MTH 307 - Fall 2004

Introduction to Mathematical Rigor

Sec. 1 Meets MWF:  10:00-10:50

Sec. 2 Meets MWF:  11:00-11:50

Instructor:  Dr. Nancy Eaton

E-mail me:  eaton@math.uri.edu
Phone:  874-4439
Office:  Rm. 222, Tyler Hall
Office hours:  Monday 12:30-2 and

Tuesday 11:30-1

Visit my web page:  WEB PAGE

Updated: December 14, 2004

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

 Course Content Text Grade for course Working with Others Dates to Remember Solutions to selected exercises Portfolio assignment Solutions Sec 1 Solutions Sec 2 Other Web Resources Rubric used for homework problems What is Math – Your feedback

Course Content:

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

• Hypothesis
• Conclusion
• Implications
• Converse
• Contrapositive
• Negation
• If and only if statements
• Truth tables
• Equivalence of statements
• Venn diagrams
• Induction
• Strong induction
• Method of descent
• Direct proof
• Proving the contrapositive
• Pigeonhole principle
• Combinatorial Reasoning
• Method of counting elements of a set in two different ways
• Basic method to prove equality of sets
• Basic method to prove that a given function is an injection
• Basic method to prove that a given function is a surjection

Basic Structures of Mathematics:

• Permutations
• Arrangements
• Selections
• Partitions
• Field axioms
• Ordered field axioms
• Real numbers, R
• Natural numbers, N
• Integers, Z
• Rational numbers, Q
• Well-Ordering Property
• Multiple, divides, prime, gcd, lcm
• Sets
• Sequences and series
• Cartesian product
• n-tuples, ordered pairs
• q-ary representations
• Unions and intersections of sets
• Functions – injections, surjections, bijections
• Inverses
• Composition of functions
• Bounded functions, monotone functions
• Ceiling, floor, and greatest integer functions
• Cardinality
• Finite sets
• Infinite sets – countable, uncountable
• Combinatorial graphs

Some Theorems:

• Pthagorean Theorem
• AGM Inequality
• Geometric Sum
• Binomial Theorem
• The Division Algorithm

Text:  A Discrete Transition to Advanced Mathematics by Bettina Richmond and Thomas Richmond, Thomson Brooks/Cole Pubs.

Homework - 25%
Portfolio - 25%
Three 1-hour exams - 25%
Final - 25%

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:

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≤ xy and 0 ≤ uv imply that x۰uy۰v.

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.

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 three 45-minute 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 8:00 - 11:00 AM or Thursday Dec 23 at 8:00 – 11:00 AM.  The exams will be taken from material that we covered in class or in homework assignments, so take careful notes.  Study these for the exams.

Dates to remember:

Exam 1:  Oct 6;  Exam 2:  Nov 3;  Exam 3:  Dec 1
Portfolio:
Dec 6
Final Exam:  Dec 18 or Dec 23

Homework assignments:

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(a-d), 8 1.2 1, 2, 4-8, 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(2-6) Sept 29 3 Exercises 2-9 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 1-3, 6, 9-11 6(a, b), 9(b) Oct 25 6 2.1 2, 4-8, 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 1-4, 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:

Keys to exams:

Key to Exam 1

Key to Exam 2

Key to Exam 3

Practice for exams:

Review for Exam 1

Review for Exam 3

Some Web Sites:

What is Math?

Throughout the semester, submit answers to these questions via e-mail.  I will put them up on our web-page 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 hypothisis-1. 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 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

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 thinkable-that is the mathematician’s aim.”

-C.J. Keyser