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Transition to Higher Mathematics : Structure and Proof

Transition to Higher Mathematics : Structure and Proof - 07 edition

ISBN13: 978-0073533537

Cover of Transition to Higher Mathematics : Structure and Proof 07 (ISBN 978-0073533537)
ISBN13: 978-0073533537
ISBN10: 007353353X
Cover type: Hardback
Edition/Copyright: 07
Publisher: McGraw-Hill Publishing Company
Published: 2007
International: No

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Transition to Higher Mathematics : Structure and Proof - 07 edition

ISBN13: 978-0073533537

Bob A. Dumas and John E. McCarthy

ISBN13: 978-0073533537
ISBN10: 007353353X
Cover type: Hardback
Edition/Copyright: 07
Publisher: McGraw-Hill Publishing Company

Published: 2007
International: No
Summary
  • Writing Style - Dumas/McCarthy presents the material in a friendly, conversational writing style designed to appeal to students, while maintaining the appropriate level of formality and rigor required for proper mastery of proof technique. The authors motivate understanding of formal definitions with discussions and remarks designed to help students grasp the basic direction of the arguments.
  • Attention to Detail - The authors take utmost care to present correct mathematical detail, structure, notation, and terminology at all times. Informal discussion and hints are clearly distinguished from formal definitions and proof with a ''Discussion'' label.
  • Exercises - Each chapter features numerous exercises of varying difficulty designed to direct students' attention to the reading and compel them to think through the details of the proofs.
  • Flexible Topic Coverage - Dumas/McCarthy covers a wide and well-developed range of topics, and the chapters are self-contained enough to allow instructors to easily pick and choose the topics they wish to cover. The material can be structured to accommodate the time constraints of a quarter course as well as a semester-long course.

Table of Contents

Chapter 0. Introduction

0.1. Why this book is
0.2. What this book is
0.3. What this book is not
0.4. Advice to the Student
0.5. Advice to the Instructor
0.6. Acknowledgements

Chapter 1. Preliminaries

1.1. ''And'' ''Or''
1.2. Sets
1.3. Functions
1.4. Injections, Surjections, Bijections
1.5. Images and Inverses
1.6. Sequences
1.7. Russell's Paradox
1.8. Exercises
1.9. Hints to Get Started on Some Exercises

Chapter 2. Relations

2.1. Definitions
2.2. Orderings
2.3. Equivalence Relations
2.4. Constructing Bijections
2.5. Modular Arithmetic
2.6. Exercises

Chapter 3. Proofs

3.1. Mathematics and Proofs
3.2. Propositional Logic
3.3. Formulas
3.4. Quantifiers
3.5. Proof Strategies
3.6. Exercises

Chapter 4. Principle of Induction

4.1. Well-Orderings
4.2. Principle of Induction
4.3. Polynomials
4.4. Arithmetic-Geometric Inequality
4.5. Exercises

Chapter 5. Limits

5.1. Limits
5.2. Continuity
5.3. Sequences of Functions
5.4. Exercises

Chapter 6. Cardinality

6.1. Cardinality
6.2. Infinite Sets
6.3. Uncountable Sets
6.4. Countable Sets
6.5. Functions and Computability
6.6. Exercises

Chapter 7. Divisibility

7.1. Fundamental Theorem of Arithmetic
7.2. The Division Algorithm
7.3. Euclidean Algorithm
7.4. Fermat's Little Theorem
7.5. Divisibility and Polynomials
7.6. Exercises

Chapter 8. The Real Numbers

8.1. The Natural Numbers
8.2. The Integers
8.3. The Rational Numbers
8.4. The Real Numbers
8.5. The Least Upper Bound Principle
8.6. Real Sequences
8.7. Ratio Test
8.8. Real Functions
8.9. Cardinality of the Real Numbers
8.10. Order-Completeness
8.11. Exercises

Chapter 9. Complex Numbers

9.1. Cubics
9.2. Complex Numbers
9.3. Tartaglia-Cardano Revisited
9.4. Fundamental Theorem of Algebra
9.5. Application to Real Polynomials
9.6. Further Remarks
9.7. Exercises
Appendix A. The Greek Alphabet
Appendix B. Axioms of Zermelo-Fraenkel with the Axiom of Choice
Bibliography
Index