MTH 307 Fall 2002
Introduction to Mathematical Rigor

## Instructor:  Dr. Nancy Eaton

E-mail me:  eaton@math.uri.edu
Phone:  874-4439
Office: Rm 222, Tyler Hall
Office hours: Monday 2:00-5:00, or by apt.
Visit my web page:  WEB PAGE

Section 01-Kingston
Meets MWF: 11:00-11:50, Wales 224

Homework
Solutions to selected homework problems
Keys to exams and quizzes
Samples from Portfolios

Students who require accommodations and who have documentation from
Disability Services (874-2098) should make arrangements with me as soon as possible.

Text:  Mathematical Thinking, by D'Angelo and West, 2nd Edition

Course Content and Goals:
As you progress in  the study of mathematics, it becomes necessary to learn the theory of mathematics which requires being able to understand the proofs of mathematical theorems and to prove results yourself.  In this course, you will learn proof techniques and use these techniques to prove theorems in many areas of mathematics.  We will cover the following topics: axioms of number systems, sets, fields, quantifiers, logical statements, induction, the Well-Ordering Property, proof by contradiction, injective functions, surjective functions, and permutaions. Also, we will cover selected topics from number theory, abstract algebra, discrete math, and continuous math in our overview of mathematics.

While learning the skills of proving theorems, you will also be achieving the goals of learning to communicate mathematics both in writing and verbally.

Motivation:
Motivation is a key concept in the way we will approach this course.  It is my belief that there are many theorems, examples, and puzzles that are fun and interesting, even amazing, that are elementary enough to be proved by a student who is just learning proof techniques. In order to prove the statements, it becomes necessary to learn proper techniques. Hopefully many of these problems will interest you and motivate you. I will choose examples to illustrate and practice a wide variety of proof techniques.  The book Mathematical Thinking contains many fun examples, so that you will be motivated to learn sound proof technique.  We will draw problems from many mathematical disciplines so that you will also get an overview of the many different areas of mathematics. Here are some examples:

Example:
A positve integer is palindromic if reversing the digits of its base 10 representation doesn't change the number.  Why is every palindromic integer with an even number of digits divisible by 11?  What happens in other bases?

Example:
Repeatedly pushing the ``x^2" button on a calculator generates a sequence tending to 0 if the initial positive value is less than 1 and tending to infinity if it is greater than 1.  What happens with other quadratic functions?

Example:
Suppose each dot in an n by n grid of dots is colored black or white.  How large must n be to guarantee the existence of a rectangle whose corners have the same color?

I assign many problems from the book.  You are expected to do them all.  We will go over some of the solutions in class.  I will select from those assigned a few to hand in from time to time.  I will grade them and return them with written comments on your work.

From those exercises collected and returned to you, I will select 10 for you to rewrite and hand in as your portfolio for the class.  This is intended to be a collection of well thought out proofs from the course.  This will be due Nov 27.

I encourage you to work together under the following
circumstances.   Each person tries every problem before talking it over with someone else. Each
problem that is written up and handed in should be essentially your own work.   There are benefits to discussing the problems with your classmates.  If you become stuck on a problem, fresh ideas from someone else might provide you with some new angles to try.  In the academic community as well as in business and industry, people often work in teams.  So, it is good to get some practice working in groups.  Working with someone from class will help you to improve your math communication skills as well.

Also, to improve your verbal communication skills, you will be asked from time to time to present the solution of a homework problem to the class.  You will be given extra points toward your homework grade for this.

Exams: (Tests 25%, Final 25%)
There will be two one-hour tests on Oct. 21st and Dec. 2nd and a final exam on Monday Dec. 16, 8:00 - 11:00 AM.  The exams will be taken from material that we covered in class, so take careful notes.

Sections Covered:
We will draw material from the following chapters in the book.

CHAPTER& TOPIC
1   Numbers, Sets, and Functions
2   Language and Proofs
3   Induction
4   Bijections and Cardinality
5   Combinatorial Reasoning
6   Divisibility
7   Modular Arithmetic
9   Probability
10 Two Principles of Counting
11  Graph Theory

Dates to remember:
Exam 1:  Oct 21
Portfolio due:  Nov 27
Exam 2:  Dec 2
Final Exam:  Dec 16

Homework assignments: Check this list frequently to get the latest homework assignments.  We will go over some of these in class, and I will ask that you hand some in.
 Ch Section Exercises In class Hand In Due # 1 Elem Ineq 1,2,3,7,8,11,28(a),29,30 1,2,3,28(a),29 30 9/18 1 Sets 13,14,16,18,20,21,22,24, 36,40,41,44 14, 20, 24, 41 22 9/18 1 Functions 45,46,49, 50 49 50 9/18 1 Real  Numbers prop 1.43(f)  prop 1.44(O3)  prop 1.45(F3,F4)  prop 1.46(b,g) prop 1.43(f)  prop 1.45(F3)  prop 1.46(g) prop 1.44(O3)  prop 1.45(F4)  prop 1.46(b) 9/25 2 EXTRA CREDIT dominoe puzzle dominoe puzzle 9/30 2 Logic 2,3,4,5,6,8,9,10 10 EXTRA CREDIT 11 11 10/2 Compound Remark2.20 (b,d,e,f) b,f d,e 10/4 3 Proof Tech 17,19,24,25,28,31,32,33,35,36,38 31,35,37,38 17,19,25 10/9 4 40,44,45,47,48,50,51,52,53 44 45 10/9 4 3 Induction 5,10,14,15,16,17,18,21,23,29,33 5,10,14,16,33 10/18 5 EXTRA CREDIT 34 34 10/18 4 Bijections 2,4,6,7,8,9,10,11,12, 22, 23, 25, 29, 33 7,9,11 11/4 6 Q-ary 15,13 22,23,29,33 11/15 7 Cardinality 21,43, 45, 47, 49 47 Prisoner's Dilemma Prisoner's Dilem 11/1 extra credit 13, 15, 11/18 4.43 11/25 8 5 1, 5, 11 1,5,11 11/25 8 EXAM 2 12/4 PORTFOLIO 12/6 11 3,6 3,6 12/9 9

Solutions to selected homework problems and examples.

Keys to Quizzes and exams:

Portfolio:

The portfolio is due on Dec 6.   Give complete questions to each problem.  Write the solution as formally as you can.  Use complete sentences.  Write it as if it would be published.  In fact, I will publish selected solutions on the web!  Either type or write very neatly.  If you type, it is okay to type the sentences and leave spaces for writing equations by hand.  If you have access to an equation editor, that would be best.

Do:
Chapter 1:   29, 50    Sample solutions Fall 2002
Chapter 2:  45, 51     Sample solutions Fall 2002
Chapter 3:  10 (only the product),  16   Sample solutions Fall 2002
Chapter 4:  12(a,b,c,d),  33,  47  Sample solutions Fall 2002
Chapter 5:  11