MTH 307 - Spring 2008
Introduction to Mathematical Rigor

Meets MWF:  10:00-10:50

Instructor:  Dr. Nancy Eaton

E-mail me: mailto:eaton@math.uri.edu
Phone:  874-4439
Office:  Rm. 208, Tyler Hall
Office hours:  Mon, Tues, Fri: 2:00-2:30

Visit my web page: http://math.uri.edu/~eaton/

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

Course Description:

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.

• 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 n-tuples.
• Recognize and form partitions of sets and name the corresponding equivalence classes.
• Distinguish the different number systems, including natural numbers, integers, rational numbers, real numbers, and complex numbers and explain their respective constructs.
• 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.
• Explain the meaning of the equal sign in terms of equivalent expressions that can be substituted, one for the other in other equations.
• Identify sets as ordered or partially ordered.
• Recall the field and ordered field axioms and apply them in the justification of some basic identities and inequalities for real numbers.
• Express the summation and product of multiple terms, using sigma and pi notation and manipulate expressions that contain these symbols.
• Classify relations as functions or not and functions as injections, surjections, bijections, or not.
• Identify and employ the basic operations on functions, including addition, multiplication, and composition
• Identify and formulate the inverse of a function and interpret the meaning in terms of the inverse under composition of functions.
• Distinguish between the various uses of negative one in the exponent.
• Establish cardinality of sets, both finite and infinite.
• 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.

• Produce proofs of set equalities and inclusions using basic set theory principles.
• Translate between mathematical statements, written in ordinary English and using quantifiers, both universal and existential.
• Distinguish the types of mathematical statements, such as implications and if and only if statements.
• Use truth tables to establish equivalence between mathematical statements.
• Identify the hypothesis and the conclusion of an implication.
• Form the converse, contrapositive, and negation of statements containing implications and quantifiers.
• Form the negation of any mathematical statement and form counterexamples to false statements.
• Employ any of the proof methods: direct proof, induction, contradiction, and proving the contrapositive, to formulate a proof of an elementary result.
• Apply basic proof techniques, the Pigeonhole Principle, and combinatorial reasoning to prove some elementary counting theorems that are used in probability and combinatorics.
• 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.

• Set Theory (e.g. De Morgan’s Laws, cardinality)
• Geometry:  Axioms and theorems, Pythagorean Theorem
• Algebra (e.g. Algebraic identities, Quadratic Formula, Geometric Sum Identity, Binomial Theorem)
• Real Functions (e.g. Ceiling, floor, and greatest integer functions, bounded functions, monotone functions)
• Real Numbers (e.g. Ordered field axioms and theorems)
• Discrete Mathematics (e.g. Counting formulas and principles, graph theory)
• Number Theory (e.g. Divisibility, q-ary representations of numbers, Division Algorithm)
• Real Analysis (e.g. Sequences, series, and limits)

Texts:

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, by Bettina Richmond and Thomas Richmond, Thomson Brooks/Cole Pubs

Homework Assignments - 25%
Portfolio - 15%

Worksheets – 10%
Three 1-hour exams - 30%
Final - 20%

Expected work:

o       In order to achieve the most from this course you are expected to do the following:

o       Take notes in class.

o       Do the Suggested Exercises. These are not to be handed in.  Check answers in the back of the text and/or Solutions Manual.  Ask about those you do not understand.

o       Complete the Homework assignments that are to be handed in, including worksheets.

o       It is recommended that you ask questions during my office hours about the assignments BEFORE handing them in.

o       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.

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:

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.

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.

A) Write out the entire question.

B) Write neatly or type your assignment. Do not use torn paper.

C) The format should be single column.

D) Use only one side of the paper.

E) Clearly mark the section number and due date on the top of the front page.

F) Arrange the exercises in order by problem number. Attach pages together.

G) One point will be deducted from the assignment for each class day that it is late. In the case that it is late, you must put the date that you handed the assignment in, or no credit will be given.

I) Explain carefully, using proper mathematical logic.

J) Use complete sentences.

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 Monday April 28th. 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 Friday April 11th.

Portfolio Problems:

1.    Extra credit problem from Exam 2

2.    2.2 (2f)

3.    4.4 (14)

4.    6.1 (12)

5.    6.1 (6a)

6.    6.1 (19.  Do part b 2 ways.)

7.    6.2 (14)

8.    6.2 (5b,d)

9.    6.3 (3c)

10.                      6.3 (6)

Extra Credit:  8.3 (2), 8.3(4) Earn up to 2% out of 100% of your grade.

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.

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.

Exams: (Exams 30%, Final 20%)

There will be three 45-minute exams. The dates for these are Monday Feb 18th, Wed Mar 26th, and Monday Apr 21st. The final exam is will be given during the scheduled time on Monday May 12th from 11:30 – 2:30 PM.  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.

Dates to remember:

Exam 1:  Feb 18; Exam 2:  Mar 26; Exam 3:  Apr 21
Portfolio:  Apr 11 (problems announced), Apr 28 (portfolio due)

Final Exam:  May 12 (11:30 – 2:30 PM)

Syllabus:

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 Introduce Section/Topic Exams Wed 1/23 1.1 Mon 1/28 1.2, 1.3, 1.4 Mon 2/4 1.5, 1.6, Logarithms Mon 2/11 2.1, Fields Mon 2/18 2.2 Exam 1 2/18 Mon 2/25 2.3, 3.1 Mon 3/3 4.1, 4.2, 4.3 Mon 3/10 4.4, 5.1, 5.2 Mon 3/17 Spring Break Mon 3/24 5.3 Exam 2 - 3/26 Mon 3/31 6.1, 6.2 Mon 4/7 6.3, 7.1 Mon 4/14 8.1, 8.3 Mon 4/21 8.5 Exam 3 – 4/21 Mon 4/28 Game Mon 5/12 Final Exam

 Section/Topic Suggested Exercises Present In Class Hand In HK# Due 1.1 1, 2, 3, 4, 5, 8 9 - - - 1.2 2, 4, 5, 9, 10, 15 1, 6, 7 - - - 1.3 5, 9 3, 8, 11 6, 10 1 2/11 Set theory - - - 1/30 1.4 1, 3, 7, 10, 12, 14 2 8a, 9 1 2/11 1.5 1, 2, 4 3, 6 5 2 2/15 1.6 1, 3, 6, 9, 11 2, 7 10 2 2/15 Logarithms - - - 2/29 2.1 5, 8, 12, 27 2, 9a, 25 4, 7, 11, 14 3 3/5 FieldsWorksheet3 - - - - - - - - - - - - - 2, 3,  6, 8 (8pts) 5 4/2 2, 4, 5, 11 (8pts) 5 4/2 - 5, 8 (4pts) 5 4/2 2 6 4/9 3(a, b) 6 4/9 3 7(a,b) 6 4/9 5 7 4/16 10, 13 7 4/16 - - - - 3a, 6 8 4/23 8 4/23 - - - - - - -

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.

Take what you learned and write out the solution on your own, using your own words.

Solutions to selected homework problems and examples:

Practice for exams:

Practice for exam 1