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 PAGESection 01-Kingston

Meets MWF: 11:00-11:50, Wales 224

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?

Homework Assignments: (25%
of your grade)

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.

Portfolio:
(25% of your grade)

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.

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