Proofs & Fundamentals

Course: Math 261, Spring 2013

Time & Place: TR 3:10pm–4:30pm in Olin LC 115

Instructor: Maria Belk (mbelk@bard.edu)

Office Hours:

Thursday 7–9pm in RKC 101
Friday 2–5pm in the Learning Commons

Dedicated Tutors:

Matt Hughes(mh6379@bard.edu)
Fanny Wyrick-Flax (ff9267@bard.edu)

Tutor Office Hours: TBA

Math Study Room: Sunday through Thursday, 7–10 pm in RKC 111

Announcements

Takehome Final: The takehome final is due on Tuesday, May 21. You must work completely on your own, consulting only the textbook, your course notes, and your homeworks as references. If you have questions, you can come to my office hours or ask me via e-mail. Good luck!

Takehome Final (TeX, PDF)

Number Theory Notes: Our next topic in class is some elementary number theory. This topic is not in the textbook, so here are some notes on the topic:

Number Theory Notes

Reading Assignment: Read the Number Theory Notes.

Homework 10: The tenth homework assignment is due on Friday, May 10. Your solutions should be written in LaTeX.

§ 6.4 # 2
Number Theory Notes # 4, 5
§ 6.5 # 10

In-class Final: The in-class portion of the final exam will be on Thursday, May 2. Here are practice problems and solutions:

Practice Problems (and Answers)

Homework 9: The ninth homework assignment is due on Friday, April 26. Your solutions should be written in LaTeX.

§ 5.3 # 4 (part 1) and # 6
§ 6.3 # 1 (parts 1, 2, and 5)

Project Information

Part of the requirements for this class include a final project. Here is some basic information about the project:

• Though you are free to work on a project individually, I recommend working together in groups of two or three.
• The project includes a short in-class presentation as well as a write-up. The presentation must use computer slides, and should take about 5–10 minutes per student. The write-up must be in LaTeX, and should be roughly 2-3 pages per student (double-spaced).

Project Topics

There are two basic options for your project:

• You may do a small bit of research on an interesting math problem. It is your job to formulate and prove theorems about the solution, and then present your theorems to the class.

In the following list of topics, research topics are marked with an R, and book topics are marked with a B. Some topics might work either way, so these are marked RB. These topics are only suggestions—you should feel free to invent your own topic, or discuss other possibilities with me:

~~R: Which natural numbers are sums of two or more consecutive natural numbers? For example, 7 = 3 + 4 and 10 = 1 + 2 + 3 + 4.~~
R: Consider n lines in the plane. How many regions are determined by these lines? For example, two non-parallel lines will determine 4 regions. You should start by assuming that no two lines are parallel, and no point lies on more than two lines, and try to find a formula for the number of regions. If you have time, you can consider the general case.
R: Let a and b be natural numbers that are relatively prime. What is the largest natural number n such that n cannot be written as ax + by for any integers x and y?
RB: Which triples of natural numbers (a, b, c) are Pythagorean triples (meaning a² + b² = c²)?
RB: Find and prove the relationship between V, E, and R for a planar graph, where V is the number of vertices, E is the number of edges, and R is the number of regions determined by the graph.
RB: Which graphs can you draw in a continuous manner, without lifting your pencil or going over any edge twice? This comes from the Konigsberg Bridge Problem.
RB: Which prime numbers can be written as the sum of two squares? For example, 5 = 1² + 2², but 3 is not the sum of two squares.
B: Prove Fermat's Last Theorem for n = 4.
B: Prove Fermat's Little Theorem, about powers modulo a prime p.
B: Prove that ISBN numbers detect single digit errors and transposition errors.
B: Prove the Euler/Euclid Perfect Number Theorem on the correspondence between perfect numbers and Mersenne primes.

Project Timeline

Here is a timeline of due dates for the project:

Thursday, April 25: Project proposal due
May 16 and 21: Presentations in class
Tuesday, May 21: Due date for the paper

LaTeX

You are required to write homework solutions in LaTeX. Here are some links to get you started:

If you want to download LaTeX to your own computer, Ethan recommends TeXShop for Mac users, and proTeXt for PC users:

Sample Assignments

Two sample assignments in LaTeX:

Sample Assignment 1 (TeX, PDF)
Sample Assignment 2 (TeX, PDF)

Homework Solutions

Here are solutions to the homeworks:

Homework 1 Solutions (TeX, PDF)
Homework 2 Solutions (TeX, PDF)

Takehome Midterm: The takehome portion of the midterm was due on Friday, March 22. Here is the midterm and solutions:

Takehome Midterm (TeX, PDF)
Midterm Solutions (TeX, PDF)

Midterm: The in-class portion of the midterm exam was on Thursday, March 14. Here are practice problems and answers:

Practice Problems (and Answers)

Schedule

Week	Dates	Practice Problems	Homework
Week 1	Jan. 29, 31	None	None
Week 2	Feb. 5, 7	§ 1.2 # 1, 5, 7, 8, 12, 13 § 1.3 # 8, 9 § 1.4 # 2	§ 1.4 # 1 (parts 3–6)
Week 3	Feb. 12, 14	§ 1.5 # 1, 2, 6 § 2.2 # 2, 3, 4, 5	§ 1.5 # 11 (part 3) In paragraph form: § 1.4 # 1 (parts 4 and 6) In both two column and paragraph form: § 2.2 # 6, 7
Week 4	Feb. 19, 21	§ 2.4 # 2, 4, 7	§ 1.5 # 8 § 2.3 # 3, 5 § 2.4 # 3, 8
Week 5	Feb. 26, 28	§ 2.5 # 3, 5, 7 § 3.2 # 2, 3, 4, 7, 8	§ 2.3 # 7 § 2.5 # 4, 6, 8
Week 6	March 5, 7	§ 3.3 # 1, 3, 4, 13 § 3.4 # 1, 2	§ 3.2 # 14 § 3.3 # 5, 18, 19
Week 7	March 12, 14
Week 8	March 19, 21		Takehome Midterm (TeX, PDF)
Spring Break
Week 9	April 2, 4	§ 4.1 # 1, 2, 3 § 4.2 # 1, 3 § 4.3 # 1 § 4.4 # 1, 2	§ 4.3 # 3, 5, 9
Week 10	April 9, 11		§ 4.4 # 4, 11, 14
Week 11	April 16, 18	§ 5.1 # 3 § 5.2 # 1	§ 5.2 # 4, 5, 6, 8
Week 12	April 23, 25	§ 5.3 # 1, 2, 3, 8, 9	§ 5.3 # 4 (part 1) and # 6 § 6.3 # 1 (parts 1, 2, and 5)
Week 13	May 2
Week 14	May 7, 9		§ 6.4 # 2 Number Theory Notes # 4, 5 § 6.5 # 10
Week 15	May 14, 16
Week 16	May 21

Course Policies

Introduction

This course is an introduction to the language and methodology of modern mathematics. Topics covered include an introduction to mathematical logic, axiomatic systems, sets, relations and functions, cardinality, and mathematical structures. In addition, you will be learning how to communicate using the precise language of mathematicians, including the writing of formal proofs. This material is fundamental to any serious study of mathematics beyond the level of calculus.

Prerequisites

The prerequisite is Math 142 (Calculus II) or permission of the instructor. We will not actually be using much calculus, but the course will require the sort of mathematical maturity and familiarity with abstract reasoning that tends to be developed in a calculus class. It would also be helpful to have taken another 200-level math course beforehand such as Math 213 Linear Algebra with Ordinary Differential Equations.

Textbook

The textbook is Proofs and Fundamentals, A First Course in Abstract Mathematics, by our own Ethan Bloch. We plan to cover most of chapters 1–6, with selected other topics as time permits.

Homework

There will be weekly homework assignments, which will often involve the writing of formal proofs. You are encouraged to work together on the homework, but you should write up your own solutions individually, and you must acknowledge any collaborators.

Writing Your Solutions

A major goal in this course is to learn to write formal mathematical proofs. A good proof often requires multiple drafts to get right, and you will need to get into the habit of editing your homework solutions after you write them.

Because it makes editing easier, and because it is the universal standard for mathematical writing, all of your homework solutions must be written in LaTeX. If you haven't used LaTeX before, I recommend starting with Ethan Bloch's LaTeX web page for Bard students, which has links to many helpful resources.

Exams, Project, and Grading

In addition to the weekly homework assignments, much of your grade will be based on a midterm exam and final exam. Both exams will have an in-class part and a takehome part. There will also be a project due near the end of the semester, which will consist of an in-class presentation and a short write-up. These items will be weighted as follows:

Homework	40%
Midterm	20%
Final	20%
Project	20%

Feedback

Do you have any suggestions for the class? Let me know by using the following feedback form!