http://www.math.uri.edu/~eaton/Mth307F06.htm
Updated: September 6, 2006
MTH 307  Fall 2006
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: MTW 910 and
MTW 89 by apt.
Visit my web page: http://www.math.uri.edu/~eaton/
Students who require accommodations and who
have documentation from
Disability Services (8742098) should
make arrangements with me as soon as possible.
Provides
a bridge between lower level and upper level mathematics courses, where proofs
of mathematical statements are discussed.
The emphasis is on basic structures, logic, the structure of
mathematical statements, and proof techniques.
Theorems and problems from many areas of mathematics serve as motivation
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
suggested, I will assign assign many more than 40 to hand in. Follow
the guide. These will be graded and comments
will be given. Each is worth up to 4 HkPts.
(See the grading rubrik.) Your goal is to get as close to 160 HkPts as
possible.
Earning
extra HkPts:
You can earn extra Hkpts by
hanking in more than a total of 40 assigned 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.
Class work for extra credit is sometimes
given – you must be in class to receive these points.
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.
Carefully write a final version of each problem, as best you can, and keep them together in a binder.
Portfolio: (25% of your grade)
From the suggested homework problems,
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 11th. You will have two weeks advance notice as
to which problems will be included in the portfolio. The problems will be listed on this web
page on November 27^{th}. 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 MUST 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. See: link.
Portfolio
problems
Sec 1.2
– 7
Sec 1.4
– 6
Sec 2.1
– 5c, 10
Sec 2.2
– 2c
Sec 2.3
– 2 Use P.P to explain if applicable.
Sec 3.1
– 7e
Sec 6.1
– 12, 19
Sec 6.2
 12
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.
11, Nov 6, and Dec 4. The
final exam is will be given during the scheduled time on Friday Dec. 15 from
Exam 1: Oct 11; Exam
2: Nov 6; Exam 3: Dec 4
Portfolio: Nov 27 (problems announced), Dec 11 (portfolio due)
Final Exam: Dec 15
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:
AA)Write out the entire question.
BB)
Write
neatly or type your assignment. Do not use torn paper.
CC)The format should be single column,
doublespaced.
DD) Use only one side of the paper.
EE)Clearly mark the assignment number and due
date on the top of the front page.
FF)Arrange the exercises in order by problem
number. Attach pages together.
GG) One point will be deducted from the assignment for each class
day that it is late.
Explain carefully, using proper mathematical logic.
Use complete sentences.
Section 
Do all 
Hand In 
Due Date 
Assignment # 
1.1 
1, 2, 3, 5(ad), 8 
2, 5(a), 8 
Sept 13 
1 
1.2 
1, 2, 48, 12, 13, 15, 16 
4, 8, 12, 
Sept 13 
1 
1.3 
3, 6, 7, 9, 10, 12 
6, 10 
Sept 20 
2 
1.4 
All 
8, 9 
Sept 22 
2 
1.5 
1(a, c, d), 2(a, b), 3, 6 
4(b), 4(c) 
Sept 25 
3 
1.6 
13, 6, 911 
6(a), 9(b) 
Sept 25 
3 
2.1 
2, 48, 10, 13, 14, 17, 24 
6, 
Oct 2 
4 
2.1 

14, 24 
Oct 2 
4 


Exam 1 
Oct 13 

2.2 
2(a, b, d), 3, 10, 12, 13, 14(a, b), 15 
4, 10, 12 
Oct 20 
5 
2.3 
14, 7 
2, 4 
Oct 20 
5 
3.1 
2, 3, 4, 9, 15, 16, 20, 24 
2, 4, 16 
Oct 25 
6 
4.1 
1, 3, 5, 6, 7 
6(b), 6(c) 
Oct 25 
6 
4.2 
2 – 8 
4, 6 
Nov 1 
7 
4.3 
1 – 11 
2, 4, 6, 8 
Nov 1 
7 


Exam 2 
Nov 6 

5.1 
5, 6, 15 



5.2 
1, 2, 3, 4 



6.1 
3, 4, 6, 7, 9, 11, 19 
4, 6 
Nov 17 
8 
6.2 
2, 3, 5, 8 – 14 
2, 8, 6(not part e), 10 
Nov 20 
8 

Worksheet A 
Nov 27 
9 

6.3 
2, 3, 7, 9 
2 
Nov 27 
9 
7.1 
2, 3, 4, 5, 6, 11, 12 
2, 4, 6, 12 
Dec 1 
10 

Worksheet B 
Dec 1 
10 



Exam 3 
Dec 6 



Portfolio 
Dec 11 

8.1 
1, 2, 3, 14 
2, 14 
Dec 11 
11 
8.5 
2, 3, 5, 6 
2, 6 
Dec 11 
11 


Final 
Dec 15 

Packet1 Packet2 Packet3 Packet4 Packet5
Packet6 (Sec 2.1) Packet7
(Sec 2.2) Packet8 (Sec 2.3)
Packet9 (Sec
6.1) Packet10
(Sec 6.2) Packet11 (Sec 6.3)
Solutions to selected homework
problems and examples:
Key to Exam 3
Practice for
exams:
Exam 2 Solution Key to Exam 2 (For exam 2 – ignore #4) (For exam 3 – see #4)
Exam 3 Solution Key to Exam 3 (For exam 2 – see #4, 5, 7) (For exam 3 – ignore #4, 5, 7)
Some Web Sites:
Fermat's
Last Theorem
Puzzles  Think.com
Recreational Math
Throughout the semester, you may submit answers to these questions via email. I will post your answers on our web page and indicate the author. If you quote someone else, 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 math 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.~H. Hazard
3. Why do universities usually have a department devoted to the study of
mathematics?
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. – J. 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 different 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 will learning mathematics 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. – J. 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