## 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: 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.

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## 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:

• 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 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: https://piazza.com/utk/fall2016/math307/home. (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:

• Chapters and Exams: Each chapter and exam has its own folder. Ask question related to each chapter or exam in the corresponding folder.
• Course 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 LaTeX, Piazza itself and Blackboard 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.

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!

### Videos

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.)

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

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## Legal Issues

### Conduct

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

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!

### Disabilities

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.

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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:
• 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.

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## Study Guide

### General Study Guide

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

• Quickly review your notes and read the book. The most important is to review the definitions, Theorems, and how to compute things.
• I'd strongly recommend that you write these important facts in a different sheet of paper so that you don't have to browse your book every time you need to refresh your memory and so that you can easily review them later.
• Do all HW problems! This is the most important thing! You can only learn by doing it.
• Don't be too quick to look for solutions (which are posted below). Sometimes it takes time to get a problem. Keep thinking about it, trying different approaches, look for similar problems/examples in the book or notes (or videos!). This is an important part of the learning process. If you just look for solutions you might not learn enough. (See next item, though.)
• If you are stuck for a while on a problem, look at the solution. But, make sure to take notice of what you were missing. (Did you forget something? Is there an idea that you did not have before?) Try to remember that. Moreover, redo the problem a few days later without looking at solutions, to make sure that you've assimilated the ideas.
• Look at all examples and solved problems (from book, class and videos!). If you have time, redo some of them (without looking at the solution).
• Do as many other problems as possible from the book.
• When you first start studying, you can look at the book or notes, but by the end, you should be able to do problems without looking.
• Look at problems from old exams, which will likely be posted in the individual sections below.
• If you have questions, bring them to class or post them on Piazza.

### Midterm 1

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

• It covers all sections of Chapter 1.
• It will be in class on Wednesday 09/07. (Usual room and time.)
• The exam has six questions, two worth 20 points (each has two parts worth 10 points) and four worth 15 points.
• The questions are very similar to HW problems, and in fact, two questions are HW problems.
• You can practice with the following problems (which might not cover all the topics you need to know):

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):

• It covers all sections of Chapter 2.
• It will be in class on Wednesday 09/21. (Usual room and time.)
• The exam has six questions, two worth 20 points (each has two parts worth 10 points) and four worth 15 points.
• The questions are very similar to HW problems, and in fact, one question is a HW problem. (Although only one, the others are very similar.)
• You can expect the usual questions: analyzing logical forms (from English or set theory), intersection/union of families, indexed or not, logical equivalences (involving quantifiers), etc.
• Similar to the first midterm (e.g., Problem 4), there are a couple of question that should be very quick and easy, as long as you know the corresponding basic facts/definition, but since there is so little to them, there is not much room for partial credit, so be careful.
• I would say this exam is a bit harder than our first, since the material is also. But there are, for those well prepared, at least 45 "easy" points. Also, in many of the other problems, would be fairly easy to get some partial credit.
• You can practice with the following problems (which might not cover all the topics you need to know):

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):

• It covers all sections of Chapter 3, except 3.7.
• It will be in class on Wednesday 10/12. (Usual room and time.)
• Feel free to bring questions to class on Monday (10/10). I will answer as many questions as I can that day.
• The exam has five questions, each worth 20 points, all proofs.
• The questions are very similar to HW problems, and in fact, two questions are HW problems. The others are very similar to HW problems.
• You can expect the do all types of proofs we've worked on: contradiction/contrapositive (do you know how to negate statements?), proving statements with "and(s)" (possibly "if and only if" or equality of sets), proving statements with "or(s)", using statements with "or(s)" (proof broken in cases), existence and uniqueness, etc.
• Partial credit (although sometimes small) will be given to choosing the appropriate proof technique (so make your approach clear!) and interpreting the notation. So, even if you cannot finish the proof, write those down to make sure you get the partial credit for them. (This should also help you figure out the proof! That's why it grants you partial credit.)
• You will be graded on style! Please be neat, direct, clear and concise! State clearly your assumptions and, in parenthesis, your goals (as I've been doing in class).
• Some proofs have "parts", like cases or different statements that need to be proved. Sometimes some of these are hard, while others are easy. Make sure you try all parts (even if you cannot do them all!), so that you can get the credit for the easier parts!
• Material from previous chapters, like families, indexed families and power sets do appear, as well as things like divisibility and odd/even (just like in the HW for this chapter).
• This might be the hardest of our exams, as writing proofs take some practice.
• You can practice with the following problems (which might not cover all the topics you need to know):

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:

• It will be in class on Wednesday 10/19. (Usual room and time.)
• The exam has four questions, each worth 25 points, all proofs.
• The questions are very similar to HW problems, and in fact, one question is a HW problem. Two of the others come from examples from either book, class or video. The remaining one is similar to previous HW problems.
• Besides all the studying suggestion above (for Midterm 3), I also suggest you try to redo Midterm 3. Look for ideas you've missed, and make sure you have good knowledge of the definitions and proof techniques that were used.

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):

• It covers all sections of Sections 4.1 to 4.4. (Section 4.6 is not in the exam.)
• It will be in class on Wednesday 11/09. (Usual room and time.)
• The exam has five questions, each worth 20 points, three of them are proofs, and two of those are HW problems.
• There is one question from 4.1, one from 4.2, one from 4.3 and two from 4.4.
• In one of the questions from 4.4 you are asked to find minimal/maximal elements, smallest/greatest elements, lower/upper bounds and least lower bound/greatest lower bound in a concrete example.
• The questions are very similar to HW problems, and in fact, two questions are HW problems. The others are very similar to HW problems.
• I will be generous with partial credit if you know your definitions, so make sure you do!
• Also, as before, partial credit (although sometimes small) will be given to choosing the appropriate proof technique (so make your approach clear!) and interpreting the notation. So, even if you cannot finish the proof, write those down to make sure you get the partial credit for them. (This should also help you figure out the proof! That's why it grants you partial credit.)
• You will be graded on style! Please be neat, direct, clear and concise! State clearly your assumptions and, in parenthesis, your goals (as I've been doing in class).
• You can practice with the following problems (which might not cover all the topics you need to know):

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):

• It covers all sections of Sections 4.6 and 5.1 to 5.3.
• It will be in class on Monday 11/21. (Usual room and time.)
• The exam has four questions, each worth 25 points, all are proofs.
• The questions are very similar to HW problems, and in fact, two questions are HW problems. The others are very similar to HW problems.
• Like in the last exam, I will be generous with partial credit if you know your definitions, so make sure you do!
• Also, as before, partial credit (although sometimes small) will be given to choosing the appropriate proof technique (so make your approach clear!) and interpreting the notation. So, even if you cannot finish the proof, write those down to make sure you get the partial credit for them. (This should also help you figure out the proof! That's why it grants you partial credit.)
• You will be graded on style! Please be neat, direct, clear and concise! State clearly your assumptions and, in parenthesis, your goals (as I've been doing in class).
• You can practice with the following problems (which might not cover all the topics you need to know):

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

### Final

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

• It will be on 12/07 (Wednesday), from 8am to 10am. (Usual room.)
• In principle it is comprehensive, covering all we've studied in this semester. On the other hand, emphasis will be given to Chapters 4, 5 and 6, as the previous sections are heavily used in these chapters.
• The exam has 8 questions, 4 worth 12 points, 4 worth 13 points. One from Chapter 3 (a problem from the book, involving families), two from Chapter 4 (one on ordering relations, one on equivalence relations), two from Chapter 5 and three from Chapter 6.
• Note that there are three induction problems, worth 39 points in total.
• Note that it would be impossible to cover all topics from the course with only 8 questions, so many important things (on which you might work very hard) may not appear in the exam.
• I believe it covers all the "proof kinds" we've studied in Chapter 3.
• The questions are very similar to HW problems, and in fact, two questions are HW problems and other three are either examples done in class or in a video. The other three are very similar to HW problems.
• Like in the last two exams, I will be generous with partial credit if you know your definitions, so make sure you do!
• Also, as before, partial credit (although sometimes small) will be given to choosing the appropriate proof technique (so make your approach clear!) and interpreting the notation. So, even if you cannot finish the proof, write those down to make sure you get the partial credit for them. (This should also help you figure out the proof! That's why it grants you partial credit.)
• You will be graded on style! Please be neat, direct, clear and concise! State clearly your assumptions and, in parenthesis, your goals (as I've been doing in class).
• You can now do all problems from all the exam from past 307/504 courses. Of course you should focus on problems from Chapters 4 to 6. (Previous final exams had explicit material from Chapters 1 to 3.) You can practice with the following problems (which might not cover all the topics you need to know):
• We should have a review session before the exam. Please study before coming for the review, so that you can ask questions that would help you! Watch for an announcement (through Blackboard/e-mail) soon.

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

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## LaTeX

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:

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## 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!)

CHANGE LOG:

• 11/29 -- 3:50pm Solutions for Chapter 6 posted.
• 11/16 -- 3:35pm Solutions for Sections 4.6 and Chapter 5 posted.
• 11/01 -- 9:20am Solutions for Sections 4.1 to 4.4 posted.
• 09/30 -- 3:30pm Solutions for Chapter 3 posted.
• 09/13 -- 5:15pm Solutions for Chapter 2 posted.
• 08/26 -- 3:10pm Solutions for Chapter 1 posted.

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## 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.

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