Set Theory With Applications

✍ Scribed by You-Feng Lin, Shwu Yeng T. Lin

• Publisher: Mariner Publishing Company
• Year: 1981
• Tongue: English
• Leaves: 229
• Edition: 2nd
• Category: Library

✦ Synopsis

Preface from second edition
It has been over six years since the appearance of Set Theory: An Intuitive Approach, the predecessor of this book. During these years much has happened in mathematics curricula, and the demand for new math has widened. Here at the University of South Florida, the classes of set theory, traditionally attended by mathematics and mathematics education majors, now draw half of their students from the College of Engineering, especially electrical engineering and computer science majors. Because of this, we decided to include a new chapter on Boolean algebra with applications to logic circuits and digital circuit designs. This new chapter offers practical applications of the early chapters on logic and sets. Since no engineering background is required, we hope that the nonengineering reader too will enjoy seeing the applications of a subject that is as pure as set theory in our modern technology.
Presenting set theory within the framework of an axiomatic system is, without doubt, more satisfactory when one intends to be rigorous. But there is no axiomatization of set theory that is simple and easy enough for the beginning student. Therefore, we have utilized, cautiously we hope, an intuitive approach to the subject.
In addition, we have endeavored to make the book self-contained. We have included the necessary basic material for the student who wishes to go on to such courses as topology, analysis, modern algebra, etc. But it is our hope that the material in this book will prove interesting to the student and teacher in its own right. It would be unfortunate to discover that a subject as interesting as set theory be regarded as useful only as a prerequisite course.
The prerequisites for the study of this book include some contact with college mathematics but not necessarily great mathematical skill. Ideally, the reader will have taken one quarter or one semester of calculus. The book contains enough material for a one-semester or two-quarter course in introductory set theory for the undergraduate. It can also be used in a one-quarter or short summer course by omitting the last few chapters. We suggest that the instructor go faster in the beginning two chapters and proceed more slowly later in the course.
Various minor errors and obscurities in the earlier edition have been corrected, and exercise problems have been expanded to cover a wider range of difficulty and to provide greater variety.
Our hearty thanks go to a number of colleagues who took the trouble to write kind words, point out errors, and suggest revisions. These letters were very encouraging and very helpful in the preparation of the revised edition.
We would like to thank Luke Lin for reading the chapter on Boolean algebra with applications and making many valuable suggestions and Winston Lin for proofreading parts of the manuscript.
Finally, we are grateful to the staff of Mariner Publishing Company for their assistance and encouragement.
Temple Terrace, Florida Shwu- Y eng T. Lin
You-Feng Lin

✦ Table of Contents

Chapter One / Elementary Logic
1. Statements and Their Connectives
2. Three More Connectives
3. Tautology, Implication, and Equivalence
4. Contradiction
5. Deductive Reasoning
6. Quantification Rules
7. Proof of Validity
8. Mathematical Induction

Chapter Two / The Concept of Sets
1. Sets and Subsets
2. Specification of Sets
3. Unions and Intersections
4. Complements
5. Venn Diagrams
6. Indexed Families of Sets
7. The Russell Paradox
8. Historical Remark

Chapter Three / Relations and Functions
1. Cartesian Product of Two Sets
2. Relations
3. Partitions and Equivalence Relations
4. Functions
5. Images and Inverse Images of Sets
6. Injective, Surjective, and Bijective Functions
7. Composition of Functions

Chapter Four / Boolean Algebra with Applications
1. Boolean Algebra
2. Boolean Functions
3. Boolean Functions and Logic Gates
4. Applications to Digital Computer Circuits

Chapter Five / Denumerable Sets and Nondenumerable Sets
1. Finite and Infinite Sets
2. Equipotence of Sets
3. Examples and Properties of Denumerable Sets
4. Nondenumerable Sets

Chapter Six / Cardinal Numbers and Cardinal Arithmetic
1. The Concept of Cardinal Numbers
2. Ordering of the Cardinal Numbers-The SchrOder-Bernstein Theorem
3. Cardinal Number of a Power Set-Cantor's Theorem
4. Addition of Cardinal Numbers
5. Multiplication of Cardinal Numbers
6. Exponentiation of Cardinal Numbers
7. Further Examples of Cardinal Arithmetic
8. The Continuum Hypothesis and Its Generalization

Chapter Seven / The Axiom of Choice and Some of Its Equivalent Forms
1. Introduction
2. The Hausdorff Maximality Principle
3. Zorn's Lemma
4. The Well-Ordering Principle

✦ Subjects

Set Theory Foundation of Mathematics