This course continues the study of discrete mathenatics where Math 231 left off. The course has two main topics, Recurrence relations and generating functions, graphs and trees, partial orderes and equivalence relations.

1. Week 1-3: RECURRENCE RELATIONS AND GENERATING FUNCTIONS. Sections 8.1, 8.2, 8.4. Solving homogeneous linear recurrence relations. This could be introduced quickly with examples, and then done again using generating functions and partial fractions techniques (although this is challenging since it relies on algebra skills). Lots of examples and proofs by induction along the way.

2. Weeks 3-8: GRAPHS AND TREES. Parts of chapters 10 and 11. Graphs, basic properties such as degrees of vertices, counting edges. Lots of examples of proof by induction. Euler's identity for planar graphs, graphs of regular polygons and their duals, K,sub>n and K_m,n, statement of Kuratowski's theorem. Optimization problems involving weighted graphs and algorithms to solve them.

3. Weeks 8-9: PARTIAL ORDERS AND EQUIVALENCE RELATIONS. Selection of topics from sections 4.4, 4.5, 4.6, 9.1, 9.5, 9.6. The idea of a relation, e.g., functions are special relations. Definition of partial order and equivalence relation. Building new sets as equivalence classes. Examples including integers modulo n, integers constructed formally from natural numbers, or rational numbers constructed from integers, via equivalence relations; well-definedness of operations

4. Week 10: REVIEW.

Remote Learning Format

Here is the plan how we will run this remote course. I am optimistic that this online format will be as effective for learning the material as a traditional class, though it will take a little more effort for all us to stay focused and engaged.

Lectures

Lectures will be given over Zoom. I will write in a my ipad using Goodnotes5 and share my screen while doing so. You can (and are encouraged to) take your own notes during class, but you will also have access to a video recording of most classes. I encourage each of you to attend every class and ask questions during the class (via voice or chat); watching videos is not so efficient. In particular, you must have a working microphone. Don't skip lecture because you are counting on being able to watch the recording. Be prepared to interact with me and other students.

There will be 6 points given for your class attendance..

An invitation to the Zoom lectures ("meetings") will be posted to Canvas.

To prepare, before the quarter starts:

1. Make sure you have Zoom working on your computer or tablet. (A phone is also an option in the worst case, but it may be hard to see what I write.) Make sure you know how to mute and un-mute your microphone.
2. Find a place that you are comfortable joining the lecture with your video turned on: you will probably find it easier to pay attention if your video is on, and the visual feedback I receive that way helps me pace the lectures.

Tuesdays will typically be used for review and homework help, as well as catching up if (when) we get behind on material.

Office Hours

Office hours will also be via Zoom, the link to be provided. They will not be recorded.

Homework

Homework is due in class on Wednesdays, beginning January 13. Late homework will not be accepted. There will be 9 homework assignments, the last of which will not be graded. Your lowest homework score will be dropped. You may collaborate with other class members on your homework, although you must each write up your solutions independently and in your own words. To avoid falling behind, you should do the reading and homework as the material is presented in class, rather than leaving it all until the last minute.

The homework should be uploaded to Canvas by the deadline. You can solve the homework by:

1. Solving the problems on paper and then scanning your solutions. You can use a scanning app like Adobe Scan, Office Lens, Scanner Pro, Scanbot, or others, but your scans should be easy to read and contained in a single file. If you do not have a tablet, this is probably the easiest option.
2. Writing solutions using a tablet (iPad, Surface, etc.) and exporting and uploading the result. If you have a tablet, this is probably the easiest option.
3. Typing your solutions using LaTeX, Word, or Pages. You need to show your work, so typing your solutions is a substantial amount of effort. I would do this only if you're trying to learn to type mathematics in LaTeX.

Graded homework will be returned electronically, using Canvas.

Exams

There will be two in-class midterm exams and a final exam. All exams will be open-note and open-book. They are to be completed individually -- you may not consult with anyone else (except me) about the material while taking the exams.

During the exams you must have your video on.

To summarize, you need:

• A computer or tablet from which you can access Zoom, including a microphone you can use to talk on Zoom.
• A way to create electronic copies of your homework solutions -- either by scanning paper or writing your solutions electronically.
• A webcam and a place to participate in lectures and office hours where you are comfortable being on video. (Note: if you have a sufficiently powerful computer, Zoom can obscure your background.)
• A tablet and stylus, or other way of writing in OneNote or Zoom's whiteboard (or using Zoom's annotation tools).