MATH 261: Proofs and Fundamental
Fall 2013 Course Information
Professor: Lauren Rose, Albee 305, rose@bard.edu
Class schedule: T/Th 11:50  1:10, Hegeman 204

Textbook: Proofs and Fundamentals, by Professor Ethan Bloch, 2nd Edition.
Course Requirements: There will be a midterm and a final exam. Your grade will be determined roughly by homework(30%), midterm(25%), final exam(25%), and a final project, attendance and class participation(20%).
Homework: Homework will be discussed in class, and several proofs will be handed in each week. You will also be expected to do homework problems at the board.
Attendance: Attendance is required. More than 2 absences will affect your grade.
Most Important: READ THE BOOK! If you’ve never read a math book before, this is the time to start. I will expect you to have read the relevant sections BEFORE I discuss them in class.
This course provides a bridge between the calculus sequence and the more abstract mathematics in upper level courses. You will learn about the language of mathematics, the structure of mathematical proof, and how to communicate mathematics to others. We will cover topics that arise in most branches of mathematics, such as sets, functions, relations, and cardinality.
TENTATIVE SCHEDULE: subject to change
Assignments: bold = hand in, (purple) = challenge problems, grey = tentative.)
Week  Dates  Notes  Due for class T/Th  Hand in Friday 
1  Read 1.2  1.2:115 odd parts  
2  Read 1.3, 1.4  1.3: 1,2,4,5,8,10,12 odds 1.4: 1, 2 odd parts  
3  Read 1.5, 2.2, 2.3  1.5: # 2, 4, 6, 8 2.2.3, 2.2.6  2.2.7, 2.3.3  
4  Read 2.4  2.4.2, 2.4.4  2.3.4, 2.3.5, 2.4.7 (Challenge: 2.3.7, 2.3.8)  
5  Oct 1/3  Read 2.5  2.5.4, 2.5.5, 2.5.7  2.4.6, 2.5.6 (Challenge: 2.4.3) 
6  Oct 8/10  Read 3.  3.2: 3, 5, 8, 12, 3.3: 2, 3, 4  3.3.5, 3.3.15 ,3.3.16 
7  Oct 15/17  Ch 3  4  3.4: 1, evens only 4.1: 1, 3, 4, 5  3.2.14, 3.3.18 
8  Oct 22/24  Ch 4  4.2: 1, 3, 4  4.2.10, 4.2.12 
9  Oct 29/31  Midterm  Quiz on Tu, Midterm given 4.3: 3, 4, 7(1) on Th  Work on midterm 
10  Nov 5/7  Midterm due  4.4: #1, 2, 4, 5 due Th in class  4.3.6, 4.4.7 due Mon 11/11 
11  Nov 12/14  5.1: 1, 3, 4 odd parts  Film Response due Sun 11/17 5.2: 4, 5, 6, 7 due Wed, 11/20  
12  Nov 19/21  5.2: 1, 2, 5.3: 1, 2 odd parts  6.3.1(6), 6.3.8 due... before Thanksgiving  
13  Nov 26/28  Thanksgiving  no class Thursday  
14  Dec 3/5  Projects: Preliminary outlines due Wednesday  6.5: #1, 6 6.7. #1 for Tuesday in class  
15  Dec 10/12  Final given out  Detailed outlines due Wednesday  Powerpoints due Sunday. 
16  Dec 17/19  Final due Friday Writeups: Sat  Presentations  Presentations 
Project Topics for Math 261
Project Guidelines: 3 people per group, 1 group can have 2 people.
1) Class presentation (2025 min) introduction, background, motivation, etc.
2) Proof write up (1 per group): brief introduction, cite sources including websites.
R = research project. Discover a theorem, figure out a proof, present to class with examples, etc. B = book project: Read a proof from a book, present to the class, give examples and motivation. RB = both: discover and prove the theorem on your own, or find it in a book. 
Note: Other topics are possible with permission from the instructor.
Mathematical Theorems: A theorem is a statement that has been proven to be true.
1. The Pythagorean Theorem
2. The Fundamental Theorem of Algebra
3. The Fundamental Theorem of Arithmetic
4. The Fundamental Theorem of Calculus
5. The Quadratic Formula
6. Fermat’s Last Theorem
7. The Four Color Theorem
8. The Product Rule from Calculus
9. The Intermediate Value Theorem
Mathematical Statements: A statement can be either true or false, although it is not always easy to tell its truth value. What are the truth values of the following statements?
1. sin2x + cos2x = 1, for all real numbers x.
2. The sum of two even numbers is even.
3. The sum of two odd numbers is odd.
4. The square root of 2 is irrational.
5. The Euler constant e is rational.
6. There are finitely many prime numbers.
7. A number is divisible by 8 if and only if the sum of its digits is also divisible by 8.
8. A number is divisible by 9 if and only if the sum of its digits is also divisible by 9.
9. If x and y are rational, so is x+y.
10. If x and y are irrational, so is x+y.
11. Is p is prime, so is 2p – 1.
12. x2+ y2 = z2 has infinitely many integer solutions.
13. x3+ y3 = z3 has infinitely many integer solutions.
14. Every polynomial with real coefficients has a root.
15. If n is divisible by 3, so is n2.
16. If n2 is divisible by 3, so is n.
17. x2 > x, for all real numbers x.
18. Every prime number can be written as the sum of two squares.
19. Every even number can be written as the sum of two prime numbers.
20. There are infinitely many pairs of consecutive odd prime numbers.
Extras
SelfReferential Multiple Choice Quiz