Introductory Concepts for Abstract Mathematics / Edition 1

by Kenneth E. Hummel

Overview

Beyond calculus, the world of mathematics grows increasingly abstract and places new and challenging demands on those venturing into that realm. As the focus of calculus instruction has become increasingly computational, it leaves many students ill prepared for more advanced work that requires the ability to understand and construct proofs.

Introductory Concepts for Abstract Mathematics helps readers bridge that gap. It teaches them to work with abstract ideas and develop a facility with definitions, theorems, and proofs. They learn logical principles, and to justify arguments not by what'seems right, but by strict adherence to principles of logic and proven mathematical assertions - and they learn to write clearly in the language of mathematics

The author achieves these goals through a methodical treatment of set theory, relations and functions, and number systems, from the natural to the real. He introduces topics not usually addressed at this level, including the remarkable concepts of infinite sets and transfinite cardinal numbers

Introductory Concepts for Abstract Mathematics takes readers into the world beyond calculus and ensures their voyage to that world is successful. It imparts a feeling for the beauty of mathematics and its internal harmony, and inspires an eagerness and increased enthusiasm for moving forward in the study of mathematics.

Product Details

ISBN-13:	9780367398835
Publisher:	Taylor & Francis
Publication date:	09/05/2019
Pages:	344
Product dimensions:	6.12(w) x 9.19(h) x (d)

About the Author

Hummel, Kenneth E.

Table of Contents

Preface vii

Section I Logic and Proof 1

Chapter 1 Logic and Propositional Calculus 3

Chapter 2 Tautologies and Validity 15

Chapter 3 Quantifiers and Predicates 29

Chapter 4 Techniques of Derivation and Rules of Inference 41

Chapter 5 Informal Proof and Theorem-Proving Techniques 53

Chapter 6 On Theorem Proving and Writing Proofs 69

Chapter 7 Mathematical Induction 79

Section II Sets 91

Chapter 8 Sets and Set Operations 93

Chapter 9 Set Union, Intersection, and Complement 107

Chapter 10 Generalized Union and Intersection 123

Section III Functions and Relations 133

Chapter 11 Cartesian Products 135

Chapter 12 Relations 141

Chapter 13 Partitions 151

Chapter 14 Functions 161

Chapter 15 Composition of Functions 173

Chapter 16 Image and Preimage Functions 179

Section IV Algebraic and Order Properties of Number Systems 187

Chapter 17 Binary Operations 189

Chapter 18 The Systems of Whole and Natural Numbers 195

Chapter 19 The System Z of Integers 201

Chapter 20 The System Q of Rational Numbers 215

Chapter 21 Other Aspects of Order 223

Chapter 22 The Real Number System 227

Section V Transfinite Cardinal Numbers 235

Chapter 23 Finite and Infinite Sets 237

Chapter 24 Denumerable and Countable Sets 243

Chapter 25 Uncountable Sets 253

Chapter 26 Transfinite Cardinal Numbers 259

Section VI Axiom of Choice and Ordinal Numbers 273

Chapter 27 Partially Ordered Sets 275

Chapter 28 Least Upper Bound and Greatest Lower Bound 285

Chapter 29 Axiom of Choice 289

Chapter 30 Well Ordered Sets 299

Reading List 305

Hints and Solutions to Selected Exercises 307

Index 327

Editorial Reviews

As an offshoot of an undergraduate foundations course the author has taught for years at Trinity College, this text aims to bridge the gap between basic calculus studies and abstract algebra and real analysis. Seven sections cover logic and proof, naive set theory, functions and relations, algebraic and order properties of number systems, transfinite cardinal numbers, and axiom of choice and ordinal numbers. Parenthetical comments are interspersed to help students connect concepts. Concludes with a reading list, and hints/ solutions to selected exercises. Annotation c. Book News, Inc., Portland, OR (booknews.com)

Booknews

"... very clearly written. Sophomore-level undergraduates should have no difficulty with the book."
-Zentralblatt fur Mathematik

From the Publisher