CRN: 23673

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
• Generalize familiar mathematical concepts to abstract settings

To reach this goal you will acquire 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
• recursive definitions and proofs by induction
• proofs that use Zorn's Lemma

The specific subject matter used to teach you this will be primarily taken from number systems and real analysis:

• an axiomatic approach to the properties of natural numbers, integers, fractions and real numbers will be presented.
• You will learn how to compare the sizes of infinite sets (cardinality).
• Convergence and continuity will be explored with mathematical rigor in the context of real numbers.
• Convergence, continuity and other concepts involving closeness and distance will be generalized to metric and topological spaces.

General course info / Math 330 Section 5 web site:

There is a course website for Math 330 Section 5, and this syllabus is published both as part of that site, but also as a standalone version on the Mycourses (BrightSpace) site. You can link to my Math 330 site from my instructor's home page. Here is a direct link to the Math 330 Section 5 web site.

Prerequisites:    

Math 223-227 (single variable calculus) on a C- level or permission of the instructor. If you did not pass each one of those courses with a grade better than D then 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.

Helpful:

Basics of linear algebra: vector spaces and subspaces, linear independence, (linear) span, basis, and Euclidean space Rⁿ as a vector space. See the bottom of the Course material page of the Math 330 website for more.

Course material:    

• This instructor's lecture notes: Math 330 - Lecture Notes. We abbreviate this as MF or the MF doc. (Required) A downloadable PDF of this document and additional background material can be found on the Course material page of the Math 330 website.
• The Art of Proof: Basic Training for Deeper Mathematics by Matthias Beck and Ross Geoghegan (Springer, 2010). We abbreviate this as B/G or AoP. (Optional, but strongly recommended)

Lectures:

The Homework page of the course website contains a link to the complete set of homework assignments that were given out during the previous semester. The reading assignments of those homework sets constitute a complete record of the material that was taught during that semester. There will always be changes, but those reading assignments reflect to a large extent what I will teach during this semester.

Lectures during campus wide class cancellations:

BU admin leaves it at the discretion of the instructor whether or not lectures will be held in case of a cancellation due to inclement weather or other circumstances that transportation to/from campus is suspended.

My policy for this course is as follows.

• I will hold lecture via Zoom. I will create a separate session for each such meeting and publish the link for the meeting via email. This may happen on short notice, so check your email for an announcement in case there is a campus wide cancellation of classes.
• I plan to record those sessions and publish the links on the course materials page.
• If a midterm or quiz is scheduled during such a date then it will be administered during the next in-class lecture date.

MyCourses (Brightspace):

I do NOT plan to use Brightspace. Instead I will assign to each one of you a course internal ID which is known only to you and me. I will periodically email a spreadsheet of everyone's grade, one row per student, and each row will contain that internal ID. That way your grades will remain anonymous.

Record keeping:

Stay on top of your grades! If your grade is incorrect you must contact me immediately.

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, May 6, 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.

Students With Disabilities:

Students requesting disability-related accommodations should register with the Services for Students with Disabilities office (SSD). They are the appropriate entity on campus to determine and authorize disability-related accommodations. For more info please click here. Once you are registered with SSD you should approach me during office hours or after lecture so that you and I can discuss the implementation of your accommodations.

Your grade:

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

TENTATIVE grading scale (unlikely to be adjusted):

Covid:

An entire web page has been created for this issue. You can navigate to it from any of the other pages by using the nav bar at the top of the page. For your convenience here is a direct link .

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. The document should be issued to you at the day of the test. 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 MyCourses 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
• Homework assignments only: only having your name on the PDF but not understanding the proofs.
• I will make spot checks of exams, quizzes and homework to ascertain that the student obtained the correct answers on her/his own.

To eliminate suspicion, only writing/erasing utensils will be permitted on desks during an in-class exam.

Best wishes for a successful semester!

Michael Fochler