A Bridge to Advanced Mathematics

by Dennis Sentilles

Overview

This helpful workbook-style "bridge" book introduces students to the foundations of advanced mathematics, spanning the gap between a practically oriented calculus sequence and subsequent courses in algebra and analysis with a more theoretical slant.
Part 1 focuses on logic and number systems, providing the most basic tools, examples, and motivation for the manner, method, and concerns of higher mathematics. Part 2 covers sets, relations, functions, infinite sets, and mathematical proofs and reasoning.
Author Dennis Sentilles also discusses the history and development of mathematics as well as the reasons behind axiom systems and their uses. He assumes no prior knowledge of proofs or logic, and he takes an intuitive approach that builds into a formal development. Advanced undergraduate students of mathematics and engineering will find this volume an excellent source of instruction, reinforcement, and review.

Product Details

ISBN-13:	9780486482194
Publisher:	Dover Publications
Publication date:	08/18/2011
Series:	Dover Books on Mathematics
Edition description:	Unabridged
Pages:	416
Product dimensions:	6.40(w) x 9.20(h) x 1.00(d)

About the Author

Dennis Sentilles is Professor Emeritus in the Department of Mathematics at the University of Missouri.

Table of Contents

Preface to the Second (Dover) Edition iii

Preface v

Part 1 Starting Points 3

Chapter 1 Logic, Language and Mathematics 9

1 Introduction 9

2 Logic and Symbolic Logic 11

3 Statements, Propositions, Disjunction and Conjunction 13

4 The Negation of a Proposition 17

5 Logical Equivalence 20

6 The Kernel of Logical Thought: Implication 22

7 A Final Connective for Symbolic Logic: Equivalence 25

8 The Proof of Implications: Direct Method 28

9 The Proof of Implications: Indirect Methods 32

10 Tautology 36

11 Some Odds and Ends 39

12 Negation in the Mathematical Idiom 40

13 Quantifiers and Propositional Functions 44

14 Quantifiers That Aren't There and Other Sleights-of-Mind 54

Chapter 2 The Foundations of Mathematics 65

1 Introduction 65

2 Mathematics as a System of Thought 67

3 A Very Brief History of Mathematics 69

4 The Development of Non-Euclidean Geometry 75

5 The Axiomatic Method in Mathematics 89

6 The Natural Number System 98

7 The Real Number System 104

8 The Natural Numbers as a Part of (R 111

9 Infinity 117

10 Some Problems and Properties of Axiom Systems 135

Part 2 The Strategic Attack in Mathematics 147

Chapter 3 A Formally Informal Theory of Sets 155

1 Introduction 155

2 Fundamental Set Operations 161

3 Subsets 164

4 Set Theory: Second Floor 166

5 Cross Products of Sets 170

6 Operations with Arbitrarily Large Collections of Sets 173

7 The Axiom of Choice 180

Chapter 4 Topology and Connected Sets 185

1 Introduction 185

2 Basic Concepts: Open Sets and Closed Sets 193

3 The Closure of a Set 200

4 Topology Meets Set Theory 205

5 Connectedness in a Topological Space 209

6 The General Theory of Connected Sets 216

Chapter 5 Functions 227

1 Introduction 227

2 Relations and Functions 229

3 Idealizing the Function Concept 243

4 Functions Acting on Sets 248

5 Inverse and Composite Functions 257

Chapter 6 Counting the Infinite 267

1 Introduction: Counting the Finite 267

2 Extension: Counting the Infinite 272

3 Countably Infinite Sets and Uncountably Infinite Sets 275

4 Beyond the Countably Infinite: (R and the Answer to Question I 282

5 Cantor's Proof That (R Is Uncountable 291

6 The Schroder-Bernstein Theorem 294

7 The Continuum Hypothesis 296

Chapter 7 Equivalence Relations 299

1 Introduction 299

2 Equivalence Relations and Equivalence Classes 303

3 Cardinal Number 307

4 A Characterization of Open Sets in (R 309

5 A Factorization Theorem 313

Chapter 8 Continuity, Connectedness and Compactness 317

1 Introduction 317

2 Continuity, or, Distortion without Tearing 318

3 The Main Theorem: Answer to Question II 330

4 A Source of Application—Compactness 34

5 Euclidian n-Dimensional Space 351

6 Connectedness in En 359

7 Compactness in En 364

Appendix A 373

Index 384