MATH 220 – Section 906

CATALOG DESCRIPTION: Foundations of mathematics including logic, set theory, combinatorics, and number theory. Prerequisite: MATH172.

Instructor: Dr. Eric Rowell. email: rowell@math.tamu.edu, webpage: www.math.tamu.edu/~rowell. Office Blocker 510B

Place and Time: Section 906 meets TR 12:45-2:00pm in Blocker 148.

Office hours: TBD or by appointment or drop in (if I am not too busy...)

Course Objectives: Understand and communicate in the language of higher mathematics. This course is designed to provide a foundation for further study of mathematics beyond calculus. A major part of this foundation is learning to write proofs, and will be the main objective of the course. A secondary objective is to develop a basic appreciation for higher mathematics. As this course has a W-designation a key objective is to learn to communicate mathematics in clear, correct English.

Text: The Tools of Mathematical Reasoning, by Tamara Lakins (Caution: this is a different book than last year!)

Material Covered: Most of chapters 1-6. Additional topics may be added.

Grading: Your grade will be based upon 2 Midterms (25pts each), Homework (13pts), a Writing Project (25pts) and quizzes/extended problems (12pts). The grading scale is the usual one: 90-100% A, 80-89 % B, 70-79% C, 60-69 % D, 0-59% F, with one exception: if a student does not pass the writting portion of the course (at least a C) he/she cannot pass the course (i.e. will receive a D or F). Dates of exams/assignments will be posted below. Note that the the Writing Project takes the place of a final exam.

Exams: There are two midterm exams, tentatively scheduled for Thursday October 12 and Thursday November 30. There is NO in-class final exam.

Writing Project: The writing project will be a research paper of 5-10 pages on a topic in mathematics. You may include images, figures and indented/block quotes, but these do not count towards the page constrants (so an 11 page paper with lots of images. might be ok, but a 5 page paper with images might not be). Moreover, the works cited/bibliography does not count towards the page constraints. This will be turned in three times: once as a draft for peer-review on November 9, again as a draft for Prof. Rowell to grade on November 21, and a final version in lieu of a final exam due on December 13, by 5pm Central Time. In the peer-review you will exchange papers with a classmate (in class on November 9) and return them with comments on November 16--this counts as a quiz (half for turning in a draft, half for providing comments on your classmate's paper). The draft due on November 21 is worth 12pts and the final version an additional 13pts. See below for further information.

Quizzes/Extended Problems: Most Thursdays there will be either a short, in class quiz or an extended problem due. Extended problems are more involved exploratory problems that do not come from the text, independent research projects etc. The quizzes may be short writing assignments or exam-type problems, and one quiz/extended-problem grade will be dropped.

Homework: Weekly homework sets will be turned in to be graded. Typically homework will be assigned on Thursdays and due the following Thursday. As writing proofs takes considerable practice, the homework sets will likely be quite time-consuming. One homework score will be dropped. Late homeworks count as a 0, and may or may not be graded.

Course Policies: Late homework and make-ups for missed exams will only be allowed for a university approved excuse in writing. Wherever possible, students should inform the instructor before an exam is missed. Consistent with University Student Rules, students are required to notify an instructor by the end of the next working day after missing an exam. Otherwise, they forfeit the right to a make-up.

An Aggie does not lie, cheat, or steal or tolerate those who do. Copying work that was done by others is an act of scholastic dishonesty and any instance of it will be prosecuted according to University Student Rules.

The Americans with Disabilities Act (ADA) is a federal anti-discrimination statute that provides comprehensive civil rights protection for persons with disabilities. Among other things, this legislation requires that all students with disabilities be guaranteed a learning environment that provides for reasonable accommodation of their disabilities. If you believe you have a disability requiring an accommodation, please contact Disability Services, currently located in the Disability Services building at the Student Services at White Creek complex on west campus or call 979-845-1637. For additional information visit http://disability.tamu.edu.

Writing Project/Plagiarism: Your Writing Project should be the product of your own research. This means you should cite references properly (use any standard format you choose). See this page by Prof. Gregory of Washington and Lee University for a description of plagiarism (which should be avoided, as it is heavily penalized). Long quotes (4+ standard lines) should usually be indented/blocked, but remember these do not count towards your page constraints. I reserve 5% of the writing project grades to reward good and interesting writing. At least 3 sources must be cited, and at least 2 must be static (i.e. unchanging, unlike Wikepedia). Help is available for general writing questions at The University Writing Center. Typically, the three main ways students lose points on their papers are: 1. not long enough (discounting images, block quotes, bibliography etc.) 2. Insufficiently credited (inline citations, bibliography) and 3. Poor Grammar/Spelling (This is not twitter. No covfefe, please!).

Grade Grubbing: Google it, and never do it. It only serves to reduce your self-respect. If you feel there was a grading error on a specific assignment you must submit a regrade request within 2 business days. I do not entertain requests to change course grades without significant justification.

Copyright Policy: All printed materials disseminated in class or on the web are protected by Copyright laws. One photocopy (or printout from the web) is allowed for personal use. Multiple copies or sale of any of these materials is strictly prohibited.

Important Announcements:

Help Sessions: MW 8-10pm Blocker 202 (Beginning Sept. 4)

Key Dates (tentative):

Writing project topic approved by: Sept. 12
Exam I: October 12
Rough Draft for Peer-Review due: November 9
Peer-Review due: November 16
Rough Draft of Writing Project Due: November 21
Q-drop Deadline: November 17
Exam II: November 30
Writing Project due: December 13, by 5pm Central Time (via email, class does not meet).

Homework Assignments:

HW 1 (Due Thurs. Sept. 14 (not Sept. 7th--Thanks Harvey!)): Lakins section 1.1: 1-4,7.8,9,11,13b,14(b,c) Suggestion problems (do not turn in): 15,16
HW 2 (Due Thurs. Sept. 21): Lakins: Section 1.2: 1,3 (you may use either definition of odd: "not even" or "of the form 2k+1.") Section 2.1: 1,2,7,11. Suggested problems 21. Section 2.2: 3,5
HW 3 (Due Thurs. Sept. 28): Section 2.2: 3,5,6,8,9, Section 2.3: 1,2
HW 4 (Due Thurs. Oct. 5): Section 3.1: 1,3,4,6,8,10,12,14,17,18
HW 5 (Due Thurs. Oct. 19): Section 3.2: 1,2,4. In addition, do the following problems. A: Determine the smallest N>0 so that every n>N can be written as n=3x+7y for some non-negative integers x,y. B: Define a sequence inductively by A₁=1, A₂=2 and for n>2 A_n=2A_n-2 -A_n-1 (so A₃=2*1-2=0, for example). Show that A_n=1/3(4-(-2)^n-1). C: Let choose(n,r)=n!/(r!(n-r)!) be the binomial coefficient. Show that choose(n,0)+choose(n,1)+...+choose(n,n)=2^n for all n>0. (Actually you may start at n=0 if you wish...)
HW 6 (Due Thurs. Oct. 26): Section 4.1: 1,2,4,5 Section 4.2: 3,4,9,17,21,23acdef
HW 7 (Due Thurs. Nov. 2): Section 4.2: 12,13 Section 5.1: 1abdf, 2,5,6
HW 8 (Due Thurs. Nov. 9): Section 5.2: 1,2 Section 5.3: 1abimo,2,3 also do the following problems. D: Let A,B be sets with |A|=n and |B|=m. (i) How many different relations are there between A and B? (ii) How many functions are there from A to B? (iii) If n<m, how many injective functions are there from A to B? (iv) If n>m how many surjections are there from A to B? (v) if n=m, how many bijections are there from A to B?
HW 9 (Due Thurs. Nov. 16): Section 5.4: 1,2 Section 6.1: 1,2 Section 6.2: 1 and (i): Prove that if gcd(a,b)=1 and a divides bc then a divides c.

Long Problem Assignments:

(Due Thursday Sept. 14) Describe your writing project topic, including a specific reason you are attracted to this topic and what you already know about this project. This should be 1 page in length.
(Due Thursday Oct. 12): Consider the game of SET that we played in class on 9/26 (If you missed class, I can loan you the game). a) What is the (minimum) number of comparisons/operations necessary to verify that a collection of 12 cards does not contain a SET? b) Design a version of SET using 4-tuples of numbers so that a computer can check if 3 cards is a SET or not. c) Call any collection of cards that does not contain a SET a noSET. What is the maximum number of cards in a noSET? d) Consider the 9 and 27 card version of the game. What is the maximum noSET? (This is both a writing prompt and a math problem. Write your answers in paragraph form, explaining what you tried, what you discovered, including (partial) answers to all questions. Grading is based on completeness, effort, exposition and correctness, with emphasis/points in that order.)
Due Tuesday Nov. 28): An archimedean solid is a polyhedron that is not a Platonic solid, is not a prism (one n-gon and 2 squares at a vertex) and is not an anti-prism (one n-gon and 3 triangles at a vertex). Recall that an archimedean solid is determined by its vertex figure, i.e. an ordered list (a,b,c,d,..) of integers with a smallest, indicating an a-gon, a b-gon etc. at each vertex in (say) clockwise order. a) consider the vertex figure (3,4,3,4): how many vertices, edges and faces can there be? b) consider the vertex figure (3,4,5): can an archimedean solid have this vertex figure? c) can (3,3,5,6) be the vertex figure of an archimedean solid? d) what is the maximum number of faces an archimedean solid may have at a vertex? e) what is the largest n-gon that can appear in an archimedean solid? Prove and illustrate your answers to each of a)-e)!