MTH 307  Fall 2004
Introduction to Mathematical Rigor
Sec. 1 Meets MWF:
Instructor: Dr. Nancy Eaton
Email me: eaton@math.uri.edu
Phone: 8744439
Office: Rm. 222, Tyler Hall
Office hours: Monday 12:302 and
Tuesday 11:301
Visit my web page: WEB PAGE
Students who require accommodations and who
have documentation from
Disability Services (8742098) should
make arrangements with me as soon as possible.
The
purpose of this course is to provide a bridge between lower level computational
mathematics courses and upper level theoretical math courses. In this
course, we will emphasize basic structures, logic, and the structure of
mathematical statements and proofs. In this course, you will learn about logic, proof techniques, and
basic structures of mathematics. You will apply these new tools to prove
theorems and solve problems in many areas of mathematics. The theorems
and problems serve as motivation for learning proof techniques and come from a
wide range of topics including number theory, abstract algebra, discrete math,
and continuous math. As a result, you will be learning the language of
mathematics in both written and oral form. Once learned, you will be ready to
appreciate the beauty of mathematics and to use these skills in all of
your future math courses including Abstract Algebra, Topology, Real Analysis,
and Complex Analysis. Specifically,
we will cover topics from the following list.
Logic
and Proof techniques:
Quantifiers  universal and existential
Mathematical statements
Basic
Structures of Mathematics:
Some
Theorems:
Text:
A Discrete Transition to
Advanced Mathematics by
Bettina Richmond and Thomas Richmond, Thomson Brooks/Cole Pubs.
Homework  25%
Portfolio  25%
Three 1hour exams  25%
Final  25%
Homework Assignments: (25% of your grade)
I assign many problems from the book
and other sources. You are expected to do them all.
From
those assigned, I will select some to hand in. Always write out the entire question and
the solution in complete sentences. These will be graded and comments will be given. The
problems marked (EC) are optional and will count as extra credit toward your
homework grade.
During
the semester, you will be asked to present some of your solutions on the
board. You will always be given a chance to prepare your
presentation. This will serve to improve your verbal communication
skills. You should try to go over your solution with me during my office
hours before you present it to the class. You will be graded on your
presentation and this will count toward your homework grade.
Here are some tips in doing homework.
Try each problem on your own.
Check your answer with the back of the book
or someone in class.
Go over each problem with someone in the class.
Ask me about the ones that you still do not completely understand.
Take notes whenever we go over a problem in class.
Rewrite each problem as best you can and keep them together in a binder.
Portfolio: (25% of your grade)
From the homework problems that you
do during the semester, I will select 10 for you to rewrite and hand in as a
portfolio of work representing what you learned in this course. This is
intended to be a collection of your best work. This will be due on Monday December 6th. You will have two weeks advance notice as
to which problems will be included in the portfolio. The intention here is that
you will make sure you understand how to solve each homework problem as we go
along in case it will be selected to be in the portfolio.
Write out
the entire statement of the exercises. 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!
This should be typed. 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.
Ten exercises will be chosen from the homework and will be
posted here on Nov 22
Section
1.2, Number 8.
Proposition
1 (3).
Section
1.4, Number 9.
Number
11 from the handout. Be sure to use only the axioms, the propositions,
and the AGM Inequality. State every theorem that you use in its
entirety. For instance, if you use P3 (4), say “Using Proposition 3 (4)
which states that 0≤ x ≤ y and 0 ≤ u
≤ v imply that x۰u ≤ y۰v.
Section
1.6, Number 6(a, b).
Section
2.1, Number 7
Section
2.2, Number 2(b).
Section
2.2, Number 4
Section
3.1, Number 20.
Section
4.3, Number 6. We did this one in class. Use the Binomial Theorem.
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 with others. Working with someone from class will
help you to improve your math communication skills as well.
I encourage you to work together under the following circumstances.
Begin the problem on your own, do as much as you can.
Ask someone from the class to explain the basic outline of a solution.
If working with someone else, sometimes they will also ask you for your understanding of the basic outline.
Take what you learned and write out the solution on your own, using your own words.
Never copy word for word from anyone else’s paper. In fact it is better not to look at anyone else’s completed written solution or to show yours to anyone else.
If you still do not completely understand the solution, you can ask your professor to look at what you wrote and try to clear up any parts of the solution that are not completely clear and accurate.
The following is an
excerpt from the University Manual.
8.27.11 A student's
name on any written exercise (theme, report, notebook, paper, examination)
shall be regarded as assurance that the work is the result of the student's own
thought and study, stated in the student's own words and produced without
assistance, except as quotation marks, references and footnotes acknowledge the
use of other sources of assistance. Occasionally, students may be authorized to
work jointly, but such effort must be indicated as joint on the work
submitted. Submitting the same paper for more than one course is
considered a breach of academic integrity unless prior approval is given by the
instructors.
There will be three 45minute
exams. The dates for these are Oct.
6, Nov 3, and Dec 1. You may
take the final exam on either Saturday Dec. 18 at
Exam 1: Oct 6; Exam
2: Nov 3; Exam 3: Dec 1
Portfolio: Dec 6
Final Exam: Dec 18 or Dec 23
Homework
assignments will be given in class and recorded here for your reference. You
must do the following for
full credit on the assignments that you hand in.
Write out the entire question.
Explain carefully, using proper
mathematical logic.
Use complete sentences.
Write neatly or type your assignment.
The format should single column, double
spaced.
Use only one side of the paper.
Clearly mark the assignment number and due
date on the top.
Arrange the exercises in order by problem
number.
Staple each assignment.
Section 
Do all 
Hand In 
Due Date 
Assignment # 
1.1 
1, 2, 3, 5(ad), 8 



1.2 
1, 2, 48, 12, 13, 15, 16 
8 
Sept 13 
1 
1.3 
3, 6, 7, 9, 10, 12 





Redo 1.2(8) 
Sept 24 
Ex Cred 

Proposition 1 from handout 
P1(1) 
Sept 24 
2



P1(26) 
Sept 29 
3


Exercises 29 from handout 
2

Oct 8 
4

1.4 
All 
8, 9 
Oct 8 
4 

Exercises from the handout 
11, 12, 13 
Oct 18 
5 
1.5 
1(a, c, d), 2(a, b), 3, 6 
4(b, c) 
Oct 25 
6 
1.6 
13, 6, 911 
6(a, b), 9(b) 
Oct 25 
6 
2.1 
2, 48, 10, 13, 14, 17, 24 
6, 7, 13, 14, 24 
Nov 12 
7 
2.2 
2(a, b, d), 3, 10, 12, 13, 14(a, b), 15 
2(b), 4, 10, 12, 4(a, b),
15

Nov 12 
7 
(2.2) 

2(d), 
Nov 24 
8 
2.3 
14, 7 
2 
Nov 24 
8 
3.1 
2, 3, 4, 9, 15, 16, 24 
20 
Nov 24 
8 
4.1 
1, 3, 5, 6, 7 
6(b, c) 
Nov 24 
8 
4.2 
2 – 8 
6 
Nov 24 
8 
4.3 
1 – 11 



6.1 
3, 4, 6, 7, 9, 11, 19 
6, 11 
Dec 10 
9 
6.2 
2, 3, 5, 8 – 14 
3(b, c), 10 
Dec 10 
9 
6.3 
2, 3, 7, 9 
7(a) 
Dec 10 
9 
7.1 
2, 3, 4, 5, 6, 11, 12 



8.1 
1, 2, 3, 14 



8.5 
2, 3, 5, 6 



Solutions
to selected homework problems and examples:
Example using
associativity of addition and induction
Review
for Exam 3
Some Web Sites:
Fermat's
Last Theorem
Puzzles  Think.com
Recreational Math
Throughout the semester, submit answers to these questions via email. I will put them up on our webpage and indicate the author. If you quote someone, please indicate. I will read them to the class as they come in.
1. What is mathematics?
Math is a language which we use in society to
communicate quantities and the intermediate operations which are used to obtain
these quantities. Like with English, Spanish, German, or Russian, there are
many things we can say about the different things we encounter; but when it
comes to expressing numbers, humans have found that there is a need for an
additional language, math.  Jason Stockford
Mathematics
is the Science of numbers and of space configurations. Webster Dictionary
To the
question what is math: "My hypothisis1. Mathematics is the
language of nature. 2. Everything around us can be represented and understood
through numbers.3. If you graph these numbers, patterns emerge.Therefore: There
are patterns everywhere in nature." Maximillian Cohen from the movie Pi
Mathematics is an arithmetic
interpretation and study of the physical world. Math is a natural language that defines the actions of natural
occurrences and it gives us the ability to interpret those actions. –Amy Brown
2. How does it benefit society?
"In most sciences one generation tears down what another has built and what one has established another undoes. In Mathematics alone each generation builds a new story to an old structure." ~Hermann Hankel
Math benefits society in many ways which people who are poorly educated in math
do not realize. For instance, we can speak of one of the most wellknown areas
of math, Calculus. Calculus is something that many students fear, but what few
realize is that Calculus does not complicate that which is simple, but rather
just the opposite. Calculus enables people from a broad array of professions to
tackle difficult problems with much more ease then would be possible if
everything needed to be done using brute force. For example, engineers often are
required to find areas, volumes, or centers of mass in irregular shaped
objects, and with the relatively simple concept of the definite integral these
engineers who must find precise answers may do so without the pure drudgery of
computing a huge number of Riemann sums.  Jason Stockford
It is the basis of
interpretation of science. Mathematics
is repetitive, unique, and it is used in development of all technological
advances since the beginning of history.
It is used to define and explain the universe, solar system, and the
world. Math has been used as society
has evolved monetarily, technically, and scientifically. – Amy Brown
3. Why do universities usually have a department devoted to its study?
Math is something that is still developing, and
everyday mathematicians are finding new solutions to old problems and applying
old problems to new applications. Colleges and Universities are at the core of
almost all research done in the world today, so it only makes sense that
colleges and universities would have departments devoted solely to the study of
math.  Jason Stockford
Math is essential to all the related physical sciences. A separate department is devoted for math because of the need for future research and development. It is the basis of explaining the world.
“To think the thinkablethat is the mathematician’s aim.”
C.J. Keyser
4. How does learning it help you in your career other than that it is required
for your degree?
Being able to figure out math problems requires
a certain amount of logic and, companies which are hiring new graduates may
consider people with degrees in math to be more valuable then people with other
degrees for that reason. People who are good in math tend to be good decision
makers and they tend to analyze problems that arise, in a logical manner. These
are both positive things which competitive employers look for when choosing
applicants.  Jason Stockford
Math is logical, methodical reasoning, problem solving and a thought process. It is important in understanding the structure of nature. Math provides an overall well rounded intelligence and gives a basic understanding of all other disciplines. Without math we would not have a society. Currency, economics, biology, and medicine are just a few fields who’s foundation is based on math.
“A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made of ideas.”
 G.H. Hardy