Math 307: Introduction to Abstract Mathematics (Honors) - Spring 2013

GO TO:

## Important Notes

• (04/26 - 5:00pm) I have written (a first draft for) the final exam. Here is some info about it:

• It is this coming Thursday (05/02) from 8am to 10pm, as scheduled by the university, in our regular classroom.
• I will have office hours Monday from 10am to 11am and Wednesday from 2pm to 3pm. You can also try to make an appointment if you cannot come at those times.
• No notes, books, index cards, etc. will be allowed!
• It covers all sections covered in class: Sections 1.1-5, 2.1-3, 3.1-6, 4.1-4 and 4.6 (no 4.5), 6.1-4.
• The exam has ten questions, two worth 8 points, six worth ten points and two worth 12 points. We have two questions (worth 16 points) on formal logic (as in Chapters 1 and 2), two questions (worth 20 points) on sets (as in Chapters 3, mostly), two questions (worth 22 points) on partial orders, two questions (worth 22 points) on relations and equivalence relations (last four question as in Chapter 4) and two questions (worth 20 points) on induction (as in Chapter 6).
• The questions are very similar to HW problems and examples from the book (and class).
• Look at my finals from when I taught M300 in the Fall 2008 and Fall 2009. The textbook was different, so it has somethings we did not cover and we covered things it did not (such as formal logic). From the former you can do problems 1, 2, 3, 5, 6 (this uses more inequality properties than we did, but you can attempt it), 7, and from the latter you can do problems 2, 3, 4, 5, 7. This might be good practice (besides redoing HW problems, extra problems from the books, examples from class and book, etc.).
• Please read my recommendations and comments on how to study (and how to get partial credit) from my recommendations for the last two exams below!
• (04/04 - 3:15pm) I will allow you to make corrections to your Midterm 2. Here is how this works:

• YOU SHOULD CONSIDER IT AS TAKE HOME EXAM!!!! This means you CANNOT talk to ANYONE at all about the exam yet, you cannot collaborate or use any references other than our textbook and your notes. (No Internet search or other texts!) If I catch anyone cheating, the person will get zero and be reported.
• It's due Tuesday (04/09) at the beginning of class! You will turn in your corrections together with your original exam. (If you have not picked up your exam from class, please come by my office to do so. I suggest you make an appointment to not lose your trip.)
• A copy of the exam is available at the section Handouts below.
• You can redo any problem you've missed for HALF of the missing points. (E.g., if you got 5 out of 15 in a problem and turn in a perfect correction of the problem, you can get another 5 points (half of the 10 you've missed), for a total score of 10 out of 15. If you got 10 out of 15, you can get up to 12.5 final score.)
• Obviously, you will not get points for the same things you already got points in the exam. You will only get points for new (correct) additions to the problem.
• Since you will have your textbook and notes, I won't be able to be generous just for getting definitions correct (like in the actual exam).
• Also, since you will have time, I'd expect you to write clean and concise proofs.
Let me know if you have any questions.

• (03/21 - 7:50pm) I have written (a first draft for) the second exam. Here is some info about it:

• It is the Tuesday after the break (04/02), during class. You will have the whole 75 minutes to work on it.
• I have office hours Monday from 10am to 11am. You can also try to make an appointment if you cannot come at that time.
• No notes, books, index cards, etc. will be allowed!
• It covers sections Sections 3.1-6 and 4.1-3 from the text. (Of course, these sections require knowledge of the past material, like formal logic and set theory!)
• Note that there is no HW on the day of the exam, but Section 4.3 is part of the exam, so I urge you to do HW10 anyway. I also urge you to only look at solutions (see below) after you try to do the problems by yourself.
• Since you did not get HW9 back and HW10 will not be graded, solutions for HWs 9 and 10 are posted in the section Handouts below. (See the disclaimer, though.) Look at the solutions carefully!
• The exam has six questions, five worth 15 points each, and one, with five (short) parts, worth 25 points. Roughly, half of the exam is about Chapter 3 and the other half about Chapter 4, although there are more points for Chapter 4 than for Chapter 3.
• The questions are very similar to HW problems and examples from the book (and class).
• Remember to try to get all easy partial credit points by showing the strategy of your proof (e.g., to prove sets are equal show that the first is contained in the second and vice-versa) and explicitly writing what the symbols mean (i.e., using the definitions).
• Your scratch, if done carefully and is readable, can give you partial credit too! So, try to be more careful with it, especially if you are having trouble with a problem!
• To study, I recommend:
• Review all definitions! As you might have noticed, a lot of proofs follow straight from the definitions. Moreover, you have no hope to prove something that involves a term that you don't know the definition.
• It is much easier to memorize the definitions if you understand them! Write them all in a separate piece of paper for quick reference and to help you memorize them.
• Review your old HWs. Redo all problems that you've missed. (You should always learn from your mistakes!) Always look for exactly what went wrong so that you do not repeat the same mistake!
• Learn how to do every single problem you could not do in HW. Try to find out what you were missing! (Did you get a definition wrong? Did you make a logic mistake? Did you have the wrong strategy? Did you miss an idea?)
• Look over examples and solved problems (class and book). Try to redo some of them yourself.
• Do as many other problems from the book as possible.
• 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.
• (02/20 - 12:05pm) I've posted some information about our first midterm below: Midterm 1.

• (02/18 - 12:40pm) I've posted solutions for HW5 below (in the section Handouts). Please read the disclaimer!

• (02/15 - 5:30pm) I have written (a first draft for) the first exam. Here is some info about it:

• It is this coming Tuesday (02/19), during class. You will have the whole 75 minutes to work on it.
• I have office hours Monday from 10am to 11am. You can also try to make an appointment if you cannot come at that time.
• No notes, books, index cards, etc. will be allowed!
• It covers sections Chapters 1 and 2 from the text. (Chapter 3 is not part of the exam at all.)
• Note that there is no HW due next week, so concentrate on studying rather than doing problems from Chapter 3.
• The exam has six questions, two worth 10 points each, and four worth 20 points each. Three of the questions have two parts. Roughly, half of the exam is about Chapter 1 and the other half about Chapter 2.
• The questions are very similar to HW problems and examples from the book (and class).
• To study, I recommend:
• Quickly review your notes and read the book. Make sure you know the definitions and logical rules. If necessary, write those in a different sheet of paper so that you don't have to browse your book every time you need to refresh your memory.
• Look at all examples and solved problems (from book and class!).
• Review your old HWs. Redo all problems that you've missed. (You should always learn from your mistakes!) Always look for exactly what went wrong so that you do not repeat the same mistake!
• Learn how to do every single problem you could not do in HW.
• Look over examples and solved problems (class and book). Try to redo some of them yourself.
• Do as many other problems from the book as possible.
• 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.

• (01/05 - 5:00pm) Please, check this section often. I will put announcements and important info here.

Back to the TOP.

### Instructor Contact and General Information

Instructor: Luís Finotti

Office: Ayres Hall 243

Office Hours: M, W 10am-11am or by appointment (subject to change!!)

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

Prerequisite: Math 142 or Math 148.

Class: Tu, Th 9:40am to 10:55pm at Ayres 110. (Section 001.)

Exams: Midterms: 02/19 (Tu) and 04/02 (Tu), in regular classroom and time; Final: 05/02 (Thu) from 8:00am to 10:00am also in our regular classroom.

Grade: 25% for quizzes/HW + 20% for each Midterm + 35% for the Final. Note the weight of the Quizzes/HWs!

Back to the TOP.

### Exams:

#### Midterm I

The exams were graded and will be returned in class on Thursday (02/21). (You can check your grades in Blackboard.) If you are not in class that day, you have to come by my office to pick up yours. (I will not take them to class anymore.) The average was 69.44, the median was 69.5, and the highest grade was 100 (two of them). (You can see more statistics at Blackboard.)

The results were not very good, although we had some good grades. If I were to curve these grades, roughly, an A would be 90 and above, a B would be 80 and above and a C could be either 65 or 60 and above. (I'd have a hard time actually deciding. I don't think 60 is a grade good enough for an exam like this, but maybe I would consider it.) Note that this is just to give you an idea on how I curve grades. In the end I will just curve (if necessary) your final averages, and will not even consider this curve given here.

Back to the TOP.

#### Midterm II

The exams and corrections were graded and will be returned in class on Thursday (04/11). (You can check your grades in Blackboard.) Your official Midterm II grade will be the one with the points from corrections added. If you are not in class that day, you have to come by my office to pick up yours. (I will not take them to class anymore.) The average was 74.96, the median was 80 and the highest grade was 100.

If I were to curve these grades, roughly, an A would be 89 and above, a B would be 80 and above and a C would be 65 and above. Note that this is just to give you an idea on how I curve grades. In the end I will just curve (if necessary) your final averages, and will not even consider this curve given here.

Back to the TOP.

#### Final

Here is the final: Final, and Final Solution.

Back to the TOP.

### Course Information

#### Content

Math 300/307 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.

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. We will use also basic notions of real and integer numbers.

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.

#### Course Structure

The goal is to cover the whole text. Chapter 7 is not crucial but would be interesting to cover it. Some sections might eventually be skipped or only briefly discussed if we are pressed on time, though. In particular, I might go over Chapter 3 (on proof strategies) quickly, leaving to discuss the corresponding strategies as they appear in a more natural context.

#### Homework and Quizzes

Homeworks will be assigned after every class and the selected problems and due dates will be posted at the section Homework of this page. No paper copy of the HW assignments will be distributed in class. It is your responsibility to check this page often! Besides HW assignments, other important information will be posted here. (Check the section Important Notes often!)

The HWs will be due on Tuesdays, when you will either turn in your HW or take a quiz on problems taken directly from the HW set due on that day. In the former case, two or three problems will be graded and count the same as a quiz. In the the latter, the HW will not be collected at all. You will not know in advance if the HW will be collected or if there will be a quiz, and hence you should be prepared for either one.

The quizzes will take place at the beginning of the class. You will not have extra time if you are late. You will have only ten or fifteen minutes to take a quiz, and so if you hadn't already solved the HW problems, you might not have enough time to come up with a solutions.

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!

Calculators will not be allowed! (This includes HW, quizzes, and exams!) As you will see, you won't miss them either.

I will do my best to post solutions to the most difficult problems. If I do, they will be posted in this page.

If you like to do your HW early, you can do problems in the list of Problems Likely To Be Assigned below, even before I assigned them. I might change a problem or two for the actual assignment, but that is not very likely. I will try to update it early enough for those who would like to start before we covered the material.

Also, some times I might get too ambitious in posting problems, i.e., I might think we will cover a section during the week, put exercises from it in the next assignment, and then end up not being able to finish it. In this case I might have to take a few problems off the assignment. The bottom line is the following: the assignment is not final until I remove the "More to come" from it. (If you've done problems which were removed, just saved them for the following week.)

Finally, if there is still a "More to come" in an assignment on a Thursday, please write me right away so that I can update it. If I only realize on Monday (or Tuesday morning!) that an assignment was not complete (with the "More to come" still there) and nobody tried to contact me, I may add new problems just then, giving you little time to finish the assignment. (If I delay in replying, you can proceed with the Problems Likely To Be Assigned.)

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 try to come to my office hours if you are having difficulties with the course. I will do my best to help you. Please try to come during my scheduled office hours, but feel free to make an appointment if that would be impossible.

Finally, it is your responsibility to keep all your graded Quizzes, HW, and Midterms! It is very important to have them in case there is any problem with your grade. You can check all your scores at Blackboard. (Blackboard will be used only for scores. This is the official site for the course.)

#### Missed Work

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

#### E-Mails

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

I have an On-line Feedback Form where you can anonymously send me your comments and suggestions. I will consider your comments and try to do whatever I can to resolve possible problems before it is too late. So, please, feel free to use it whenever you have any constructive comment or suggestion. (In fact, I would greatly appreciate it.) If you don't want you comments to be anonymous, just send me an e-mail or come by my office and we can discuss the problem.

Back to the TOP.

### Legal Issues

#### Conduct

All students should be familiar and maintain their Academic Integrity: from
Hilltopics 2012/2013, 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 2012/2013, 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.

#### 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.

Back to the TOP.

Back to the TOP.

### Handouts

• Campus Syllabus.
• Solutions for HW5. (UPDATE: I've copied 2.3, 2(b) wrong, with ``contained'' instead of ``belongs to''. I will try to fix it tonight.) Disclaimer: Note that I've written these quite quickly, so watch out for mistakes (and ``typos'')! Please contact me if you think there is a mistake and I will check and fix it if necessary.
• Solutions for HWs 9 and 10. Disclaimer: Note that I've written these quite quickly, so watch out for mistakes (and ``typos'')! Please contact me if you think there is a mistake and I will check and fix it if necessary.
• Solutions for HW11. Disclaimer: Same as above.
• Solutions for HW12. Disclaimer: Same as above.
• Solutions for HW13. Disclaimer: Same as above.
• Solutions for HW14. Disclaimer: Same as above.
• Exam 2.

Back to the TOP.

### Problems Likely To Be Assigned

This list is subject to change without prior notice. The official assignments will be posted below.

Section 1.1: 1, 3, 6, 7.

Section 1.2: 2, 3, 8, 9, 12, 17.

Section 1.3: 2, 4, 6, 8.

Section 1.4: 2, 6, 7, 9, 11, 15.

Section 1.5: 3, 4, 5, 9.

Section 2.1: 3, 5, 6, 9.

Section 2.2: 2, 5, 7, 10.

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

Section 3.1: 2, 3, 6, 8, 10, 15, 16.

Section 3.2: 2, 3,4, 7, 9, 12.

Section 3.3: 2, 4, 10, 13, 15, 18, 21, 24.

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

Section 3.5: 3, 6, 9, 13, 17, 21, 24.

Section 3.6: 2, 3, 7, 10.

Section 4.1: 3, 7, 9, 10.

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

Section 4.3: 2, 4(a), (c), 9(a), 12, 14, 16, 21.

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

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

Section 6.1: 2, 4, 9, 16.

Section 6.2: 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: 5, 9, 16.

Section 6.4: 7(a), (b), (c).

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 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 for HWs 9 and 10.
• Solutions for HW11.
• Solutions for HW12.
• Solutions for HW13.
• Solutions for HW14.
• Back to the TOP.

### Homework

HW1 - Due on Tuesday 01/15:

Section 1.1: 1, 3, 6, 7.

HW2 - Due on Tuesday 01/22:

Section 1.2: 2, 3, 8, 9, 12, 17.

HW3 - Due on Tuesday 01/29:

Section 1.3: 2, 4, 6, 8.

Section 1.4: 2, 6, 7, 9, 11, 15.

HW4 - Due on Tuesday 02/05:

Section 1.5: 3, 4, 5, 9.

Section 2.1: 3, 5, 6, 9.

HW5 - Due on Tuesday 02/12:

Section 2.2: 2, 5, 7, 10.

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

HW6 - Due on Tuesday 02/26:

Section 3.1: 2, 3, 6, 8, 10, 15, 16.

Section 3.2: 2, 3,4, 7, 9, 12.

Section 3.3: 2, 4, 10, 13, 15, 18, 21, 24.

HW7 - Due on Tuesday 03/05:

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

HW8 - Due on Tuesday 03/12:

Section 3.5: 3, 6, 9, 13, 17, 21, 24.

Section 3.6: 2, 3, 7, 10.

HW9 - Due on Tuesday 03/19:

Section 4.1: 3, 7, 9, 10.

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

HW10 - Not to be turned in! Just practice for the exam:

Section 4.3: 2, 4(a), (c), 9(a), 12, 14, 16, 21.

HW11 - Due on Tuesday 04/09:

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

HW12 - Due on Tuesday 04/16:

Section 4.4: 20, 22.

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

HW13 - Due on Tuesday 04/23:

Section 6.1: 2, 4, 9, 16.

HW14 Not to be turned in! Just practice for the final:

Section 6.2: 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: 5, 9, 16.

Section 6.4: 7(a), (b), (c).

And that's all!

PLEASE, HIT ``REFRESH'' (OR ``RELOAD'') IN YOUR BROWSER WHEN VISITING THIS PAGE!!!!!!! I usually get messages asking for the update in the HW when it has already been updated. Since I change this page often, some times the browser don't see the changes. But, if you hit refresh and there is still problems missing, feel free to write me.

If it is already Thursday afternoon and there still is a ``More to come'' after the HW assignment due on the coming Tuesday, write me an e-mail at lfinotti@utk.edu, and I'll update it and let you know.

Back to the TOP.