MTH 307 Fall 2003
Introduction to Mathematical Rigor
Meets MWF: 11:00-11:50, Wales 223

Instructor:  Dr. Nancy Eaton

E-mail me:  eaton@math.uri.edu
Phone:  874-4439
Office: Rm. 222, Tyler Hall
Office hours: Monday 1:30-3:00, Tuesday 10:30-12:00
Visit my web page:  WEB PAGE

Students who require accommodations and who have documentation from
Disability Services (874-2098) should make arrangements with me as soon as possible.

Text
Course Content
Grade for course
Working with Others
Dates to Remember
Homework assignments
Solutions to selected exercises
Practice for Exams
Keys to Exams
Portfolio assignment
Other web resourse
Rubric
Inequality Fact Sheet

NOTE - HK due date changes - nothing due Nov 19 - see below.
Portfolio problems are listed below.

Text:  Mathematical Thinking, by D'Angelo and West, 2nd Edition
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  
9   Probability
10 Two Principles of Counting
11  Graph Theory


Course Content and Goals:
This course serves as a stepping stone between the lower level math courses and the upper level math courses. 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 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. The tools used to prove theorems and understand mathematical theory include quantifiers, mathematical statements, truth tables, equivalence of statements, venn diagrams, induction, the Well-Ordering Property, direct proof, proof by contradiction, permutations, arrangements, selections, and partitions.  We will also cover some of the basic structures of mathematics including, field axioms, ordered field axioms, number systems, sets, cartesian product, n-tuples, q-ary representations, basic set theory, functions, cardinality, and combinatorial graphs.

While learning the skills of proving theorems, 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 you future math courses including Abstract Algebra, Topology, Real Analysis, and Complex Analysis.

Motivation:
We will use some interesting, puzzling, and even amazing examples to motivate you to learn proof techniques and mathematical theory. In order to prove the statements it becomes necessary to learn the background material and proper techniques. Hopefully many of these problems will inspire you to become more fluent using the tools of mathematics to solve problems. The book Mathematical Thinking contains many fun examples.  We will draw problems from many mathematical disciplines so that you will also get an overview of the many different areas of mathematics.

Examples:
The Checkerboard Problem.  Counting squares of sizes 1 x 1 through 8 x 8, an ordinary 8 x 8 checkerboard has 204 squares.  How can we obtain a formula for the number of squares of all sizes on an n x n checkerboard?


The Handshake Problem.    Consider n  married couples at a party.  Suppose that no person shakes hands with his or her spouse, and the 2n-1 people other than the host shake hands with different numbers of people.  With how many people does the hostess shake hands?


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?


The Weights Problem.  A balance scale has left and right pans;  we can place objects in each pan and test whether the total weight is the same on each side.  Suppose that five objects of known integer weight can be selected.  How can we choose the weights to guarantee being able to check all integer weights from 1 through 121?


Nonnegative Integer Solutions.  Suppose that each resident of New York City has 100 coins in a jar. The coins come in five types (pennies, nickels, dimes, quarters, half dollars).  We consider two jars of coins to be equivalent if they have the same number of coins of each type.  Is it possible that no two people have equivalent jars of coin


  Chinese Remainder Problem.  A general in ancient China wanted to count his troops.  Suppose that when his soldiers were split into three equal groups there was one soldier left over, when split into five equal groups there were two left over, and when split into seven equal groups there were four left over.  Whis is the minimum number of soldiers that makes this possible?

Calculation of Grade:
Homework - 25%
Portfolio - 25%
Two 1-hour exams - 25%
Final - 25%

Homework Assignments: (25% of your grade)
I assign many problems from the book.  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 the solutions will be posted.  The problems marked (EC) are optional and will count as extra credit toward your homework grade.

Also, to improve your verbal communication skills, you will be asked to present the solution of some homework problems to the class.  You should always 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.

I will present the remaining problems to the class or post them on the web.

 Try each problem on your own.

 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 1st. 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 17
            HERE they ARE  Chapt 1:  13, 29   Solutions
                                        Chapt 2:  28, 45, 53  Solutions
                                        Chapt 3:  10(just the product), 16  Solutions
                                        Chapt 4:  10,   Solutions
                                                       12, 
                                                       42 (Don't use induction - instead explain how injective implies m >= n and
                                                              surjective implies m <= n)
                                                       43

            There are 11 here - so you are free to pick a subset of size 10, or do all 11 for extra credit.


Guide to working with others:
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.

Exams: (Exams 25%, Final 25%)
There will be two one hour exams. The dates for these are Wednesday Oct. 15 and Monday Nov. 17. The final exam will be given on Friday Dec. 19 at 8:00 - 11:00 AM.  The exams will be taken from material that we covered in class, so take careful notes.  Study these for the exams.  There will be no practice exams.

Dates to remember:
Exam 1:  Oct 15 
Exam 2:  Nov 17
Portfolio due:  Dec 1

Final Exam:  Dec 19
 

Homework assignments:   Check here regularly as due dates are subject to change.

Remember to write out the question, explain carefully, and use complete sentences.  
Write neatly using one side of the paper.
Clearly mark the assignment number and due date on the top.
Arrange the exercises in order by problem number and staple each assignment. 

Chapter
Section
Exercises
Hand In
Due/Assignmt#
1

Elem Ineq

1, 2, 3, 7, 11, 18, 28, 29, 30

3, 11, 30
Sept12 / #1

 

Sets

13, 14, 16, 36, 40, 41, 44

13
Sept12 / #3

 

Functions

22, 24, 45, 46, 49, 50

22(EC), 24(EC)
Sept17 / #2

 

Real 
Numbers

prop 1.43 (f) 
prop 1.44 (O3) 
prop 1.45 (F3,F4) 
prop 1.46 (b,g)



2

Logic

3, 4, 5, 6, 8, 9, 10, 11

3, 4, 5, 11(EC)
Sept24 / #3

 

Compound

Remark 2.20 (b,d,e,f)



 

Proof Tech

24, 28, 31, 32, 33, 35, 36, 37, 38

28, 33(EC)
Oct1 / #4

 

 

40, 44, 45, 48, 50, 51, 52, 53

40a(EC), 44, 45, 51, 52
Oct15 / #5

3

Induction

5, 10, 14, 15, 16, 17, 18, 21, 23, 29, 33, 34, 42, 55, 62

10, 16, 34(EC), 62(EC)
Oct29 / #6

4

Bijections

4, 6, 7, 8, 9, 10, 11, 12, 20, 22, 23, 25, 29, 33

10, 12, 25
Nov14 / #7

 

Q-ary

2, 15, 17



 

Cardinality

21, 43, 45, 47, 49

43, 45
Nov14 / #7

5

 Comb Reason

1, 5, 8, 11, 12, 14, 21, 31

21, 31

9

 Probability

 1, 2, 3, 4, 5, 6, 7, 11

7




Portfolio
Dec 1-8

10

 Pigeonhole

 2, 3, 5, 14

5(EC), 14

11

 Graph Theory

3, 5, 6, 12, 15, 16, 17

15, 16

Solutions to selected homework problems and examples.

Chapt 1 - Theorems

Chapter 2 - 44

Chapter 2

Chapter 2 - word problems from students in class

Chapter 3

Keys to exams:

Sequel to EXAM 1   -Solutions

Key to Exam 2

Final Exam - If you want to see the solution key - Please contact me in my office.  Happy holidays.

Practice for exams:

Solutions to Prac Exam 1

Practice for Exam 2

            Solutions to Prac for Exam 2