Fundamentals of Mathematics: An Introduction to Proofs, Logic, Sets, and Numbers / Edition 1

by Bernd S. W. Schröder

Overview

An accessible introduction to abstract mathematics with an emphasis on proof writing

Addressing the importance of constructing and understanding mathematical proofs, Fundamentals of Mathematics: An Introduction to Proofs, Logic, Sets, and Numbers introduces key concepts from logic and set theory as well as the fundamental definitions of algebra to prepare readers for further study in the field of mathematics. The author supplies a seamless, hands-on presentation of number systems, utilizing key elements of logic and set theory and encouraging readers to abide by the fundamental rule that you are not allowed to use any results that you have not proved yet.

The book begins with a focus on the elements of logic used in everyday mathematical language, exposing readers to standard proof methods and Russell's Paradox. Once this foundation is established, subsequent chapters explore more rigorous mathematical exposition that outlines the requisite elements of Zermelo-Fraenkel set theory and constructs the natural numbers and integers as well as rational, real, and complex numbers in a rigorous, yet accessible manner. Abstraction is introduced as a tool, and special focus is dedicated to concrete, accessible applications, such as public key encryption, that are made possible by abstract ideas. The book concludes with a self-contained proof of Abel's Theorem and an investigation of deeper set theory by introducing the Axiom of Choice, ordinal numbers, and cardinal numbers.

Throughout each chapter, proofs are written in much detail with explicit indications that emphasize the main ideas and techniques of proof writing. Exercises at varied levels of mathematical development allow readers to test their understanding of the material, and a related Web site features video presentations for each topic, which can be used along with the book or independently for self-study.

Classroom-tested to ensure a fluid and accessible presentation, Fundamentals of Mathematics is an excellent book for mathematics courses on proofs, logic, and set theory at the upper-undergraduate level as well as a supplement for transition courses that prepare students for the rigorous mathematical reasoning of advanced calculus, real analysis, and modern algebra. The book is also a suitable reference for professionals in all areas of mathematics education who are interested in mathematical proofs and the foundation upon which all mathematics is built.

Product Details

ISBN-13:	9780470551387
Publisher:	Wiley
Publication date:	08/16/2010
Pages:	348
Sales rank:	317,866
Product dimensions:	6.30(w) x 9.40(h) x 0.90(d)

About the Author

BERND S.W. SCHRÖDER, PhD, is Edmundson/Crump Professor, Academic Director, and Program Chair of the Program of Mathematics and Statistics at Louisiana Tech University. He has authored more than thirty journal articles in his areas of research interest, which include ordered sets, probability theory, graph theory, harmonic analysis, computer science, and education. Dr. Schröder is the author of Mathematical Analysis: A Concise Introduction and A Workbook for Differential Equations, both published by Wiley.

Table of Contents

Preface vii

Questions x

1 Logic 1

1.1 Statements 1

1.2 Implications 5

1.3 Conjunction, Disjunction, and Negation 11

1.4 Special Focus on Negation 19

1.5 Variables and Quantifiers 25

1.6 Proofs 30

1.7 Using Tautologies to Analyze Arguments 42

1.8 Russell's Paradox 46

2 Set Theory 51

2.1 Sets and Objects 52

2.2 The Axiom of Specification 56

2.3 The Axiom of Extension 59

2.4 The Axiom of Unions 67

2.5 The Axiom of Powers, Relations, and Functions 73

2.6 The Axiom of Infinity and the Natural Numbers 83

3 Number Systems I: Natural Numbers 89

3.1 Arithmetic With Natural Numbers 89

3.2 Ordering the Natural Numbers 98

3.3 A More Abstract Viewpoint: Binary Operations 103

3.4 Induction 111

3.5 Sums and Products 120

3.6 Divisibility 133

3.7 Equivalence Relations 142

3.8 Arithmetic Modulo m 147

3.9 Public Key Encryption 153

4 Number Systems II: Integers 161

4.1 Arithmetic With Integers 161

4.2 Groups and Rings 167

4.3 Finding the Natural Numbers in the Integers 175

4.4 Ordered Rings 179

4.5 Division in Rings 185

4.6 Countable Sets 195

5 Number Systems III: Fields 201

5.1 Arithmetic With Rational Numbers 201

5.2 Fields 205

5.3 Ordered Fields 211

5.4 A Problem With the Rational Numbers 213

5.5 The Real Numbers 216

5.6 Uncountable Sets 226

5.7 The Complex Numbers 230

5.8 Solving Polynomial Equations 233

5.9 Beyond Fields: Vector Spaces and Algebras 243

6 Unsolvability of the Quintic by Radicals 249

6.1 Irreducible Polynomials 250

6.2 Field Extensions and Splitting Fields 255

6.3 Uniqueness of the Splitting Field 260

6.4 Field Automorphisms and Galois Groups 269

6.5 Normal Field Extensions 273

6.6 The Groups S_n 276

6.7 The Fundamental Theorem of Galois Theory and Normal Subgroups 281

6.8 Consequences of Solvability by Radicals 292

6.9 Abel's Theorem 298

7 More Axioms 301

7.1 The Axiom of Choice, Zorn's Lemma, and the Well-Ordering Theorem 301

7.2 Ordinal Numbers and the Axiom of Replacement 308

7.3 Cardinal Numbers and the Continuum Hypothesis 311

A Historical Overview and Commentary 317

A.1 Ancient Times: Greece and Rome 318

A.2 The Dark Ages and First New Developments 321

A.3 There is No Quintic Formula: Abel and Galois 323

A.4 Understanding Irrational Numbers: Set Theory 326

Conclusion and Outlook 328

Bibliography 329

Index 333

Editorial Reviews

"This is a lovely introduction to mathematics. The book gives an elegant construction of the familiar number systems while at the same time introducing the student to mathematical logic, set theory and rigorous mathematical proof." (Zentralblatt MATH, 2011)

From the Publisher