Math 256, Introduction to Higher Mathematics, Fall 2008
This site updated May 12, 2008
(Recent updates will usually be above the first horizontal line. The syllabus is here.)

For the Spring 2008 site, see here.  

Syllabus for Introduction to Higher Mathematics, Math 256, at Montana State University.

Time and Room:  1:10-2:00 pm, MWF, in Wilson 1-139 (Fall 2008)

Goals:  You will learn to read, write, and think like an advanced mathematician. You will learn to read symbolic mathematics with comprehension, express mathematical thoughts clearly, reason logically, recognize and employ common patterns of mathematical thought, and read and write proofs.

Instructor:  Dr. Warren Esty, 994-5354, Wilson 2-238 (East wing, South wall).  westy at math dot montana dot edu  Phone calls and e-mails are both fine. Appointments are easier to arrange on the phone. For regular office hours, click here.

Required text:  Proof: Introduction to Higher Mathematics, third edition, third printing (available at the bookstore in August), by Warren W. Esty and Norah C. Esty. (The second printing, used Spring 2008, has been superceded.) A web site about the text is here. There is no solutions manual for students.
    This course has almost nothing to do with calculation, so no calculator is required. 

Course Content:  We will proceed straight through the text, covering every section through Chapter 5.
    Chapter 1:  Preview of proof, sets, logic for mathematics (including truth tables and important logical equivalences that provide alternative forms).  Chapter 2: generalizations, existence statements, negations, reading symbolic mathematics with full comprehension, logical form and deduction, and practice with alternative forms in the context of rational and irrational numbers.  Chapter 3:  Proof of theorems about inequalities and absolute values, theory of proofs, proofs by contradiction or contrapositive, proofs by mathematical induction, and common types of mistakes in proofs.  Chapters 1 through 3 complete the theory.
   The rest of the course provides practice in several content areas of mathematics. Chapter 4 is Set Theory. Chapter 5 is about the concepts of one-to-one and onto, functions applied to sets, and cardinality. We will cover through Chapter 5.

Prerequisite:  Math 182 (two semesters of calculus). The mathematical sophistication provided by additional mathematics such as Math 221 (Matrix Theory) and Math 224 (Calculus of Functions of Several Variables) would be very welcome, but the material covered in those courses is not a prerequisite.   
    This course is primarily for students who wish to be math teachers or math majors. It is an introduction to the most important types of thought processes in mathematics.

Etiquette. Proper etiquette is required. During class, students will not engage in any potentially distracting behavior such as reading a newspaper or whispering about non-math subjects. Cell phones must be turned off and unavailable. Pagers or watches that make a sound, however quietly, must have the sound off. No type of earphones is allowed.

Attendence:  Attendence every day is expected. More than a couple unexcused abscences is unacceptable. Of course, excuses for academic reasons, illness, participation in university sporting events, and significant life events will be accepted.
    Every day in class you will learn about common mistakes and how to avoid them. It is not possible to recogonize your own errors in logic, so you must take every opportunity to see deceptive errors in reasoning explained and to get feedback about your own and your classmates errors in reasoning. Students who miss a day are missing a significant lesson that cannot easily be recovered from the text alone.

Homework.  There will be homework due almost every day. It is important that it be attempted on time. The work you hand in need not be all correct, but it must display serious effort. More than a few late homeworks is not acceptable.
    You are expected to work, on average, about two hours outside class for each class hour.
    You must read the assigned sections. Learning to read math with full comprehension is one of your goals, and you learn to read by reading. Reading is part of those two hours.
    Bring your text to class every day. We will use it in class regularly.
 
Exams and Grading.  There will be unit exams, frequent quizzes, regular homework, class participation, and a comprehensive final. 
   To receive full credit, daily homework must be handed in on time. Homework handed in late will receive half credit.
Exam dates will be announced on this site.
Homework and its due dates will be announced on this site.
 
Conflicts.  You are required to take all exams and the final exam at the scheduled hours (unless you have another exam or class scheduled at that hour, in which case we will make arrangements). Any exceptions must be approved well in advance, and in no case will exceptions be made for two exams.

Attitude.  This course requires a change in attitude about what mathematics is and how it is learned. It is hard for anyone to change their attitude about anything, so this part may be difficult for you. One goal is to have you learn to read with comprehension. Then you will be able to learn without relying on the teacher. How can we help you reach this goal?  By making you read and work with material even before there is a lecture on it. You learn to read by reading. So, expect to learn by reading. Lectures will clarify things, but not always introduce things.

Success.  Higher mathematics requires a significantly different way of thinking. There is a much greater focus on learning by reading. There is a much greater focus on the truth, or falsehood, of statements and connections between facts. There is much less focus on algorithms (methods for doing problems).  
    To succeed you will need to read far more than you did in previous math courses. You learn to read by reading.
    Advice on how to learn math in this course is here. 

Here is how Spring 2008 began. Fall 2008 will be similar, but the HW might be different.

Homework due Friday, Jan. 18:  Read all of Section 1.1. This is a reading and writing course. You learn to read and write by reading and writing. Read the text thoroughly. Every section, learn the terms listed at the end of the section's conclusion. For this section, hand in at the beginning of class Friday: A1, 3, 5, 7, 11, B1, 5, 7, 21, 23.  Each day, homework will be put on the board by (self-selected) students. Occasionally put a problem on the board before handing it in.
    Label your homework with your name at the very top of the page and the section number of the homework just below it.

    The first day in class you will get an important handout listing some special emphases of this course. Here is a link to a copy in HTML (here is a copy in MS Word.)

Monday is a holiday.

Due Wednesday, Jan. 23:  Section 1.2:  A28, 32, B22, 24

Due Friday, Jan. 25:  (1.2, part 2): B42, 44, (1.3) A2, 5, 12, 18, B1, 3, 8, 12

Read each section. Do not skip the harder parts. In fact, when the going gets rough you need to slow down and read it several times until it makes sense. If it remains unclear, ask me! 
    If something on your homework is wrong, please make sure you understand why. Do not treat your homeward as just part of your grade. Treat it as an occasion to learn. Anything you got wrong must be looked at again and studied much harder than anything you easily got right. Some things are easy. It is not a thrill if you can learn the easy stuff. Some things are harder. Put substantial effort into making sure you understand the harder stuff too. 
    I am soliciting your help. If you find a typo, please let me know and if you are the first you will get a minor reward. (The current list of typos is posted here.) You can e-mail it to me (westy at math dot montana dot edu) or tell me about it before or after class. If you find a particular paragraph unclear, please let me know. I can go over it in class and revise it for the next edition.

Here is advice about how to learn math in this course.

Be sure you can pronounce everything, and grasp all the "grammar" exercises. Little typographical differences can make a big difference in the referent (the thing being named or referred to). For example, there is a major difference between a and A and a major difference between (, [, and {.

Page 17, Example 10:  In English, "is in" and "is contained in" might be synonymous. But, it Math, they are not. I avoid "is contained in" because it refers to the concept of subset, not membership, which "is in" refers to.  2 is in {1, 2, 3, 4}.  {2} is a subset of {1, 2, 3, 4}.
    All intervals are sets. Some sets are intervals.  {1, 2} is a set that is not an interval.
    0 is not in (0, 3).  If the endpoints are included, we use square brackets, not parentheses.
    To show (0, 3) is not a subset of [1, 4], we must exhibit an element of (0, 3) that is not in [1, 4]. x = 1/2 will serve, but x = 0 will not.

Due Monday, Jan 28:  (1.3, part 2)  B2, (1.4) A1, 2, 8, B15

Quiz Wednesday, Jan 30, on 1.1 and 1.2.

Due Wednesday, Jan. 30:  (1.4, part 2)  B1, 2, 6, 10, (1.5)  A1, 24

Mathematics is a written language. To get good at math, you must read it. Read!
If you are coming to class and feel yourself slipping even the slightest bit behind, please come see me in the office. I want to help!
Fortunately, we will use the language of mathematics every day and we never drop any topic, so you will see every usage and hear every pronunciation again and again. Pay attention and notice what is giving you trouble. Let me know and I will help. 

On the homewowrk the truth tables requested in 1.3, B1 and B8, were often written incorrectly. Here are the rules:
1)  Basic components are listed on the left (see Examples 16 and 22) in the usual order, on page 50.
2)  Each connective gets its own column (do not skip steps and go straight from A and B to "(not A) or B" or any two-connective form).
3)  If the form does not begin with A, for example, if it is "not B => not A", your table still begins with A and B on the left columns. Then you make columns in order. In this case, after the columns for A and B comes one for "not B", then "not A" and then "not B => not A".  [I am part way through grading and so far, everyone who skipped columns has actually gotten it wrong. Each connective makes its own column!  Look again at all the parts of 1.3, B1.  It says it takes 9 columns. 1) H    2) C    3) H => C    4)  not C    5)  not H    6)  not C => not H    7)  (not H) or C    8)  C => H    9)  (not H) => (not C).  NINE!

Friday, Jan. 25, in class we did many of the pronunciation and grammar exercises from Section 1.2. Expect a quiz on 1.1-1.3 on Monday (but not yet including truth tables).
Sections 1.3 through 1.6 cover truth-table logic. Yes, I want you to know how to make truth tables, but even more important is that you understand how the connectives work. You must understand the theorems (logical equivalences) so well that they make perfect sense and using them becomes almost automatic. Advanced mathematics uses the theorems (repeated on pages 83-85) very frequently, and usually without comment, because you are expected to fully grasp them. If they don't come easily, work hard to master them. (Of course, you may ask me for help.)

Future homework will be posted on this page, usually above the first horizontal line.