# Mathematics

## MATH 260: Introduction to Abstract Mathematics

Spring 2006, 4 credits, MWF 9:00-9:50 MRC 348, R 10:00-10:50 MC 419

Instructor: Richard J. Maresh, Associate Professor of Mathematics, Department Chair

Office: MC 521, Hours: MWF 10-11, R 12-1, or by appt.

Email: rjmaresh@viterbo.edu, Office Phone: 796-3655 (Home Phone: 526-4988)

Final Exam: Tuesday, 9 May 2006, 12:50-2:50 pm

Course Description: Sentential and quantifier logic, axiomatic systems, and set theory. Emphasis is on the development of mathematical proofs. Pre-requisite: grade of C or higher in Math 180.

(Note: This course is a prerequisite for MATH 344, 420, and 450, all of which require the ability to write proofs.)

Text: A Gateway to Higher Mathematics, by Jason H. Goodfriend, Jones & Bartlett Publishers, 2006.

Course Goals

1.    The over-arching goal in this course is to learn to read and write mathematical proofs.

2.    A second major goal is to be able to communicate mathematical concepts and mathematical proofs, using appropriate mathematical notation and language.

3.   While this course is more about process than content, it is still a goal of this course to consider content which is important to further study of mathematics: logic, set theory, relations and functions, mathematical induction, the real number system.

Course Objectives: These are basically the topics we will be covering; throughout, the emphasis will be on reading, writing, and communicating proofs.

Proof:

1.   Students shall demonstrate a basic understanding of axiomatic-deductive systems.

2.   Students shall understand proofs and be able to judge the correctness of an argument.

3.   Students shall demonstrate the ability to reason inductively.

4.   Students shall demonstrate the ability to reason deductively.

Problem Solving:

5.   Students shall demonstrate the ability to apply appropriate mathematical tools and methods to novel or non-routine problems.

6.   Students shall demonstrate the ability to use various approaches to problem solving, and to see connections between these varied mathematical areas.

Communication

7.   Students shall use the language of mathematics accurately and appropriately.

8.   Students shall present mathematical content and argument orally.

9.   Students shall present mathematical content and argument in written form.

Course Procedures:

This course is a very significant step for every mathematics major, and probably more than any other single course, certainly up to this point in your course work, lets you see what it is like to be a mathematician and whether you can find fulfillment as, or have the ability to be, a mathematician. I have vivid memories of my own equivalent course as a student!

All mathematics educators believe in the adage, “Mathematics is not a spectator sport!” In this course, more than in most, however, your contributions will MAKE the course. I intend to allow a healthy fraction of our class time to consist of student presentations of attempted proofs. In addition to learning to read or listen to proofs, with comprehension, as well as learning to problem solve and write proofs, it is important to acquire a disinterested objectivity - like a scientist, we must strive to view a proof in its own terms and not take corrections personally. The “craft” of mathematics is in one sense very personal but at the same time your work must be held to the highest standard of objectivity.

I have chosen this textbook largely for its philosophy; it is designed to be used in a student-oriented class. I will make assignments virtually on a daily basis and it is very important that you work on these as best you can. I hope that at least once a week, on average, each of will get a chance to present some of your work. It is important that we not miss classes; this will be almost a “seminar” course - the size of the group and the philosophy will allow us to do that.

Because it is important that we learn to WRITE proofs which are correct and clear, as well as to present them orally, I will collect many of the daily assignments and grade them. There will also be three in class, written, exams: Feb 20, Apr12, and the final exam on May 9; I do believe in seeing what students can do on their own and “under pressure”. At the same time, there will be frequent take-home assignments, both daily homework and other problem sets.

Grading: I will use the traditional 90-80-70-60 scale as a framework for assigning grades. I do realize the artificiality of such a scale - but I usually try to make the assignments reasonable enough so that students can earn an appropriate grade. It is harder in this course in particular to make things seem “objective”, since writing proofs is something like writing essays in an English course - there is “correct” and “complete”, but there is also “elegant” and “insightful”. I will try to keep you informed along the way about your progress in the course.

Portfolios: Because this course is all about developing skills, albeit subtle and substantial skills, it is especially important to monitor your development. In fact, we are trying to compile a portfolio of the work of each mathematics major as they work their way through the courses in the major. I would like to ask you to turn in a collection of your best efforts during the course, probably five problems, which can be added to your portfolio. These portfolios should not only give you a chance to pay attention to your progress, but they also will help give the department a picture of how well our students are learning what we want them to learn.

Americans with Disability Act: If you are a person with a disability and require any auxiliary aids, services or other accommodations for this class, please see me or Wayne Wojciechowski (MC 320, 796-3085) within ten days to discuss your accommodation needs.

The Schedule: For many courses I teach, I construct a fairly detailed schedule, with assignments and material to be covered each day throughout the semester. For several reasons, however, in this course we will be playing it a bit more by ear. It is difficult for me to know just how much text material we can comfortably cover (I do hope to at least cover at least 5 or 6 chapters). The emphasis I want to give to looking at your work also requires a certain flexibility in schedule - and it is difficult to say just what effect this will have. I do think it is much more important that you LEARN the material than that we “cover” it, “depth of understanding” rather than “breadth”.

As a general framework, nonetheless, I offer the following:

1.   Logic and the Techniques for Proofs: p 1 – 20                       (2 weeks)

1.1   Statements and Truth Tables

1.2   Logical Equivalence

1.3   Quantifiers

1.4   Some Proof Techniques

2.   Elementary Set Theory: p 21 – 69                                              (3 weeks)

2.1   Introduction to Sets

2.2   Some Operations on Sets

2.3   Relations

2.4   Some Important Types of Relations on Sets

2.5   Functions

2.6   Mathematical Induction

2.7   The Axiom of Infinity

20 Feb 2006: EXAM #1 (80 Points)

3.   The Development of the Integers: p 71 – 137                          (4 weeks)

3.1   Peano’s Postulates

3.2   The Recursion Theorem, Addition and Multiplication

3.3   Order on the Positive Integers

3.4   The Uniqueness of Peano Spaces and Counting

3.5   Introduction to Rings

3.6   Postulates for the Integers and Initial Properties

3.7   Proof of the Existence of the Integers

4.   Properties and Applications of the Integers: p 139 – 195      (3 weeks)

4.1   Sequences, Cartesian Products, and the Generalized Recursion Theorem

4.2   Applications to Binary Operations

4.3   Introduction to Number Theory

4.4   Finite and Infinite Sets

4.5   Infinite Sets: Countable and Uncountable

12 Apr 2006: EXAM #2  (80 Points)

5.   Fields and Rational Numbers: p  197 – 221                              (2 weeks)

5.1   Introduction to Fields

5.2   The Rational Numbers and the Field of Quotients

6.   The Development of the Real Numbers: p 223 – 258              (2 week)

6.1   Postulates for the Real Number System and Some Initial Properties

6.2   The Uniqueness of the Real Number System

6.3   Construction of a Set Obeying the Real Number Postulates

6.4   Initial Properties of Real Numbers

FINAL EXAM:  Tuesday, 9 May 2006, 12:50-2:50 pm