http://www.math.uri.edu/~eaton/MTH307S07.htm
MTH 307  Spring 2007
Introduction to Mathematical Rigor
Instructor: Dr. Nancy Eaton
Email me: mailto:eaton@math.uri.edu
Phone: 8744439
Office: Rm. 208, Tyler Hall
Office hours: Mon, Tues, Fri: 1:302:30
Visit my web page: http://hypatia.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. De Morgan’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)
A Discrete Transition to Advanced Mathematics by Bettina Richmond and Thomas Richmond,
Thomson Brooks/Cole Pubs.
Student
Solutions Manual for A Discrete Transition to Advanced Mathematics, Bettina
Richmond and Thomas Richmond, Thomson Brooks/Cole Pubs.
In order to achieve the most from this
course you are expected to do the following:
In addition to the Homework assignments, portfolio, worksheets, and
exams, you are expected to read the sections in the book that we cover
and do all of the Suggested Exercises from the book. These problems have
answers in the back of your book and/or solutions in the Student Solutions
Manual. You are expected to ask about
anything that you do not understand.
THIS IS HOW YOU ARE EXPECTED TO LEARN THE MATERIAL. If you do these things BEFORE turning in
assignments and taking tests, your grades are likely to be much better.
Summary:
1.) Read Text.
2.) Take notes in class.
3.) Do the Suggested Exercises, these are not
to be handed in.
4.) Check answers in the back of the text
and/or Solutions Manual.
5.) Ask about those you do not understand.
6.) Complete the Homework assignments that are
to be handed in, including worksheets.
7.) Study all of the above for exams. But if you keep up with at least 4 hours of
working the above steps each week, outside of class, studying for an exam should
consist of going over what you have learned, versus learning it for the first
time.
Homework Assignments
 25%
Portfolio  15%
Worksheets – 10%
Three 1hour exams
 30%
Final  20%
Homework Assignments:
(25% of your grade)
There will be 40 exercises, selected from out text, to hand in. These will be graded and comments will be given. Each is worth up to 4 HkPts. (See the grading rubrik.) See the guide below.
Earning
extra HkPts:
Class work for extra credit is
sometimes given – you must be in class to receive these points.
During the semester, each
student will be asked to present solutions to two exercises 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 4 HkPts
for each solution. A formally written,
typed solution must be handed in at the time of your presentation. Sign up by choosing 5 problems from the list below.
Send me an email (with
subject line: 307 – board) indicating your five top choices. I will select two from your list for you to do.
Be sure to check your solution over with me before presenting it to the
class. I will post your solution for the
rest of the class to download if they like. Solutions
posted here.
At any time during the
semester, you may resubmit (once) an exercise that has been returned to you
with a grade of 0, 1, or 2 pts, provided that you originally handed it in by
its due date. You must resubmit the
entire question and answer. No more than
one exercise per sheet of paper. Proper
identification of the problem must be given at the top of the page.
Your goal is to get as close
to 160 as you can. Any points over 160
will be ignored.
Computation of this portion of your grade: Total HkPts divided by 160, or 100%, which
ever is smaller.
Guide: You must do the following for full credit on the assignments
that you hand in.
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.
DD) Use only one side of the
paper.
EE) Clearly
mark the section 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.
HH)
In
the case that it is late, you must put the date that you handed the assignment
in, or no credit will be given.
II)
Explain carefully, using proper
mathematical logic.
JJ)
Use complete sentences.
Portfolio: (15%
of your grade)
I will
assign 10 exercises similar to ones that you have worked on all semester for
you to 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 Friday May 4th. (Provide a stamped addressed
envelope so that I can mail it back to you when I am done grading it.) 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 by April 16th. You may hand
it in on Friday April 27 for a check. I
will give it back to you on Monday April 30 with my initial comments.
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. This MUST BE TYPED. If you have access to an equation
editor, that would be best. See: link.
1)
6.2#1
2)
6.2#9
3)
6.3#2
4)
6.3#3
5)
7.1#12
– You must use Euler’s Theorem for this. 5) 6.1#19
6)
8.3#4
7)
8.5#5
7) 8.3#2
8)
8.5#6 8) Let n be a natural number. Prove that the number of odd sized subsets of
[n] is the same as the number of even
sized subsets of [n].
9)
Fibonacci
Sequence: Let F_{0} = 0, F_{i}
=1, and for all i>1, F_{i} = F_{i1 }+ F_{i2 . } Prove that F_{1} + F_{2} + F_{3}
+ … + F_{2i1} = F_{2i+1} 1.
10)
The Handshake Problem: Consider n
married couples at a party, on
couple is called the host and hostess.
Suppose that no person shakes hands with his or her spouse, and the 2n1 people other than the host shake
hands with different numbers of people.
With how many people does the hostess shake hands.
Worksheets: (10%
of your grade)
Three
worksheets will be given during the semester.
These will contain exercises that are not covered in the book. You should complete the problems on the
worksheets and hand them in by their due dates.
There will be three 45minute exams. The dates for these are Feb. 16th, Mar 30th, and Apr 27th. The final exam is will be given during the scheduled time on Friday May 4th from 8:00 – 11:00 AM. The exams will be taken from material that we covered in class, exercises in the book, and worksheet problems. Take careful notes and study them for the exams. No PDAs, cell phones, laptops, or calculators will be allowed on your desk during exams.
Exam 1: Feb 16; Exam
2: Mar 30; Exam 3: Apr
27
Portfolio: Apr 16 (problems announced), Apr 30 (portfolio due)
Final Exam: May 4 (811 AM)
In the
following syllabus, approximate dates are given for starting each section of
the text that we will cover. Exact due
dates for Homework assignments and worksheets will be given in advance of each
assignment. I suggest that you read
ahead in the book and start working on exercises from the book as soon as we
start each section.
Week of 
Section 
Suggested Exercises 
In Class 
Hand In 
Exams 
Due Date 
Mon 1/22 
1.1 
1, 2, 3, 4, 5, 8 
9 

Mon  1/29 


1.2 
2, 4, 5, 9, 10, 15 
1, 6, 7 



Mon 1/29 
1.3 
5, 9 
3, 8, 11 




1.4 
1, 3, 7, 10, 12, 14 
2 
8a, 9 

Fri 2/9 

1.5 
1, 2, 4 
3, 6 
5 

Fri 2/9 
Mon 2/5 
1.6 
1, 3, 6, 9, 11 
2, 7 
5, 8 

Fri 2/9 




Logarithms 

Fri 2/23 

2.1 
5, 8, 12, 27 
2, 9a, 25 
4, 7, 11, 

Wed 2/28 




13, 15, 18 

Fri 3/2 
Mon 2/12 
Reals 



Fri 3/9 






Exam 1 
2/16 
Wed 2/21 
Reals 





Mon 2/26 
2.2 
2e, 9, 15 
4, 14a 
2b, 2d, 

Mon 3/26 

2.2 


3, 11a, 12 

Wed 3/28 
Mon 3/5 
2.3 
3, 7, 12 
5 
2, 6 

Wed 3/28 
Mon 3/12 
4.1 
4, 6, 8 
7 
1, 3 

Fri 4/13 

4.2 
1, 4, 7 

2, 6 

Fri 4/13 

4.3 
1, 3, 7, 9 
8 
4, 6, 11 

Fri 4/13 
Mon 3/19 

Spring Break 




Mon 3/26 
5.1 
3, 6, 10, 13 
7, 11 
12 

Wed 4/18 

5.2 
2, 5, 8 
1 
4a, 6a 

Wed 4/18 

6.1 
2, 4, 10, 17 
3, 6, 7, 11, 19, 
5 

Wed 4/18 





Exam 2 
3/30 
Mon 4/2 
6.2 
2, 4, 6, 18 
3, 10 
1, 8 







Mon 4/23 

Mon 4/9 
6.3 
6, 13 
7 
2, 3 



7.1 
4, 6, 13 
2, 5, 14 
11, 12 


Mon 4/16 
8.1 
3, 6, 12 
1 
2 



8.3 
3, 8, 12 
5, 14 
2, 4 


Mon 4/23 
8.5 
2 

5, 6 







Exam 3 
4/27 
Mon 4/30 



Portfolio 

4/30 





5/4 (811) 
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.
Sections
1.1, 1.2 Sections
1.3, 1.4, 1.5, 1.6
Section
2.1 Section
2.2 Section
2.3 Godel
Solutions to selected homework problems and examples:
Practice for Exam 1 Solutions to Practice for Exam 1 (Question 5 is from chapter 2 which will not be included on our Exam 1)
Practice for Exam 2 Solutions to Practice for Exam 2
Practice for Exam 3 Solutions to Practice for Exam 3
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
Section 
Num 
Name 
Download 
1.1 
9 
S. Alvarez 

1.2 
1 
J. Moniz 

1.2 
6 
A. Heissan 

1.2 
7 
J. Albanese 

1.3 
3 
J De Roche 

1.3 
8 
J DeRoche 

1.3 
11 


1.4 
2 
M. Greffin 

1.5 
3 
C. DiLibero 

1.5 
6 
J. Vitale 

1.6 
2 
Lizz Yorio 

1.6 
7 
S. Mc Donald 



C. DiLibero 

2.1 
2 


2.1 
9a 
Bill Jamieson 

2.1 
25 
Andrew Poggie 

2.2 
4 
Fran Dempsey 

2.2 
14a 
John Vitale 

2.3 
5 
S. McDonald 

2.3 
12 


Pigeon 
1 
Charlene DiLibero 

Pigeon 
2 
Sean Alvarez 

Pigeon 
5 
Matthew Greffin 

Worksheet2 
4 
Andrew Poggie 

4.1 
7 
Jessica DeRoche 

4.3 
8 
Sabrina Chevrette 

5.1 
7 
Charlene DeLibero 

5.1 
11 
John Vitale 

5.2 
1 
Fran Dempsey 

6.1 
19 


6.2 
3a 
Jessica DeRoche 

6.2 
3b 
Matthew Greffin 

6.2 
3c 
Sabrina Chevrette 

6.2 
3d 
Sean Alvarez 

6.2 
3e 
Kelly Keegan 

6.2 
3f 
Kerry Holden 

6.2 
3g 


6.2 
3h 


6.2 
3i 


7.1 
2 
Joe Piecuch 

7.1 
5 
Shelley McDonald 

7.1 
14 
Patrick Kenney 

8.1 
1 
Kelly Keegan 

8.3 
5 
Bill Jamieson 

8.3 
14 

