http://www.math.uri.edu/~eaton/Mth307F05.htm
Updated: September 6, 2005
MTH 307  Fall 2005
Introduction to Mathematical Rigor
Sec. 2 Meets MWF:
Sec. 1 Meets MWF:
Instructor: Dr. Nancy Eaton
Email me: eaton@math.uri.edu
Phone: 8744439
Office: Rm. 222, Tyler Hall
Office hours: Wednesday 3:30 – 4:30 and
Thursday 1:00 – 3:00
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.

CHECK HERE (WEB PAGE) OFTEN FOR UPDATES:
Homework Assignments See note Hk#8 

Course Description:
Provides
a bridge between lower level and upper level mathematics courses where proofs
are discussed. The emphasis is on basic
structures, logic, and the structure of mathematical statements and proof
techniques. Theorems and problems from
many areas of mathematics serve as motivation for learning proof techniques and
come from a wide range of topics including set theory, number theory, abstract
algebra, geometry, discrete math, and continuous math.
1. Become familiar
with and correctly use the basic structures of mathematics.
a. Recognize and interpret the meaning of the terms and
symbols used in set notation for unions and intersections of any number of
sets, for the complement of a set, and for the Cartesian product of sets and
ntuples.
b. Recognize and form partitions of sets and name the
corresponding equivalence classes.
c. Distinguish the different number systems, including
natural numbers, integers, rational numbers, real numbers, and complex numbers
and explain their respective constructs.
d. Interpret addition, subtraction, multiplication, and
division as binary operations on numbers and explain how subtraction and
division can be defined in terms of addition and multiplication, respectively.
e. Explain the meaning of the equal sign in terms of
equivalent expressions that can be substituted, one for the other in other
equations.
f.
Recall the field and ordered
field axioms and apply them in the justification of some basic identities and
inequalities for real numbers.
g. Express the summation and product of multiple terms, using
sigma and pi notation and manipulate expressions that contain these symbols.
h. Relate the basic definitions of functions, injections,
surjections and bijections.
i.
Identify and employ the
basic operations on functions, including addition, multiplication, and
composition
j.
Identify and formulate the
inverse of a function and interpret the meaning in terms of the inverse under
composition of functions.
k. Distinguish between the various uses of negative one in
the exponent.
l.
Establish cardinality of
sets, both finite and infinite.
m. Recall definitions and properly apply counting terms and
principles.
2. Learn the many
aspects of logic and correct reasoning to both understand and prove theorems.
a. Produce proofs of set equalities and inclusions using
basic set theory principles.
b. Translate between mathematical statements, written in
ordinary English and using quantifiers, both universal and existential.
c.
Distinguish the types of
mathematical statements, such as implications and if and only if statements.
d.
Use truth tables to
establish equivalence between mathematical statements.
e.
Identify the hypothesis and
the conclusion of an implication.
f.
Form the converse,
contrapositive, and negation of statements containing implications and
quantifiers.
g. Form the negation of any mathematical statement and form
counterexamples to false statements.
h. Employ any of the proof methods: direct proof, induction,
contradiction, and proving the contrapositive, to formulate a proof of an
elementary result.
i.
Apply basic proof
techniques, the Pigeonhole Principle, and combinatorial reasoning to prove some
elementary counting theorems that are used in probability and combinatorics.
j.
Apply basic proof techniques
to statements about the injectivity and surjectivity of functions.
3. Additionally, topics from the following list will be
covered in the course.
a. Set Theory (e.g. DeMorgan’s Laws, Cardinality)
b. Geometry: Axioms
and theorems, Pythagorean Theorem
c. Algebra (e.g. Algebraic identities, Quadratic Formula,
Geometric Sum Identity, Binomial Theorem)
d. Real Functions (e.g. Ceiling, Floor, and Greatest Integer
Functions, bounded functions, Monotone functions)
e. Real Numbers (e.g. Ordered field axioms and theorems)
f.
Discrete Mathematics (e.g.
Counting formulas and principles, graph theory)
g. Number Theory (e.g. Divisibility, qary representations of
numbers, Division Algorithm)
Text:
A Discrete Transition to
Advanced Mathematics by
Bettina Richmond and Thomas Richmond, Thomson Brooks/Cole Pubs.
Homework  25%
Portfolio  25%
Three 1hour exams  30%
Final  20%
Homework Assignments: (25% of your grade)
I assign many problems from the book
and other sources. You are expected to do them all.
HkPts System:
From those
assigned, I will select some to hand in. Follow the guide.
These will be graded and comments will be given. There will be a few more than 40 problems
selected for you to hand in. Each is
worth up to 4 HkPts. Your goal
is to get as close to 160 HkPts as possible.
Earning
extra HkPts:
·
You can earn
extra Hkpts by doing more than a total of 40 problems.
·
Also, during
the semester, you will be asked to present solutions to the assigned problems
on the board. These will be chosen from
those that you do not hand in. You will always be given a chance to
prepare your presentation. This will serve to improve your verbal
communication skills. You will be graded on your presentation and earn up
to 2 HkPts for each solution.
·
Answer the
“What is Math?” questions. See below.
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 5th. 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.
The problems are:
o
1.2: 8
o
2.1: 14
o
2.2: 4, 12
o
3.1: 2, 4
o
4.3: 6 (First, use
induction to show that the nth row of Pascal’s Triangle corresponds to the
choice numbers: C(n,0) C(n,1) C(n,2) . . . C(n,n) – Second, use the binomial
theorem to show that the sum of this row is 2^n)
o
6.1: 6 (Don’t answer the
part about the maximal subset of the domain.)
o
6.2: 10
o
6.3: 2
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 and 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.
7, Nov 4, and Dec 2. You may
take the final exam on either Friday Dec. 16 at
Exam 1: Oct 7; Exam
2: Nov 4; Exam 3: Dec 2
Portfolio: Dec 5
Final Exam: Dec 16 or Dec 19
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.
Guide:
·
Write out the
entire question.
·
Explain
carefully, using proper mathematical logic.
·
Use complete
sentences.
·
Write neatly
or type your assignment.
·
The format
should be 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 
2, 5(a), 8 
Sept 16 
1 
1.2 
1, 2, 48, 12, 13, 15, 16 
4, 12, 8 
Sept 23 
2 
1.3 
3, 6, 7, 9, 10, 12 
6, 10 
Sept 23 
2 
1.4 
All 
8, 9 
Sept 30 
3 
1.5 
1(a, c, d), 2(a, b), 3, 6 
4(b), 4(c) 
Sept 30 
3 
1.6 
13, 6, 911 
6(a), 9(b) 
Oct 14 
4 
2.1 
2, 48, 10, 13, 14, 17, 24 
6, 
Oct 14 
4 
2.1 

14, 24 
Oct 21 
5 
2.2 
2(a, b, d), 3, 10, 12, 13, 14(a, b), 15 
4, 10, 12 
Oct 21 
5 
2.3 
14, 7 
2, 4 
Oct 28 
6 
3.1 
2, 3, 4, 9, 15, 16, 20, 24 
2, 4, 16 
Oct 28 
6 
4.1 
1, 3, 5, 6, 7 
6(b), 6(c) 
Nov 11 
7 
4.2 
2 – 8 
4, 6 
Nov 11 
7 
4.3 
1 – 11 
2, 4, 6, 8 
Nov 11 
7 
6.1 
3, 4, 6, 7, 9, 11, 19 
4, 6 
Nov 18 
8 
6.2 
2, 3, 5, 8 – 14 
2, 8, 6(not part e), 10 
Nov 18 
8 
6.3 
2, 3, 7, 9 
2 
Nov 23 
9 
7.1 
2, 3, 4, 5, 6, 11, 12 
2, 4, 6, 12 
Nov 23 
9 
8.1 
1, 2, 3, 14 
2, 14 
Dec 9 
10 
8.5 
2, 3, 5, 6 
2, 6 
Dec 9 
10 
Solutions
to selected homework problems and examples:
Practice Probs for Exam 2 More Solutions
Practice
Probs for Exam 3 Solutions
Some Web Sites:
Fermat's
Last Theorem
Puzzles  Think.com
Recreational Math
Throughout the semester, submit answers to these questions via email. You may receive up to 4 HkPts. I will put them up on our web page and indicate the author. If you quote someone, please indicate.
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
Mathematics is the study of quantity, structure, space and change. It developed, through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the study of the shapes and motions of physical objects. (Wikipedia encyclopedia)
Math is the representation
of numbers in any universe and dimension. It is the understanding of numerical
expressions and their relationship to everything in our lives, from time, to
the numbers of chairs at a wedding, to complex ideas such as their discrete and
infinite properties. It is the understanding of patterns in nature and on
paper. –A.Koster
Mathematics is "the study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols"dictionary.com
Math is a
language that describes the laws of nature. Professor Gerry Ladas.
Mathematics
is a science (or group of related sciences) dealing with the logic of quantity
and shape and arrangement and it is a study of welldefined concepts.
www.dictionary.com and Daniel Henry Gottlieb
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 well known 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
Mathematics
benefits society because it is the job of predicting and calculating things
that are important or valuable for us to know, the job of being sure those
predictions and calculations are right, and the job of discovering new truths
that may one day be essential to ensuring a better life for us all. (http://www.easimath.com/Math.htm)
Math is used
all over the world all of the time. Although not everyone understands the more complex
equations and problems, everyone uses math every day. From calculating a tip at
a restaurant, to determining how much you are saving when a shirt goes on sale
25%. We are able to understand why as we drive farther in our cars, the
odometer continues to grow. In some cultures that depend on hunting, they are
able to count how much food they have left and hunt and gather depending on how
much they need. We all understand that you can never tangibly see a negative
amount of something. –A.Koster
Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries.(wikipedia.orgoline encyclopedia).
Mathematics
is involved in every aspect of life, whether it be shopping at the mall, or
discovering a cure for cancer. Math provides us with the possibility of
always acquiring new knowledge.~Heather Hazard
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
Universities
realize that mathematical expertise is more important than ever, particularly
in computers and hightechnology areas.
Having a mathematics department at a university prepares students for
employment in the mathematical sciences or for future study. Mathematics degree
concentration opportunities combine math study with philosophy, economics,
computer science, and math education. Universities know that all of these areas
are an essential part of life and that without math they would not be possible.
–H.Lally
Math is
continuously growing. New ideas are always being proven, and old ones are
constantly being replaced in order to make them more understandable. Many
people have been pulled into the idea of possibly proving something new or
improving an old idea. It is like a workout for the brain, and our society sure
it obsessed with working out any portion of our body. A university is the
easiest place to allow intelligent minds to gather and think up ideas and work
on issues together. It combines the efforts, so that more things can be done
quicker. One person might be so close to proving something but are just missing
one small variable, and after showing it to a coworker, who notices the small
mistake, this large issue has all of a sudden been solved. It is also one of
the easiest ways to get students involved and interested in the study of math.
–A.Koster
Universities have departments devoted to mathematics because mathematics is an on going learning process. While most of the information known about mathematics is grounded, there will always be someone that is able to take a particular concept further. Math departments at the university level are here to broaden the mind frame and to establish new information in its pupils. Not only this, but it is a known fact that much of our lives revolve around mathematics, and college is the stepping stone to life, so mathematics must be an integral part in our getting ready to take on life~Heather Hazard
Colleges
know that mathematics is important and they have their own departments for it
because it is used for so many differnt jobs.
You have to take different math classes for every major that you are in
but it is very important you take those classes. Since all students can't take engineering classes that have to
offer many different classes. – Jessica Weil
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
Mathematics
is a field that can be used in many different careers. Pretty much every occupation you want to get
into requires some type of math. In
your job you may have to calculate interest rates or work with money. Mathematics is used often in fields like
accounting, engineering, science, and economics. Knowing different ways to work with numbers will make ordinary
calculations a lot easier and less time consuming. Therefore allowing you to have more time to do work on more
difficult things and not have to worry about simple math. H. Lally
Math allows for
me to be able to problem solve quicker. By knowing how to do something quite
complex well, I will be able to do all of those easier concepts just that much
faster. You must be able to problemsolve and work through something until you
reach a conclusion. Employers are looking for these qualities when they go to
hire someone. There is not one job that does not involve some understanding of
math. Engineers use it to understand the forces involved in what they are
creating; Economists need to be able to understand numbers and their meanings
in statistics, etc; Hairdressers must understand exactly what a client means
when they say they only want one inch off. Personally I am a math major because
as a career I would like to work in the Power and Energy field specifically
with wind power. I must be able to understand the financing of the projects,
the statistical data that is found through the wind in the area, the dimensions
of projects, etc.
Seeing how i am going to be a math teacher, mathematics has
to be at the basis of my knowledge. Teaching to me, is being able to
teach another person how to be prepared to succeed in life. In order for
a person to succeed, they are going to need at least some minimal knowledge of
mathematics. Therefore it is important that i know all that i can to be
able to pass it on to another person.
–H.Hazard
Since I am
going to be a math teacher it is important I am skilled in all areas of
mathematics. Even though I want to
teach middle school algebra, I think it is important that I know some of the
higher math also. If the students ever
need help in later years they will know they can come to me. –Jessica Weil