## 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!) e-mail: lfinotti@utk.edu 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 4pm to 5pm, and Thursdays from 12pm to 1pm, via Zoom. Exams: Exam 1: 06/17 (due on Canvas by 11:59pm). Exam 2: 06/24 (due on Canvas by 11:59pm). Exam 3: 07/01 (due on Canvas by 11:59pm). Exam 4: 07/06 (due on Canvas by 11:59pm). Grade: Average of the exams, with lowest score having half the weight of other exams.

Back to the TOP.

## 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 time 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 Piazza.

"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.

### Meetings (Lectures)

We will meet online according to the Calendar below using Zoom.

Before each meeting, you have to read, watch the videos and do problems for the sections assigned for it, as posted in Lectures and Exams (Course Calendar). You should bring any questions you might have to class, or post them at "Class_Discussions.tex" file in CoCalc. (Questions written there will be answered in class. More on CoCalc later.)

Alternatively, you can just login to Zoom and enter Meeting ID 352717538.

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 CoCalc (previously know as "Sage Math Cloud") 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 2018).

On our meetings you will see me share my browser running CoCalc 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 CoCalc 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 CoCalc to post solutions to HW problems.)

Please watch this video for more details: Introduction to SMC (Sage Math Cloud) and How We Will Use It. (Remember, Sage Math Cloud is the previous name for CoCalc.) Note that this was made for a different course (Math 504 from Summer 2016), so somethings are a bit different (like dates, the course site, etc.) but we will use CoCalc (Sage Math Cloud) and Piazza in the same way.

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. So: in total you should work on this course (reading, watching videos, doing problems, reviewing, preparing for exams, etc.) on your own for about 20.5 hours a week and we will meet for another two hours a week.

### Recordings (and Missed Lectures)

I intend to record all our Zoom meetings in video, so you can go back to it later. I've set Zoom to automatically record on start up, so unless there is some glitch, all our meetings should be recorded.

Once the recordings are done, I will post them on Canvas. (I will post an announcement with the first one.) Feel free to download them to your computer for easier future access.

In particular, if you miss a lecture, you probably should check the corresponding video. Also, if you know in advance you will miss a meeting, you can add your questions (if any) ahead of time in the file "Class_Discussions.tex" in Cocalc, and I will answer them in class (on video). Although you will not be able to ask follow up questions, at least you will get some answer.

### 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 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 it, making it easier for you to read the answers.

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

• Chapters and Exams: Each chapter and exam has its own folder. Ask question related to each chapter or exam in the corresponding folder.
• Class Structure: Ask questions about the class, such as "how is the graded computed", "when is the final", etc. in this folder. (Please read the Syllabus first, though!)
• Computers: Ask questions about the usage of Zoom, LaTeX, CoCalc, Piazza itself and Canvas using this folder.
• Feedback: Give (possibly anonymous) feedback about the course using this folder.
• Other: In the unlikely event that your question/discussion doesn't fit in any of the above, please use this folder.

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.

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. I'd prefer you use e-mail to talk to me, unless it is a math question (in which either you or I would need to enter math symbols) that cannot be posted for all (such as an exam question). You can also send private messages to fellow students, but keep in mind that I can see those too! (So, not really that private...)

### 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:

• Chapters 1 and 2: all sections, but these will be covered quickly and skipping some parts. These are sections in formal logic, which although crucial, I find better to be introduce in more concrete settings as the need arises in the following chapters.
• Chapter 3: All sections, except 3.7.
• Chapter 4: All sections, except 4.5.
• Chapter 5: All sections, except 5.4.
• Chapter 6: All sections, except 6.5.

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 and Exam Policies

The homework sets for this course consist of problems assigned for each section. So, after you read a section (see the section Lectures and Exams to see the dates for the reading assignments) you should start on the corresponding problems for the section. You can see the problems for each section by following the link to the section in the reading assignment or by clicking on the section here.

Note, though, that homeworks will not be turned in! You should work on them to practice for exam and to bring questions to class.

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 CoCalc after the due date. You should always check your solution against the posted one and bring questions to class.

Exams will posted and collected on Canvas. (I will also posted them in the section Handouts of this page.) They should appear on the the morning of the due date (or maybe the previous evening) and are usually due at 11:59pm of the due date.

You will need to either type or scan your solutions (PDF format, please) and upload it to Canvas. Don't e-mail your solution to me unless you are having problems with Canvas and it's about the due time.

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

Points might be taken from messy solutions!

Exams will be individual, closed book/notes, with no internet or external resources, just like you'd have in a regular (not online) course. There would be not time limit, except the due time (so you'd have about one day to do it) and I will include some HW problems (which you should have done before) in it. (Of course, you could not look at your own solution, nor at the provided one!) Since I cannot monitor you long distance, we will rely on the honors code. (I don't foresee cheating problems in this course, but please be aware if find anyone teaching, the student will get a zero and I will do my best to have them removed from the MM program.)

### 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 Account button on the top left, 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!

### Feedback

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).

Back to the TOP.

## Legal Issues

### Conduct

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

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).

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!

### Disabilities

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.

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:
• develop analytic and critical thinking;
• broaden their problem solving techniques;
• learn how to concisely and precisely communicate arguments and ideas.

### Student Learning Outcomes

At the end of the semester students should be able to:
• write coherent, concise and well-written proofs with proper language and terminology;
• use counting arguments for solving concrete numerical problems and as tools in abstract proofs;
• master standard proof techniques such as direct proofs, by contradiction or contrapositive, proofs by induction, proofs of and/or statements, proof of equivalencies, among others;
• master the terminology and notation of basic set theory (such as membership, containment, union, complement, partition, among others);
• master the terminology and notation of basic fucntion theory (such as injective/one-to-one, surjective/onto, bijective, invertible, etc.);
• understand and be familiar with examples of equivalency relations and its relation with partitions.

### Learning Environment

• Type: 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.
• Where: Students will work from home in activities such as reading, watching video, participating in video conferences and long distance office hours. A lot of the discussions should happen on the Piazza discussion board.
• Student and Faculty roles:
• Students will have to be more active in the learning process than in regular courses, as they will do most of the reading and learning on their own.
• The instructor will be a facilitator, answering questions and offering advice and guidelines, answering questions and providing feedback.
• Students Responsibilities:
• Keep up with the schedule, i.e., read the assigned sections, watch the recommended videos and solve assigned problems according to the schedule. This is crucial in this flipped format!
• Carefully work on assigned problems.
• Carefully review graded work to learn from past mistakes.
• Check the course site often (at the very least once a day) for assignments and announcements.
• Search for help if having difficulties!
• Provide feedback to improve the course.
• Instructor Responsibilities:
• Be available for help.
• Provide examples and solve problems.
• Be open to discussions concerning content, format and evaluations.
• Provide relevant problems and exercises for homework, quizzes and exams.
• Provide feedback to the students.

Back to the TOP.

## LaTeX

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 you do need to register):

We will use the first one, CoCalc 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 2018).

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:

Back to the TOP.

Back to the TOP.

Back to the TOP.

## Lectures and Exams

### Lecture 1: 05/31 from 12pm to 1pm

Reading: Course Info (this site), Introduction (from the textbook).

### Lecture 2: 06/04 from 4pm to 5pm

Reading: Sections 1.1, 1.2, 1.3, 1.4.

### Lecture 5: 06/14 from 12pm to 1pm

Reading: Sections 3.3, 3.4, and catching up.

### Midterm 1: 06/17 by 11:59pm

Chapters 1 and 2.

Chapter 3.

Chapter 4.