Instructor Contact and General Information

Instructor: Luís Finotti
Office: Ayres Hall 251
Phone: 974-1321 (don't leave messages! -- e-mail me if I don't answer!)
Office Hours: MW 11-12, or by appointment via Zoom.
Textbook: D. J. Velleman, "How to Prove It: A Structured Approach",
3rd Edition, Cambridge University Press, 2019.
Prerequisite: Math 142/148.
Class Meeting Time: MWF 2:15-3:05 via Zoom.
Exams: Midterm 1: 09/09 (Wed).
Midterm 2: 09/30 (Wed).
Midterm 3: 10/21 (Wed).
Midterm 4: 11/11 (Wed).
Final: 12/04 (Fri) from 10:30am to 12:45pm.
Grade: 20% for each midterm with lowest dropped + 40% for the final.
See here for letter grade ranges.

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Course Information

Course Content

Math 300 is a basically a course on mathematical proofs. A proof is a series of logical steps based on predetermined assumptions to show that some statement is, beyond all doubt, true. Thus, there are two main goals: to teach you how think in a logical and precise fashion, and to teach how to properly communicate your thoughts. Those are the "ingredients" of a proof.

Note that you will also be graded on how well you write your proof! A poorly written correct proof will not get full credit!

Thus, the topics of the course themselves play a somewhat secondary role in this course, and there are many difference possible choices. On the other hand, since these will be your first steps on proofs, the topics should be basic enough so that your first proofs are as simple as possible. Therefore, you will be dealing at times with very basic mathematics, and will prove things you've "known" to be true for a long time. But it is crucial that you do not lose sight of our real goal: do you know how to prove those basic facts? In fact, the truth is that you don't really know if something is true until you see a proof of it! You might believe it to be true, based on someone else's word or empirical evidence, but only the proof brings certainty.

In any event, the topics to be covered in this course are: logic, set theory, relations and functions, induction and combinatorics. We will use also basic notions of real and integer numbers, but these will be mostly assumed (without proofs).


Chapters and Topics

The goal would be to cover the following:

Other topics (and digressions) might also be squeezed in as time allows.


Homework Policy

Homework problems are posted below. As soon as we finish a section in class, you should start working on the problems from the section in the list. For longer sections, you should start before we finish it. Just look for problems that have statements on the topics we have already covered.

On the other hand, HW will not be collected or graded! (Also, there are no quizzes.) The point of the HW is to learn and practice for the exams. In my opinion, doing the HW is one of the most important parts of the learning process, so even if it does not count towards your grade, I recommend you take it very seriously.

Solutions to the HW will be posted on Canvas and you can bring your questions to class. In particular, I will try to set sometime to answer HW questions the class before each exam.

Note that you should not look at the solutions before you try, for a considerable amount of time, to do the problems! But you should always check your solution against the posted one and bring questions to class (or post on Piazza).

You have to be extremely careful with this set up. If you are not responsible and motivated, and end up not taking your HW seriously (even though it is not collected), your chances of passing this course are quite slim! But, then again, you are adults, so I don't feel guilty asking you to be responsible.

Also, you should make appointments for office hours having difficulties with the HW or the course in general! (Even if it is not for grade, you have to do well on your HW!) I will do my best to help you.


Extra Practice Problems

A common question (with good reason) for this course is what extra problems students can do to get more practice. Here are some suggestions:


How to Be Successful in this Course


Piazza (Discussion Board)

We will use Piazza for online discussions. The advantage of Piazza (over other discussion boards) is that it allows us (or simply me) to use math symbols efficiently and with good looking results (unlike Canvas). It also allows anonymous posts (also unlike Canvas).

To enter math, you can use LaTeX code. (See the section on LaTeX below.) The only difference is that you must surround the math code with double dollar signs ($$) instead of single ones ($). Even if you don't take advantage of this, I can use making it easier for you to read the answers.

You can access Piazza through the link on the syllabus in Canvas or directly here: (There is also a link at the "Navigation" section on the top of this page and on the Links section.)

To keep things organized, I've set up a few different folders/labels for our discussions:

I urge you to use Piazza often for discussions! (This is specially true for Feedback!) If you are ever thinking of sending me an e-mail, think first if it could be posted there. That way my answer might help others that have the same questions as you and will be always available to all. (Of course, if it is something personal (such as your grades), you should e-mail me instead.)

Note that you can post anonymously. (Just be careful to check the proper box!) But please don't post anonymously if you don't feel compelled to, as it would help me to know you, individually, much better.

Students can (and should!) reply to and comment on posts on Piazza. Discussion is encouraged here!

Also, please don't forget to choose the appropriate folder(s) (you can choose more than one, like a label) for your question. And make sure to choose between Question, Note or Poll.

When replying/commenting/contributing to a discussion, please do so in the appropriate place. If it is an answer to the question, use the Answer area. (Note: The answer area for students can be edited by other students. The idea is to be a collaborative answer. Only one answer will be presented for students and one from the instructor. So, if you want to contribute to answer already posted, just edit it.) You can also post a Follow Up discussion instead of (or besides) an answer. There can be multiple follow ups, but don't start a new one if it is the same discussion.

Important: Make sure you set your "Notifications Settings" on Piazza to receive notifications for all posts: Click on the gear on the top right of the Piazza site, the choose "Account/Email Setting", then "Edit Email Notifications" and then check "Automatically follow every question and note". Preferably, also set "Real Time" for both new and updates to questions and notes. I will consider a post in Piazza official communication in this course, I will assume all have read every single post there!

You can also use Piazza for Private Messages. It works well for math questions, since again it can be better formatted. You can also send private messages to fellow students, but keep in mind that I can see those too! (So, not really that private...)

You should receive an invitation to join our class in Piazza via your "" e-mail address before classes start. If you don't, you can sign up here: If you've register with a different e-mail (e.g., you do not need to register again, but you can consolidate your different e-mails (like and in Piazza, so that it knows it is the same person. (Only if you want to! It is recommended but not required as long as you have access to our course there!) Just click on the gear icon on the top right of Piazza, beside your name, and select "Account/Email Settings". Then, in "Other Emails" add the new ones.


Missed Work

Unless other arrangements are made beforehand, missed midterms, with documented excuses (see below), will be made up by the part of the (comprehensive) final that corresponds to the missed midterm.

More precisely: say you missed Midterm 4, which involved sections 5.2 and 5.3, and say that in our final questions 3 and 4 (and only those) are about the material of those sections. Then, the points you get in those questions of the final will make you Midterm 4 grade.

Your justification for missing an exam has to be processed by the Office of the Dean of Students, more precisely, here. Note that, as stated in the referred site, final approval of all absences and missed work is determined by the instructor. (So, just because you've submitted a justification through the Office of the Dean of Students, it doesn't mean it will be accepted by me.)


Communications and E-Mail Policy

You are required to set up notifications for Piazza (as explained above) and for Canvas to be sent to you immediately. For Canvas, check this page and/or this video on how to set your notifications. Set notifications for Announcements to "right away"! (Basically: click on the the profile button on left, under UT's "T", then click "Notifications". Click on the check mark ("notify me right away") for Announcements.)

Moreover, I may send e-mails with important information directly to you. I will use the e-mail given to me by the registrar and set up automatically in Canvas. (If that is not your preferred address, please make sure to forward your university e-mail to it!)

All three (notifications from Piazza, notifications from Canvas, and e-mails) are official communications for this course and it's your responsibility to check them often!



Please, post all comments and suggestions regarding the course using Piazza. Usually these should be posted as Notes and put in the Feedback folder/label (and add other labels if relevant). These can be posted anonymously (or not), just make sure to check the appropriate option. Others students and myself will be able to respond and comment. If you prefer to keep the conversation private (between us), you can send me an e-mail (not anonymous), or a private message in Piazza (possibly anonymous).


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Using Zoom

Our classes and office hours will all be through Zoom. (See the links in Syllabus on Canvas.)

If you are new to Zoom, please familiarize yourself with Zoom. At the very least, read the Getting Started and Best Practices for Participants pages from OIT. See also Remote It: Just for Students.

Please login to your account and test the microphone and camera before class.

Also, please make sure you have a way to scan documents quickly (e.g., from your phone) to post your online exams. (See Section Online Exams below.)

I will be online on Zoom at around 2:00pm on our first day (08/19) in case you want to login early and check if everything is working, but the actual class starts at 2:15pm as usual.


The classes will be live, so you should plan to be "present" during those meetings. On the other hand, the classes will be recorded and posted on Canvas for later viewing or in case you had to miss it.

Instead of writing on the board, I will go over prepared notes. These will be available on Canvas ahead of time, so you can download them and see them in your own computer. Revised notes might be posted (also in Canvas) if we found any mistakes or problems.

For questions you asked during lectures (for which I don't have prepared notes), I will type an answer live (using LaTeX). I am not very fast at all typing, so this might be a bit slow. I will type just enough that, together with my explanations, all can follow. After class I will edit this document (to make it more readable) and post it as a PDF in Canvas for future reference.

The notes, videos, and PDF for questions can be found in Canvas by (after selecting our course) clicking on "Pages" on the left panel and then on the link for the date's lecture.

Office Hours

Office Hours will also be via Zoom.

Note that office hours will not be recorded! You can record it yourself if you want to or request me to do it, but I will not do it by default (unlike our lectures).

Remember that our office hours are MW (not Fridays) from 11 to 12. I should be logged in Zoom during those times, but keep in mind I may walk away for a bit if no one is there. I should not be away for long, so you can wait a little. If you don't see me, feel free to turn on your microphone and ask for me. (In case I don't notice anyone has shown up.) But please be a little patient if I don't respond, as I should be back soon.

Also, I might be in a breakout room with another student. If a student has a private question, I will likely move them to a breakout room to answer their questions. In that case, if you join our office hours, you will not see me yet. Again, please just be patient and I will help you as soon as I can.

Although not necessary in principle, if you let me know you will come to office hours (and at what time exactly), I can make sure to not be away at that time.

Also, I can take appointments for office hours at different times if necessary. In that case, please write me to make an appointment, but use the same Zoom link as for regular office hours.

Keep in mind that these office hours are shared with another course I'm teaching.

During Lectures

Here are some notes about how we will handle the lectures:

Online Exams

I decided to try to keep things as simple as possible. In order to do that, I need to trust that you will follow our code of conduct, as made explicit in the section Legal Issues. (Please, don't make me regret this.) Anyone who is caught cheating (in any way) will be reported and will receive zero for the exam!

Moreover, if I suspect there was cheating, we will move to a much more elaborate system that will require, among other things, two devices recording the exam and using/installing LockDown Browser. (You might also have to type the solutions in that case.)

The idea is to keep the exam exactly as if it were in class.

The exam will be during class time, also using Zoom. You will all be required to have your camera on at all times and with an angle I can see you clearly.

Keep your sound on, in case I need to send you or the class a chat message, or if I need to turn on my microphone to say something to the whole class. (Note: You need to keep your sound on, not your microphone, as the point is that you hear me if I need to talk to you.)

Be prepared to, at any time upon request, show your room/environment to the camera and/or share your screen(s)! I will likely do some random checks, or might ask you if I notice anything suspicious. I will do this in a private Zoom Break Room to not disturb the rest of the class. Delaying to comply will be considered cheating, and therefore will give you a zero for the exam!

At exactly 2:15pm (or as soon as all students are logged in), I will send a link to the exam via chat message. Students who come late will need to send me a private message asking for the link.

In your solutions, you do not need to copy the questions statements, just number them clearly. Also, please use a different page for each question!

If you have any questions during the exam, you can send me a private message with your question. (You can only use your computer to look at the exam or to send me private messages! If see you typing in your computer and don't receive a private message, I will have to assume you were cheating!)

Once you are done with the exam, you need to send me a private message saying that you will start scanning/uploading. (Something short, like "Scanning now", works.) You will have to upload your solutions to Canvas. You can do it by clicking on "Assignments" on the left panel of Canvas, and then selecting the corresponding exam.

Please, upload, if at all possible, a single PDF with all the pages. (A little more on this below.) Please do not just upload pictures for each page (unless you absolutely have to). Also, please don't e-mail me your solutions unless you have problems uploading it to Canvas. The scans have to be perfectly legible!

Since you have to deal with scanning and uploading, I will give you some extra time to do so: you will need to have your exam uploaded to Canvas by 3:30pm!

You can only leave the Zoom meeting after you upload your exam. You will then send me a private message, like "Done." Don't forget this step! I will match the time I get your message (and you left Zoom) with the time you've uploaded the exam. If these don't match, you will get zero!

Again, you cannot leave Zoom or turn off you camera until you upload your exam to Canvas and send me a private message saying you are done.

The exams will be graded and returned through Canvas, and you will be able to see my comments in your graded exam.


Below are some apps that can scan documents from your phone. Of course, it is not an exhaustive list and you can use anything that works well. Most of these I have no experience using, so don't consider them as a real endorsement in anyway. Especially the iOS ones, since I've never had an Apple device, so these I've seen recommended elsewhere. Anyway, here are some options:

Again, please make sure you are familiar with your scanning app (or real scanner) before the next exam!


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Legal Issues


All students should be familiar with Hilltopics' Students Code of Conduct and maintain their Academic Integrity: from Hilltopics Academics:

Academic Integrity

Study, preparation, and presentation should involve at all times the student’s own work, unless it has been clearly specified that work is to be a team effort. Academic honesty requires that the student present their own work in all academic projects, including tests, papers, homework, and class presentation. When incorporating the work of other scholars and writers into a project, the student must accurately cite the source of that work. For additional information see the applicable catalog or the UT Libraries site. See also Honor Statement (below).

All students should follow the Honor Statement (also from Hilltopics Academics):

Honor Statement

"An essential feature of the University of Tennessee, Knoxville, is a commitment to maintaining an atmosphere of intellectual integrity and academic honesty. As a student of the university, I pledge that I will neither knowingly give nor receive any inappropriate assistance in academic work, thus affirming my own personal commitment to honor and integrity."

You should also be familiar with the Classroom Behavior Expectations.

We are in a honor system in this course!



Students with disabilities that need special accommodations should contact the Student Disability Services and bring me the appropriate letter/forms.


Sexual Harassment and Discrimination

For Sexual Harassment and Discrimination information, please visit the Office of Equity and Diversity.


Campus Syllabus

Please, see also the Campus Syllabus.


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This is not necessary to our class! I leave it here in case someone wants to learn how type math, for instance to type their HW. But again, you can ignore this section if you want to.

LaTeX is the most used software to produce mathematics texts. It is quite powerful and the final result is, when properly used, outstanding! Virtually all professional math text you will ever see is done with LaTeX, or one of its variants.

LaTeX is freely available for all platforms.

The problem is that it has a steep learning curve at first, but after the first difficulties are overcome, it is not bad at all.

One of the first difficulties one encounters is that it is not WYSIWYG ("what you see is what you get"). It resembles a programming language: you first type some code and then this code is processed to produce a nice document (a non-editable PDF file, for example). Thus, one has to learn how to "code" in LaTeX, but this brings many benefits.

I recommend that anyone with any serious interest in producing math texts to learn it! On the other hand, I don't expect all of you to do so. But note that there are processors that can make it "easier" to create LaTeX documents, by making it "point-and-click" and (somewhat) WYSIWYG.

Here are some that you can use online (no need to install anything and files are available online, but need to register):

The first one, Cocalc, is more than just for LaTeX, as you can also run Sage, which can do computations with the objects we will study in this course.

If you want to install LaTeX in your computer (so that you don't need an Internet connection), check here.

A few resources:


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The videos below were made for a different course! So, if you watch them, you have to be careful with comments that I make about the course structure and what is important. They were made for Math 504 -- Discrete Mathematics for Teachers. That course is taught online and its audience is teachers. Although we cover a lot of the same material, the approach is a bit different!

On the other hand, I go over examples and solve problems, so it might be useful for you too. I also go over some computer programs, namely Sage and LaTeX, which are not part of our course, but you can learn them from the videos if you are interested.

If you are uncertain if something from the video is relevant or applies to our course, please ask! (Use a Piazza Forum, please!)

Note that the numbering for the problems in the videos are from the second edition of out book, and therefore might not match the numbering in our newer edition. I've adjusted the numbering in the links for the videos, though.

Please let me know if you find any mistake in the videos!


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Homework Problems

Note: This list is subject to change.

Read the introduction.

Section 1.1: 1, 3, 7, 9.

Section 1.2: 2, 12.

Section 1.3: 2, 4, 6, 9.

Section 1.4: 2, 6, 7, 10.

Section 1.5: 3, 4, 6, 10.

Section 2.1: 3, 5, 7.

Section 2.2: 2, 5, 7, 10, 12.

Section 2.3: 2, 5, 6, 10, 13.

Section 3.1: 2, 3, 6, 11, 16, 17.

Section 3.2: 2, 4, 8, 10, 13.

Section 3.3: 2, 4, 10, 15, 18, 21.

Section 3.4: 3, 8, 11, 17, 25.

Section 3.5: 3, 9, 10, 14, 18, 22, 25.

Section 3.6: 2, 3, 7, 10.

Section 4.1: 3, 7, 11, 12.

Section 4.2: 2, 3, 6, 7(b), 9.

Section 4.3: 2, 4, 9, 12, 14, 16, 21, 24.

Section 4.4: 2, 3, 6, 9, 15, 20, 22.

Section 4.5: 4, 5, 9, 14, 17, 21, 23.

Section 5.1: 12, 14, 16, 20.

Section 5.2: 3, 7, 8, 10, 11, 14, 21.

Section 5.3: 4, 6, 10, 12.

Section 6.1: 2, 4, 9, 17.

Section 6.2: 3, 5, 6 (use the triangle inequality from Problem 12(c) of section 3.5; you don't need to do that exercise, just refer to it), 11.

Section 6.3: 2, 5, 9, 12, 16.

Section 6.4: 5, 7, 8, 17.


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