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 9-10 or by appointment.
Textbook: D. J. Velleman, "How to Prove It: A Structured Approach", 2dn Edition, Cambridge University Press, 2006.
Prerequisite: 142 or 148 (and consent from the Math Department).
Class Meeting Time: MWF 10:10am to 11:00am. (Section 001.)
Exams: Midterm 1 (Chapter 1): 09/07.
Midterm 2 (Chapter 2): 09/21.
Midterm 3 (Chapter 3): 10/12.
Midterm 4 (Chapter 4): 11/02 Postponed to 11/09.
Midterm 5 (Chapter 5): 11/21.
Final: 12/07.
Grade: 65% for Midterm Average (lowest score dropped) and 35% for the final.

Back to the TOP.

Course Description and Information

Course Content

Math 307 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 also be graded on how well you write your proofs will affect your grade! 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.

For the outcomes and problems (as well as videos) for each individual section, check Videos, Outcomes, Problems.


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. But, 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 Blackboard and you can bring your questions to class. In particular, I will try to set sometime to answer HW questions the day before each exam.

Also, you should make appointments for office hours having difficulties with the HW or the course in general! I will do my best to help you.


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 Blackboard).

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 left panel of Blackboard 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 edited 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.

Also, you can send Private Messages in Piazza. So, if you have a math question not appropriate for the whole class, you can send me a private message instead of an e-mail. That way my reply can have the math symbols nicely formatted.

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 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.



I've recorded some videos for a course similar to this one taught online. The video has comments and solved problems from our textbook. Keep in mind that any comments about the course structure in those videos should be disregarded, as the video were not made for this course!

You can access these videos here: Videos, Outcomes, Problems. In that page you can also see the expected outcomes and problems associated to each section of the book.


E-Mail Policy

I will assume you check your e-mail at least once a day, but preferably you should check your e-mail often. I will use your e-mail (given to me by the registrar's office) to make announcements. (If that is not your preferred address, please make sure to forward your university e-mail to it!) I will assume that any message that I sent via e-mail will be read in less than twenty four hours, and it will be considered an official communication.

Moreover, you should receive e-mails when announcements are posted on Blackboard, or where there is a new post in Piazza. (Again, please subscribe to receive notifications in Piazza! Important information my appear in those.)



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, but then, of course, it won't be anonymous.


Back to the TOP.

Legal Issues


All students should be familiar and maintain their Academic Integrity: from Hilltopics, pg. 46:

Academic Integrity

The university expects that all academic work will provide an honest reflection of the knowledge and abilities of both students and faculty. Cheating, plagiarism, fabrication of data, providing unauthorized help, and other acts of academic dishonesty are abhorrent to the purposes for which the university exists. In support of its commitment to academic integrity, the university has adopted an Honor Statement.

All students should follow the Honor Statement: from Hilltopics, pg. 16:

Honor Statement

"An essential feature of The University of Tennessee 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 Office of Disability Services and bring me the appropriate letter/forms.


Sexual Harassment and Discrimination

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


Campus Syllabus

Please, see also the Campus Syllabus.


Back to the TOP.

Course Goals and Outcomes

Course Relevance

This course is clearly crucial to mathematicians, as our job is to prove things (and find things to be proved). But, this is a course also required for computer scientists, not only here at UT, but virtually everywhere. The most obvious reason is that computer programs are written using formal logic. Another relevant connection is Artificial Intelligence, where you basically have to "teach" a machine to come up with its own proofs.

Moreover, the skills taught in this course are universally important, and their benefits cannot be overstated! Everyone should be able to think clearly and logically to make proper choices in life, and you should be able to communicate your thoughts clearly and concisely if you want to convince, teach, or explain your choices to someone else. In particular, Law Schools are often interested in Math Majors, as the ability to think logically and clearly develop an argument is (or should be) the essence of a lawyer's job.

For teachers, it is important to help your students, from an early age, to be understand the importance of proofs! In my opinion, high school (at the latest!) students should be introduced to formal proofs, even if in the most simple settings. This is important to foster analytic and critical thinking and to understand what mathematics is really about.


Course Value

The students will:  

Student Learning Outcomes

At the end of the semester students should be able to:  

Back to the TOP.

Study Guide

General Study Guide

Here are some comments on how to prepare for exams. To study, I recommend:

Midterm 1

I've just written (a preliminary version of) our Midterm 1. Here is some info about it (subject to change):

Let me know if you have any questions. (Use the Piazza, if you can.)

Midterm 2

I've just written (a preliminary version of) our Midterm 2. Here is some info about it (subject to change):

Let me know if you have any questions. (Use the Piazza, if you can.)

Midterm 3

I've just written (a preliminary version of) our Midterm 3. Here is some info about it (subject to change):

Let me know if you have any questions. (Use the Piazza, if you can.)

Make Up Midterm 3

I've just written (a preliminary version of) our Make Up Midterm 3. The description and study guide is virtually the same as for Midterm 3 above, except:

Let me know if you have any questions. (Use the Piazza, if you can.)

Midterm 4

I've just written (a preliminary version of) our Midterm 4. Here is some info about it (subject to change):

Let me know if you have any questions. (Use the Piazza, if you can.)

Midterm 5

I've just written (a preliminary version of) our Midterm 5. Here is some info about it (subject to change):

Let me know if you have any questions. (Use the Piazza, if you can.)


I've just written (a preliminary version of) our Final. Here is some info about it (subject to change):

Let me know if you have any questions. (Use the Piazza, if you can.)


Back to the TOP.


This (LaTeX) is mostly irrelevant to our course! The only benefit for us is to help you post messages in Piazza with math symbols. So, feel free to skip the rest of this section.

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 available for all platforms and freely available.

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):

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

A few resources:


Back to the TOP.



Back to the TOP.



Back to the TOP.

Solutions to Selected HW Problems

Please read: I will try to post here a few solutions. The new solutions will be added to this same file. They might come with no explanation, just the "answer". If yours do not match mine, you can try to figure out again. (Also, read the disclaimer below!) You can come to office hours or ask in class if you want explanations for the answers. Be careful that just because our "answers" were the same, it doesn't mean that you solved the problem correctly (it might have been a "fortunate" coincidence), and in the exams what matters is the solution itself. I will do my best to post somewhat detailed solutions to the harder problems, though.

Disclaimer: I will have to put these solutions together rather quickly, so they are subject to typos and conceptual mistakes. (I expect you to be a lot more careful when doing your HW than I when preparing these.) You can contact me if you think that there is something wrong and I will fix the file if you are correct.

Solutions to Selected HW Problems (Click on "Refresh" or "Reload" if you don't see the changes!)



Back to the TOP.

Homework Problems

Section 1.1: 1, 3, 6, 7.

Section 1.2: 2, 12.

Section 1.3: 2, 4, 6, 8.

Section 1.4: 2, 6, 7, 9, 11.

Section 1.5: 3, 4, 5, 9.

Section 2.1: 3, 5, 6.

Section 2.2: 2, 5, 7, 10.

Section 2.3: 2, 5, 6, 9, 12.

Section 3.1: 2, 3, 6, 10, 15, 16.

Section 3.2: 2, 4, 7, 9, 12.

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

Section 3.4: 3, 8, 10, 16, 24.

Section 3.5: 3, 8, 9, 13, 17, 21, 24.

Section 3.6: 2, 3, 7, 10.

Section 4.1: 3, 7, 9, 10.

Section 4.2: 2, 3, 5, 6(b), 8.

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

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

Section 4.6: 4, 8, 13, 20, 22.

Section 5.1: 9, 11, 13, 17.

Section 5.2: 3, 6, 11, 8, 9, 18.

Section 5.3: 4, 6, 10, 12.

Section 6.1: 2, 4, 9, 16.

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), 10.

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

Section 6.4: 4, 6, 7, 19.


Back to the TOP.