CRN: 12086

Course description:

You will learn in this course how to think like a mathematician:

• Understand the nature of a rigorous mathematically proof
• Learn to write such proofs

To do so means you must acquire some knowledge in the following areas:

• logic: direct proofs vs proofs by contradiction, logical quantifiers
• the difference between axioms, definitions and theorems
• sets, functions and relations
• induction and recursion

The subject matter used to teach you this will be ••primarily taken from the theory of number systems:•• an axiomatic approach to the properties of natural numbers, integers, fractions and real numbers will be presented. You will also learn how to compare the sizes of infinite sets (cardinality).

At least 25% of the course will be devoted to the foundations of real analysis. Most of that material cannot be found in the text book and you will have to consult the additional course material. See the section on lectures below.

General course info / Math 330 section 2 web site:

There is a website for Math 330 section 2. You can link to it from the instructor's home page . Here is a direct link .

Prerequisites:    

Math 222 (Calculus 2) on a C- level or permission of the instructor. If you did not pass Math 222 with a grade better than D, this does not automatically disqualify you, but you must see me asap! You need some Calculus background in the second half of the course and you may have to catch up on some topics to follow the course. You will be dropped from this course unless you can convince me that you have knowledge or are able to learn quickly about limits, continuity, power series, derivatives as limits of difference quotients and integrals as limits of Riemann sums.

Strongly recommended:

Basics of linear algebra: vector spaces and subspaces, linear independence, (linear) span, basis, and Euclidean space Rⁿ as a vector space.

Course material:    

Text book:    

The Art of Proof: Basic Training for Deeper Mathematics by Matthias Beck and Ross Geoghegan (Springer, 2010). (REQUIRED).

Instructor's lecture notes and additional material:    

See the Course material page of the Math 330 website.

Lectures:

The "Art of Proof" text book has three parts:

• Part 1: The Discrete - Ch.1 - 7
• Part 2: The Continuous - Ch.8 - 13
• Part 3: Further Topics - Appendices A - G

The first six weeks or so of the course will be about part 1 of the text plus a lot of additional material on sets, functions and relations which is presented in the instructor's lecture notes. Also, some material on cardinality will be discussed early.

The remainder of the course will deal with part 2 and appendix A of the text and the following material which you will find in the instructor's lecture notes.

• Ordered integral domains as an abstraction of the integers and the real numbers
• Real numbers: convergence, lim inf and lim sup of sequences; continuity of functions
• Basics of metric spaces: neighborhoods, open and closed subsets
• Sequence continuity and ε - δ continuity
• Uniform continuity and uniform convergence
• Absolute and conditional convergence of series
• Sequence compactness and the Heine-Borel theorem
• Zorn's Lemma and its use to prove that every vector space has a basis

Blackboard:

Grades will be recorded in the Blackboard online gradebook. Stay on top of your grades! If your grade is incorrect you must contact me immediately.

Record keeping:

You must retain all returned papers in case of any discrepancy with your course grade. I cannot correct mistakes in grading or recording of scores without the original document. I won't review disputed points after the final. All grading issues must be settled within one week of the return of the paper.

Exams:

There will be two midterm exams and one final exam. No notes, books, cell phones, or laptops are allowed for tests. Each standard exam will last 60 minutes and is worth 100 points. Make an effort to show up 10 minutes early for those exams so they can start on time.

Exam dates can be found on the course home page. Make all arrangements necessary to take the tests at those dates as it is extremely unlikely that they will be changed.

Final exam:

The final exam counts for 200 points and it will last two hours. Date and time are not known at the time of writing of this syllabus and I will publish the info in the course home page once it becomes available.

Date and time for all finals are set by the office of records (registrar) and there is no flexibility. Do not make travel arrangements that will have you leave campus prior to the final. You can request a makeup final only if you have another final at the same time (direct conflict) or you have three final exams scheduled within 24 hours. The makeup will be given after the regular final, not before. If you want to request to take the alternate final then you must do so by Friday, April 27 by sending me an email.

Quizzes & Homework:

Quizzes:

There will be approximately 10 quizzes. The sum of points will be adjusted to 200. The number of quizzes depends on how the class is doing in knowing the axioms, definitions, main propositions and theorems as checking for this will be the main purpose of the quizzes. Additional quizzes will be given if the class needs to do better. Quizzes will often not be announced.

Homework:

Homework counts for 40% of the grade and will be graded in iterations: You will have a total of up to 3 iterations (i.e., a total of 4 submissions) for most of those assignments. The final submission date will be noted on the homework assignment and it will usually be two weeks after the date when the homework is posted. You will have less than two full weeks during the last two weeks of the semester and you may get additional time when holidays fall into that period.

Especially at the beginning I will grade your homework according to the "red line" method: I stop grading when I see a major flaw and I'll mark that spot with a red line. Sometimes I'll comment on the nature of the problem, at others you will have to figure it out on your own.

You will learn from the course material and from my presentations how to write a proof, but here are some purely technical requirements you should be clear about from the start:

• Write your proofs very neatly. Use lined paper so that your text is written in straight lines.
• Leave margins of at least 1/2 inch to the left and at least 1 1/2 inches to the right.
• Write your homework double-spaced so you and I can insert corrections in a neat and orderly fashion.
• Write legibly! I'll be spending a lot of time looking at your homework and I won't have extra time to carefully deliberate whether your variable was, e.g., a "u" or a "v".

You are allowed, even encouraged, to work in groups on your homework but each student must turn in his or her own work, not a copied solution. You must note on your assignment with whom you collaborated.

Your grade:

You can earn a total of 1,000 points in this course:

TENTATIVE grading scale (unlikely to be adjusted):

Success:

See the Advice page of the Math 330 website.

Attendance Policy and Make-up Policy:

Attendance will not be taken but you are advised not to skip lecture: If you cannot spare the time to go to lecture then you should consider dropping/withdrawing from the course before your GPA is messed up.

Make-up exams and quizzes will only be given in response to an excused absence. Excused absences include illness, religious holiday, a major tragedy in the family and participation in official BU athletic events.

To be excused, absences must be properly documented, for example with a doctor's note. The document should be issued to you at the day of the test. For example, if you missed a test on Friday then provide a Friday’s doctor’s note. The makeup will be scheduled within 3 or 4 days after the missed exam. You must request a make-up in writing by sending an email.

Except in very exceptional circumstances such as a prolongued illness, you will NOT be given the opportunity to complete old assignments at the end of the semester to improve your grades. When you receive a grade, whether on Blackboard or in class, you will have one week to discuss that grade before it becomes FINAL.

Academic Honesty:

Incidents of academic dishonesty will be dealt with severely. There is precedent for giving an "F" for the course to a student who attempts to advance his/her grade illegally. Dishonesty includes, but is not limited to: copying another student's work, letting someone copy your work, lying to or intentionally misleading an instructor, signing someone else's name to a document. To eliminate suspicion, only writing/erasing utensils will be permitted on desks during an exam.

Best wishes for a successful semester!

Michael Fochler