Department of Mathematics

Syllabus**1701.500
- Foundations of Mathematics**

**CATALOG
DESCRIPTION:**

1701.500
Foundations of Mathematics
3 s.h..

Strategies
and tools for problem solving, including computer use, will be applied
to specific problems from number theory, geometry, analytic geometry, algebra,
discrete mathematics, logic, and calculus.

**OBJECTIVES:**

The
course is intended to be a first graduate course in mathematics for students
in all of these programs.As such,
it will provide a common mathematical foundation for students in all of
the programs, drawing upon the full range of undergraduate courses in mathematics.Mathematical
connection will be emphasized in the course, allowing students to relate
topics studied separately to one another.Also
stressed will be mathematical reasoning and communication skills, as applied
to mathematics.This course will
permit students to build upon and share knowledge already acquired while
pointing out areas in which additional study may be needed.In
addition, it will develop the communication skills and understanding of
the process of doing mathematics necessary for graduate-level study.

**CONTENT:**

**1.Logic**

1.1Introduction

1.2And,
or, not, and Truth Tables

1.3Implication
and the Biconditional

1.4Tautologies

1.5Arguments
and the Principle of Demonstration

1.6Quantifiers

1.7More
Quantifiers

1.8Methods
of Proof

**2.Sets,
Relations, and Functions**

2.1Sets

2.2Truth
Sets

2.3Relations

2.4More
Relations

2.5Equivalence
Relations and Partitions

2.6Functions

2.7More
Functions

**3.Mathematical
Induction**

3.1Introduction

3.2The
Principle of Mathematical Induction

3.3 Equivalent Forms of the Principle of Mathematical
Induction

**4.Continuity**

4.1Introduction

4.2The
Real Number System

4.3Sequences

4.4Continuous
Functions

**5.Groups**

5.1Introduction

5.2Groups

5.3Groups
and Functions

**6.Cardinality**

6.1Introduction

6.2Cardinal
Numbers

6.3Infinite
Sets

6.4Infinite
Cardinal Numbers

**TEXTS:**

The
following are texts suitable for this course:

Dunham,
William, JOURNEY THROUGH GENIUS, John Wiley & Sons, Inc., NYC,1990.

Eves,
Howard, FOUNDATIONS AND FUNDAMENTAL CONCEPTS OF MATHEMATICS,
PWS-Kent Publishing Co., Boston, MA, 1991.

Kurtz,
David, FOUNDATIONS OF ABSTRACT MATHEMATICS, McGraw Hill Publishing
Co., Hightstown, NJ, 1992.

Morash,
Ronald, BRIDGE TO ABSTRACT MATHEMATICS, McGraw Hill Publishing
Co., Hightstown, NJ, 1991.