Advanced Mathematics (Fourth Edition), by D. Smith, M. Eggen and
R. St. Andre
Recommended Supplementary text(1): Discrete Mathematics (Second edition), by S. Lipschutz and M. Lipson, Shaum's Outline Series.
Pre/co-requisites: Two semesters of Calculus (or permission of the instructor)
Schedule revision: Class is scheduled to meet on Monday, Wednesday and Friday from Monday, August 30 through Friday, December 10. There will be no class or office hours on Monday, September 6 (Labor Day) or Friday, November 26 (Thanksgiving break), and I will be unable to be here on Wednesday, September 15 and on Monday, September 20. I will also not hold office hours on Tuesday, August 31, Friday September 3 or Tuesday, September 14.
In order to make up for the missed classes and to allow for extra time to go over homework and answer questions, I plan to start class at 8:15 on all Mondays and Fridays other than those listed above. (On Wednesdays, class will start as scheduled, at 8:40). I realize that some of you may not be able to make this earlier time (although I hope that you all can do so), so I will use the time only to go over homework and respond to questions.
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, 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.
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 will be expected to revise and rewrite some of these assignments, to improve your writing (and your grade).
Unless I announce a deviation, hand-in problems are due on the dates indicated on Pages 9-10 of the syllabus. Each student in the class will be allotted two grace chits for lateness. That is, twice during the semester a student may turn in a homework assignment up to one class late with no penalty, provided that I have not gone over the problems in class. Once these two grace chits are used, late problems which I have not reviewed in class and which are submitted by the following class will receive at most half credit. Problems which I have reviewed in class or which are submitted later than the following class will receive no credit but nevertheless must be submitted (otherwise there are additional penalties, as indicated below).
Problems which I have asked you to rewrite will be due one week after the problem has been returned to you and are to be resubmitted together with the originals.
Problems which were due but which have not been submitted or rewritten by the date of an exam will result in a loss of credit on the exam grade. Up to two missing problems will result in no penalty. Each missing problem beyond two will result in a loss of 5 points from your exam grade.
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 in blue or black (but not red) 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.
The term paper:
In addition to the hand-in homework problems, each student will be expected to submit a term paper discussing the notion of cardinality, a topic that I am not able (due to lack of time) to cover fully in class. You will be asked to explain how the development of this topic depends on other topics which have been discussed in class.
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; and I don't mind your discussing this material with other students in the class, so long as no one copies anyone else's work. I will also be available for consultation and assistance with these papers.
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 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 in the definition.)
2. Proof of the fact that equipotence of sets is an equivalence relation.
3. Proof of the fact that , N and E are equipotent, where N is the set of natural numbers, 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 Nk = {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 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 and hence is not equipotent with Q.
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 is 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. An outline of the paper will be due on Monday, October 25; a first draft will be due on Monday, November 15, with the final draft due at the last scheduled class of the semester - Friday, December 10.
Grading:
There will be three midterm examinations, each worth approximately 14% of the final grade, and a cumulative final examination, worth approximately 20%. Original grades for the written homework assignments will be worth approximately 20%; and the term paper, 8%; resubmitted work (see below), 5%; and class participation 5%.
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 constitute part of your grade for resubmitted work. 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. [Here, + represents 1/3, and - represents 1/3 off]. )
The grade for "resubmitted work" will be based on the ratio of total number of "improvement points" you receive to the total number of improvement points you could possibly have received. This will vary from one student to another. Thus, for example, if on a 5 point problem which I ask you to resubmit you originally receive 1+ = 1 1/3, the possible improvement points will be 3 2/3. If, when you resubmit the problem, you get a grade of 4, then your actual improvement will be 2 2/3 and your ratio will be [(2 2/3)/(3 2/3)] = 8/11. Of course, ratios will be computed on a per semester rather than on a per problem basis. They
will then be converted to letter grades using a system of cut-offs as above, the precise cut-off points yet to be determined.
Finally, the group paper and class participation will be assigned letter grades 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, B and A- on the three midterm exams, and grades of A- on the final; A- for the homework; B+ for the term paper; C+ for class participation; and A- for resubmitted work. Then, using the scale above, these will be converted to 94, 84, 90, 90, 90, 87, 77 and 90 respectively. Using the weights indicated above, the weighted average would be (.14)94 + (.14)84 + (.14)90 + (.2)90 + (.2)90 + (.08)87 + (.05)77 + (.05)90 = 88.83. 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 excessive absence can adversely affect your grade, in accordance with the following policy:
If you know in advance that you will have to miss a class, I expect you to give me a written note to that effect, in advance, briefly explaining why.
If you discover on the morning of a class that you will not be present, I expect you to contact me either by phone (you can leave a voice mail message if I am not in) or by e-mail, on the morning in question, again briefly explaining your absence and, if possible telling me how I can reach you later that morning.
If (and only if) circumstances truly prevent you from contacting me on the day in question (I will be the judge), then I expect you to do so as soon as possible thereafter.
For absence from two or more consecutive classes which is not arranged with me in advance, I will expect a doctor's note.
Any absence for which you do not adhere to the policy above will be treated as unexcused. At the end of the semester, each unexcused absence beyond two will result in a reduction in your grade by a full grade level. Thus, for example, if your grade would otherwise be B+ but you have four UNEXCUSED absences, then you will get a D+ for the course. In addition, I will not give credit for homework assignments that were due during a missed class, unless the absence is excused.
I also expect you to be on time to class. Persistent lateness will be treated as an unexcused absence. (Note, however that even though I will be starting some classes early,
as explained above, I will not consider you late until 8:40, as that is the scheduled class starting time for class.)
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 on my voice mail 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 will 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 is 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 term 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.
The teacher:
Since you have a choice of sections in this course, I thought it might be appropriate to tell you a bit about myself. I am a full professor in the Department of Mathematics. My current research interests lie in the area of "loop theory", but my training and early work is in the field of combinatorial group theory, and I have always been interested in combinatorics as well. I am a pure mathematician, with little concern about whether the mathematics I do can be applied to the "real world" or not. Mathematics is like art or music - it can be beautiful, and it is intrinsically worthwhile in its own right.
I came to Temple in July of 1968, directly after receiving my Ph.D. Except for sabbatical years at the Technion (in Haifa, Israel), at the University of British Columbia (in Vancouver, Canada) and with the National Security Agency (in Washington, D.C.), I have been at Temple ever since. Thus, I am starting my 32nd year at Temple.
Based on the course evaluations which I always give at the end of the semester and on scuttlebutt which I hear from students and colleagues, most of my students consider me to be either a "very good" or an "excellent" teacher (I was one of the recipients of the Temple University Great Teacher Award in 1995); but, most years, there are also a few students whose needs I don't meet, who don't like my teaching style or who consider me too demanding, and who give me a "poor" or "terrible" rating.
On the plus side, I explain concepts well, I am responsive to questions, I am enthusiastic about my subject matter, I genuinely care about my students and I try to be accessible outside of class.
On the minus side (from the perspective of some students), I am very demanding, assign a lot of work, set high standards (because I am a bit of a perfectionist), and give very long and comprehensive exams. On each exam, I try to include at least one question on every major topic I have covered (both to make students study everything and also to minimize the point value of each question, so that a single error should not have a major negative impact on your grade). As a result, while students who have mastered the material should have no real trouble finishing most of my exams, students who are struggling or who work very slowly often feel under time pressure to finish. (I will start each exam at 8:00 AM to allow you extra time should you need it.)
Nevertheless, I believe that most students consider me to be a fair grader. You have to convince me that you have mastered most of the material in order for me to give you an A, but, generally (no promise implied), you can get at least a C- by convincing me that you have made a serious effort and learned something, even if you haven't mastered most of the material. ("Making a serious" effort includes coming to class regularly, attempting to do the homework, asking questions about material you don't understand, and coming to me for help, when you need it.)
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
Section 1.4, page 37: 2, 6, 8a, 10(2)
Section 1.5, page 46: 1a-g, 2, 3, 4, 5a-i, 6a, 8, 92
Section 1.6, page 54: 3, 5, 112
Section 2.1, page 65: 1, 2, 3, 4, 5, 7, 8, 14, 15, 18, 192
Section 2.2, page 71: 1, 2,3, 4, 5, 6, 7, 10, 14a,b,d, 15, 172
Supplementary exercises, page 2 # 3
Section 2.3, page 81: 1, 2, 3, 4, 5, 6a, 8, 11, 12, 17a,b, 19, 202
Section 2.4, page 93: 1, 2, 4, 5, 6, 7, 8a,b,g,k,n, 9a,c, 10, 13, 152
Section 2.5, page 102: 1, 3, 5d, 8, 12
Section 2.6, page 111: 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 14, 15, 16, 17, 18
Section 3.1, page 125: 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, 202
Section 3.2, page 133: 1, 2, 4, 6, 7, 8, 12, 13, 14d2
Section 3.3, page 139: 1, 2a,b, 3, 4, 5a,c,d, 7a,b, 9, 14b2
Section 3.4, page 147: 1a,b,c,d, 2, 3, 5, 7, 8, 13, 14, 16, 17a, 18, 19, 20, 25b2
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 167: 1, 2, 3, 4, 6a,b,c, 7, 8, 9a,c, 10, 11, 14, 16
Section 4.2, page 176: 1a,b,c,e,f,g, 3a-g, 4, 7a,b, 16, 18, 20a2
Section 4.3, page 185: 1, 2, 3, 5, 7, 8, 9, 10, 11, 14, 15, 16, 172
Supplementary exercises, page 7 # 1
Section 4.4, page 193: 1a, 2a-e, 3, 4, 5, 8, 9, 10, 14, 16, 17, 18
Section 6.1: page 238: 1, 2, 3, 4, 5, 6, 7, 8, 13, 14, 15, 16, 18, 19, 21, 22, 23, 24, 25c2
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 9/1/99)
Student information sheet
Pretest
2. (Due 9/6/99)
Section 1.1, page 8: #5
3. (Due 9/13/99)
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.
Supplementary exercises, page 1 # 5
4. (Due 9/22/99)
Section 1.4, page 37: #4, 7a, 10b,d,g(3)
5. (Due 9/27/99)
Section 1.5, page 48: #1b,d,e,f,g, 9c,e,f,g,h,i3
Section 1.6, page 54 #1a,c,e, 3, 11a,b,d,e,f3
6. (Due 10/4/99)
Section 2.1, page 65: #10, 11 12, 19a,b,d,e,f3
Section 2.2, page 71: #8a,g,k,n,o,r, 9, 11, 13, 17b,e,f,g3
Supplementary exercises, page 2 # 2
7. (Due 10/11/99)
Section 2.3, page 81: #6b, 18, 20c,d,e3
Supplementary exercises, page 2 # 1, 5
8. (Due on 10/18/99)
Section 2.4, page 93: #8d,e,j,o, 9b, 11, 15b,c,e
Section 2.5, page 102: #2, 5d, 6b,d, 153
9. (Due 10/25/99)
Outline of the term paper (including bibliography)
Section 2.6, page 113: #193
Section 3.1, page 125: #3d, 20c,e,f,g3
Supplementary exercises, page 2 #4
10. (Due 11/1/99)
Section 3.2, page 135: #9, 15, 16a,b,c3
Supplementary exercises, pages 3 & 4, #1, as assigned
11. (Due 11/8/99)
Section 3.3, page 140: #14a3
Section 3.4, page 148: #6, 23, 25a,c3
12. (Due 11/15/99)
First draft of term paper (See syllabus page 3)
13. (Due 11/22/99)
Section 4.1, page 170: #17, 183
Section 4.2, page 177: #11,20a,b,d3
14. (Due 11/29/99)
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
15. (Due 12/6/99)
Section 6.1, page 239: #10, 11, 17, 20, 25a,b3
Supplementary exercises, page 10 # 14, 17, 18
16. Final draft of term paper - Due 12/10/97
Tentative examination dates: October 11, November 8, December 6
Final examination: December 15, 8:15-10:45 AM
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.