# Math 504: Discrete Mathematics for Teachers -- Summer 2014

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## 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 (For non-personal questions, please use the forums on Blackboard, so that we can all participate on the discussion and get the answers!)

Office Hours: by appointment. We can use Blackboard Collaborate (long distance) or you can come to my office.

Textbook: D. J. Velleman, How to Prove It: A Structured Approach'', 2nd Edition. Cambridge University Press, 2006.

Prerequisite: One year of calculus or equivalent.

Class Meeting Times: MWF 12:30pm to 2pm via Blackboard Collaborate. (Section 301.)

Exams: The exams will be take-home exams. (Check Conduct section below!) In principle they will be posted in the morning and you will have until the end of the day to turn it in. Midterms: 05/10 (Tuesday) and 05/24 (Tuesday), uploaded on Blackboard by midnight; Final: 07/02 (Wednesday) uploaded to Blackboard by midnight.

Grade: 25% for HW (lowest score dropped) + 20% for each Midterm + 35% for the Final. Note the weight of the HWs!

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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, the equivalent would be to study three hours a day in the summer semester!

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!

### 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.

Lectures'' will be on-line, via Blackboard, using Collaborate (under Tools -> Collaborate on the left panel of our Blackboard site). 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 not 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 the first time I am teaching a course in this format (flipped/online), so your feedback is quite important and will help shape the course.

### Forums (Discussion Boards)

I've set up four forums on Blackboard: Math Related, Course Structure, Computer Related (questions about Collaborate, LaTeX, Blackboard, etc.) and Feedback (more on this last one below).

I urge you to use these often! If you are ever thinking of sending me an e-mail, think first if it could be posted in these forums. 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.)

In all these forums 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 the forum. Discussion is encouraged here! But please be careful with Math Related 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, make sure to choose the appropriate forum for your question.

Please subscribe to all the forums to be notified of new posts!

### 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.

So, 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.
• Chapter 4: All sections, except, perhaps, 4.5.
• Chapter 5: All sections, except perhaps, 5.4.
• Chapter 6: All sections, except, perhaps, 6.5.
• Topics in Combinatorics: Some elements will be provided during the semester.
• Chapter 7: If time allows.

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

### 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.) Note that the assignment is not final until it says so. If the due date is coming, but the assignment still says tentative'', contact me immediately (preferably using the Q&A - Course Structure'' forum on Blackboard), and I will try to update it ASAP. If I fail to reply in a timely manner, proceed on doing the problems in the tentative'' assignment.

Homeworks must be turned in via Blackboard. (Please, don't e-mail 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 greater than the weight of a single midterm, and I will assume that you will work very hard on them.

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!

Finally, you can check all your scores at Blackboard.

### Missed Work

There will be no make-up quizzes or exams. If you miss a HW or exam and have a properly documented reason, your final will be used to make-up your score.

More precisely: say you missed, say HW3, which involved sections 2.2 and 2.3, and say that in our final questions 3 and 4 are about the material of those sections. Then, the points you get in those questions of the final will make you HW3 grade.

Also, the lowest HW score will be dropped, to make up for any eventual problem for which you might not have a documented excuse.

### E-Mail Policy

I will assume you check your e-mail at least once a day. 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.

### Feedback

Please, post all comments and suggestions regarding the course using Blackboard's Feedback. 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.

### Introductions

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

### Conduct

All students should be familiar and maintain their Academic Integrity: from Hilltopics 2013/2014, pg. 45:

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 2012/2013, pg. 16:

Honor Statement

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!

Our exams will be take-home exams. I will very carefully look for signs of cheating and will report any deviation from the posted instructions. I will do my best that any infraction will receive the most severe punishment.

### 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 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 the terminology and notation of basic set theory.

### 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 forums on Blackboard.
• 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.

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## 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 any with any serious intention of producing math texts to learn it! On the other hand, I don't expect all of you to do so. 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 it 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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## Handouts

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### Homework 1: 06/03 by 11:59pm

Section 1.1: 1, 3, 6, 7.

Section 1.2: 2, 12.

Section 1.4: 2, 6, 7, 9.

Section 1.5: 3, 4, 5, 9.

### Homework 2: 06/08 by 11:59pm

Section 2.1: 3, 6. (5 was removed as I've done it in a video.)

Section 2.2: 2, 5, 7, 10.

Section 2.3: 2, 9, 12. (Do, but do not turn in 5 and 6 and read the statements of 14 and 15).

### Homework 3: 06/12 by 11:59pm

Section 3.1: 2, 6 (state clearly what inequality property you are using/assuming), 8, 10, 15, 16.

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

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

### Homework 4: 06/15 by 11:59pm

Section 3.4: Turn in: 10, 16. Discuss in Meeting: 3, 8, 24.

Section 3.5: Turn in: 8, 21. Discuss in Meeting: 9, 13, 17.

Section 3.6: Turn in: 10. Discuss in Meeting: 2, 7.

### Homework 5: 06/19 by 11:59pm

Section 4.1: Turn in: 9, 10. Discuss in Meeting: 3, 7.

Section 4.2: Turn in: 5, 8. Discuss in Meeting: 2, 3(a), 6(b).

Section 4.3: Turn in: 14, 16. Discuss in Meeting: 2, 4(a), (b), 9(a), 12, 21.

### Homework 6: 06/22 by 11:59pm

NOTE: You do not need to turn in any problem from HW6 at all. (Disregard the "Turn in"'s below.) Since we have an exam on Tuesday (06/24), this might save you the time of uploading and all.

Note that you can turn in problems that you want me to give you feedback on. (I'd grade it as if it were a HW problem, assign it a grade and return it to you, but the grade will not be recorded or count towards anything, good or bad.)

Section 4.4: Turn in: 6, 22. Discuss in Meeting: 2, 3, 9, 15.

Section 4.6: Turn in: 13, 20. Discuss in Meeting: 4, 8, 16.

### Homework 7: 06/26 by 11:59pm

Section 5.1: Turn in: 9, 17. Discuss in Meeting: 11, 13.

Section 5.2: Turn in: 8(b), 9(a), 18. Discuss in Meeting: 3, 6, 11.

Section 5.3: Turn in: 10, 12. Discuss in Meeting: 4, 6.

### Homework 8: 07/01 by 11:59pm

Section 6.1: Turn in: 4, 9. Discuss in Meeting: 16.

Section 6.2: Turn in: 3, 6 (here you can use, without proving, the Triangle Inequality: if $a, b \in \mathbb{R}$, then $|a+b| \leq |a|+|b|$). Discuss in Meeting: 5, 10.

Section 6.3: Turn in: 5, 12(a-b). Discuss in Meeting: 2, 9, 16.

Section 6.4: Turn in: None. Discuss in Meeting: 4, 6(a-b), 7(b-c), 19.

### Lecture 17 and Final (by 11:59pm): 07/02

If the due date for an assignment is close but it is still marked as tentative'', post a request to update it on the Q&A - Course Structure'' forum on Blackboard, and I'll update it and let you know.