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by Bob A. Dumas and John E. McCarthy

Cover type: HardbackEdition: 07

Copyright: 2007

Publisher: McGraw-Hill Publishing Company

Published: 2007

International: No

List price: $197.75

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- 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.

**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

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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

Publisher Info

Publisher: McGraw-Hill Publishing Company

Published: 2007

International: No

Published: 2007

International: No

- 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.

**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