College of the Holy Cross, Spring 2022

Math 243 (Mathematical Structures) Problem Sets

Professor Hwang.

Last modified: January 19, 2022


Problem Sets and Portfolio Questions

Each class meeting has an activity sheet with a due date. The write-up for each activity sheet is generally due on the following Friday, about ten to fifteen questions per week. These problem sets are worth 10% of the course grade. For best results, complete each activity sheet no later than the day following the class meeting.

One primary goal of the course is to give you practice writing mathematics with feedback. Some of the problems, the portfolio candidates, will be marked with a topic from the ten listed below. These write-ups (one question of your choice from each topic) are worth 40% of the course grade.

Portfolio write-ups may only be turned in on Wednesdays. Starting February 9, you must submit at least one portfolio write-up each Wednesday unless you have positive scores on seven or more topics. This requirement will help you pace yourself. You may submit up to four portfolio write-ups on any given Wednesday. All portfolio write-ups are due on or before class time on May 4.

You may resubmit a portfolio write-up multiple times in order to raise your score on that topic. (In no event will a resubmission lower your score for a topic.) To resubmit, you must re-write the entire problem, not make corrections to an existing draft.

You are warmly invited to ask me for help writing up portfolio questions, though I'll generally provide strategies rather than details. Write up and turn in each portfolio question separately, as a stand-alone document.

Each portfolio draft will be graded on the following scale:

Exceeds expectations
(E, 100%), a write-up I could photocopy and distribute to the class.
Mastered
(M, 85%), a write-up that substantially answers the question, but is lacking in organization, clarity of exposition, and/or neatness.
In progress
(I, 70%), a write-up that mostly answers the question, and/or that requires effort to read, for reasons including logical disorganization or illegibility.
Not yet
(N, 0%) otherwise.

You may write drafts by hand on paper, use technology such as an iPad, or type them in the web-based LaTeX platform Overleaf. Submissions must include physical pieces of paper; I am unable to accept emailed scans, for example.

If you use Overleaf for portfolio questions, neatness of source code is part of the evaluation criteria. The goal is to develop your ability to write flexible, easy-to-read LaTeX. I will happily provide more detailed guidance as needed.

The topics and prospective questions are:

  1. Sets and Inclusion
  2. Induction with Formulas
  3. Induction with Properties
  4. Images and Preimages
  5. Properties of Mapping
  6. Greatest Common Divisors
  7. Equivalence Relations
  8. Well-Defined Mappings
  9. Complex Numbers
  10. Bounds and Inequalities

Academic Integrity: Accuracy and honesty, including specific, easy-to-locate citations and proper attribution, are essential skills for producing useful written work. The guidelines below are intended to help you develop habits of good scholarship, including honest collaboration, detailed citation, clarity and completeness of written expression.

You are warmly invited to ask me any questions at all. One of my main roles in the course is to be your mathematical guide. Especially in a course such as Structures, it's normal to feel unsure to a greater or lesser degree. The sooner we speak, the sooner you'll feel more certain.

Because the portfolio questions are a primary indicator of your individual work, they are subject to special rules: You may discuss question statements with classmates, but may not discuss methods of solution. On written work submitted for grading, the wording, calculations, and structure of the argument must be yours alone. I'll notify you once if work does not meet this standard. Any subsequent instances that violate this policy will result in a score of zero for that topic, and (as mandated by College policy) in my bringing a formal charge of academic dishonesty.

For non-portfolio exercises, you may collaborate with classmates, but must practice honest scholarship. When you turn in your exercises, acknowledge by name any classmates or others you worked with by writing “I worked with...” at the top of your assignment. If you use an external resource such as the textbook, give an appropriate bibliographic citation, such as the author of the resource, the complete URL, and the date on which you retrieved the content; or a book title, author, publisher, date of publication, and page number(s). My goals are to help you develop good habits of documenting your work. This allows others to verify your work more easily.

Read the College policy for definitions of plagiarism, cheating, and collusion. In this course, using wording from a classmate (or anyone else, such as a user on a math question and answer web site) on a portfolio question is cheating. The same act is, in addition, plagiarism if you do not mention the other person's name. Sharing your own portfolio work with a classmate is collusion.

To summarize, these policies are spelled out in order to clarify their scope and purpose. I want each of you to learn mathematical writing. You can only do this by actually writing, in your own words. In “real” writing, you have multiple drafts. Similarly, I expect you'll sometimes turn in E write-ups on a first try, but other times will turn in work that could use improvement in some respect. For each topic, you can resubmit write-ups until your work is as good as you like.