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: by appointment. We can use Zoom (long distance) or you can come to my office.
Textbook: D. J. Velleman, "How to Prove It: A Structured Approach", 2dn Edition, Cambridge University Press, 2006.
Prerequisite: One year of calculus or equivalent.
Class Meeting Time: Mondays from 2:30pm to 3:30pm and Thursdays from 7pm to 8pm, via Zoom. (Section 301.)
Exams: Midterm: 06/25 (due on Blackboard by 11:59pm).
Final: 07/06 (due on Blackboard by 11:59pm).
Grade: Best between 34% for HW Average and 33% (each) for Midterm and Final, and 20% for HW Average and 40% (each) for Midterm and Final. (Lowest HW score dropped in the HW average.)

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Course Information

Summer Course Warning!

This is a summer course, in which 16 weeks are squeezed into 5. So, as you can imagine, the pace is quite fast. Summer Courses are for very motivated students! If you usually study one hour a day during the regular semester (5 hours a week), the equivalent would be to study three hours a day in the summer semester! If you count (regular) lecture time (7.5 hours a week) and studying time (15 hours a week), that would amount to 22.5 hours a week dedicate to this course!

You cannot just "catch up on the weekends" in a course like this, as by then we will have covered way too much material. You should catch up immediately if you fall behind, as you will not be able to follow classes and things just start to accumulate in a faster pace than you will likely be able to catch up. I strongly recommend that you review, do problems and study every day!

I've had students taking more than one summer course in the past at the same session, and although it is possible to do it, I'd consider it a Herculean task and would usually advise against it. If you decide to do it, just make sure you are prepared for it! (Tell your loved ones you will see them in July.) :-)


Course Format

This will be a flipped course, i.e., students will learn a lot on their own, by reading the text and watching short related videos, while the times with the instructor will be spent with questions, solving problems and interactions with students.

You can always request for something you want to see in a video: a problem, some proof in the book, an example, some clarification, etc. If you think it can be done well enough in a lecture (on-line meeting) save it for then, though! If not, just post you request in the "Q&A - Math Related" forum on Blackboard.

"Lectures" will be on-line, via Zoom. I will assign reading and exercises to be done (or attempted) before our lectures. In lecture I will answer questions, solve problems and perhaps provide a few more examples. On the other hand, all (or most of) the content of the lecture will be "question driven". (But, questions such as "Can you further explain X?" or "Can you give us examples of Y?" are more than welcome.) If there are no questions or requests, the lecture will be quite short. It's essential you come to lectures prepared! Otherwise the chances of you getting anything out of this course (and passing it) are quite slim.

I recommend you attend the lectures even if you don't have any questions about the material, as I will take surveys and ask questions that might be relevant to all. You also might learn different ways of doing (or viewing) some problems.

It is only the third time I am teaching a course in this format (flipped/online), so your feedback is quite important and will help shape the course.



We will meet online Mondays from 2:30pm to 3:30pm and Thursdays from 7pm to 8pm using Zoom.

To join our class simply click here. (The link is There are also a links to access our class in Blackboard, on the Navigation area on top and in the section Links, called Class Meeting Link (Zoom).

Alternatively, you can just login to Zoom and enter Meeting ID 545 412 677.

I strongly recommend you try it out before our first meeting. Please read the LiveOnline@UT page carefully. In particular, look for Test Flights dates, when you can test Zoom before our first meeting. (Also, take a close look at Getting Started page.)

In our meetings we will use Sage Math Cloud (SMC) for our discussions. Before classes start, you should receive an invitation to collaborate on a project that I've created for this course (Math 504 -- Summer 2016).

On our meetings you will see me share my browser running SMC to answer your questions. You will be able to see and type in the same document live. (Similar to Google Docs.)

We can enter math in SMC using LaTeX. (More on LaTeX below.) The edited document with questions and answers will be stored in our project and you can look at it whenever you want/need. (I will also use SMC to post solutions to HW problems.)

Please watch this video for more details: Introduction to SMC and How We Will Use It.

Note: A regular summer course like ours (5 weeks) meets for 7.5 hours in a week. We will meet online for just about 2 hours a week. The remaining 5.5 should be spent reading the text and watching the videos. Note that these hours should not count as "studying time", but as "lecture time". Our 2 hour meetings, though, could count as "studying time". But again, relative to one hour study time a day (5 hours a week) for a regular semester, translate to 15 hours a week for a summer semester, and the meeting times count as only 2 of those!


Piazza (Discussion Board)

We will use Piazza for discussions. (Except for live meetings.) The advantage of Piazza 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! But please be careful with HW questions! You should not answer (or ask) questions about how to do a HW problem! (You can ask for hints or suggestions, though.) If you are uncertain if you can answer a (math related) post, please e-mail me first!

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.

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 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.


Course Content

Math 504 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.

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 a break down of videos, outcomes and problems for each individual section, check this page.


Homework Policy

Homeworks and due dates are posted at the section Reading and Homework of this page. (They should also appear in your Blackboard Calendar.)

Homeworks must be turned in via Blackboard. (Please, don't e-mail them to me unless strictly necessary, e.g., Blackboard is not working.) Just click on "Assignments (Submit HW)" on the left panel of Blackboard and select the correct assignment.

Scanned copies are acceptable, but typed in solutions are preferred. I recommend you learn and use LaTeX. (Resources are provided below.)

Note that you will also be graded on how well it is written, not only if it is correct! (Remember, how to communicate your proofs is part of the course.) The same applies to exams and all graded work!

In my opinion, doing the HW is one of the most important parts of the learning process, so the weight for them is quite high.

Also, you should make appointments for office hours having difficulties with the course. I will do my best to help you. Please try to first ask questions during class time (online)! I will not take appointments from students who don't attend the lectures (unless there is a good excuse, of course).

Finally, you can check all your scores at Blackboard.


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/labels (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.



You are invited to post an introduction about yourself on Blackboard. (There is an "Introductions" link.) This is not required at all! But it might make things a bit more social.


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


Campus Syllabus

Please, see also the Campus Syllabus.


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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:  

Learning Environment


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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):

We will use the first one, SageMathCloud in our course, so you have to register for it, and thus might as well use it. It is probably the best of the services anyway, and it can do a lot more than just LaTeX. You should have received, by the first day of classes, an invitation to collaborate on a project that I've created for this course (Math 504 -- Summer 2016).

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

I might need to use some LaTeX symbols when writing in our online meetings, but it should be relatively easy to follow. I will also provide samples and templates that should make it much easier for you to start.

A few resources:


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Reading and Homework (Course Calendar)


Lecture 1: 06/02 from 7pm to 8pm

Reading: Course Info, Introduction.


Lecture 2: 06/06 from 2:30pm to 3:30pm

Reading: Sections 1.1-4.


Lecture 3: 06/09 from 7pm to 8pm

Reading: Sections 1.5 and 2.1-3.


Homework 1: 06/11 by 11:59pm

Section 1.1: Turn in: 3(d), 6(b), 7(b).
Extra Problems: 1, 3(a-c), 6(a), (c), 7(a), (c-d).
Section 1.2: Turn in: 2(b), 12(b).
Extra Problems: 2(a), 12(a), (c).
Section 1.3: Turn in: None.
Extra Problems: 2, 4, 6, 8.
Section 1.4: Turn in: 6(a), 7(a), 9.
Extra Problems: 2, 6(b), 7(b).

Lecture 4: 06/13 from 2:30pm to 3:30pm

Reading: Sections 3.1-4.


Homework 2: 06/14 by 11:59pm

Section 1.5: Turn in: 5, 9.
Extra Problems: 3, 4.
Section 2.1: Turn in: 6.
Extra Problems: 3, 5.
Section 2.2: Turn in: 2(b-c), 7.
Extra Problems: 2(a), (d), 5, 10.
Section 2.3: Turn in: 2(c), 12(a-b).
Extra Problems: 2(a-b), (d), 5, 6, 9, 12(c). (Also, take a look at the statements of 14 and 15.)

Lecture 5: 06/16 from 7pm to 8pm

Reading: Sections 3.5-6, 4.1-2.


Homework 3: 06/18 by 11:59pm

Section 3.1: Turn in: 6, 10.
Extra Problems: 2, 15, 16.
Section 3.2: Turn in: 4, 7.
Extra Problems: 2, 9, 12.
Section 3.3: Turn in: 10, 15.
Extra Problems: 2, 4, 18, 21.
Section 3.4: Turn in: 10, 16.
Extra Problems: 3, 8, 24.

Lecture 6: 06/20 from 2:30pm to 3:30pm

Catch up! No reading, just questions and answers.


Homework 4: 06/21 by 11:59pm

Section 3.5: Turn in: 8, 21.
Extra Problems: 9, 13, 17.
Section 3.6: Turn in: 10.
Extra Problems: 2, 7.
Section 4.1: Turn in: 9, 10.
Extra Problems: 3, 7.
Section 4.2: Turn in: 5, 8.
Extra Problems: 2, 3, 6(b).

Lecture 7: 06/23 from 7pm to 8pm

Reading: Sections 4.3-4, 4.6, 5.1.


Midterm: 06/26 by 11:59pm

Sections: Chapters 1, 2 and 3.


Lecture 8: 06/27 from 2:30pm to 3:30pm

Reading: Sections 5.2-3, 6.1-2.


Homework 5: 06/28 by 11:59pm

Section 4.3: Turn in: 14, 16.
Extra Problems: 2, 4, 9, 12, 21.
Section 4.4: Turn in: 6, 22.
Extra Problems: 2, 3, 9, 15.
Section 4.6: Turn in: 13, 20.
Extra Problems: 4, 8, 16.
Section 5.1: Turn in: 9(b), 17(b).
Extra Problems: 9(a), 11, 13, 17(a).

Lecture 9: 06/30 from 7pm to 8pm

Reading: 6.3-4.


Lecture 10: 07/01 (Friday! -- note the date change) from 3pm to 4pm

Catch up! No reading, just questions and answers.


Homework 6: 07/01 by 11:59pm

Section 5.2: Turn in: 8(b), 9(a).
Extra Problems: 3, 6, 11, 18.
Section 5.3: Turn in: 10, 12.
Extra Problems: 4, 6.
Section 6.1: Turn in: 9(b), 16.
Extra Problems: 4, 9(a).
Section 6.2: Turn in: 3, 6 (here you can use, without proving, the Triangle Inequality: if $a, b \in \mathbb{R}$, then $|a+b| \leq |a|+|b|$).
Extra Problems: 5, 10.

Homework 7: Not to be turned in!

Section 6.3: Turn in: None.
Extra Problems: 2, 5, 9, 12, 16.
Section 6.4: Turn in: None.
Extra Problems: 4, 6, 7, 19.

Final: 07/06 by 11:59pm

Chapters 4, 5 and 6. (No Chapter 1, 2 or 3.)


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