Transition to Higher Mathematics: Structure and Proof - Rilegato

Dumas, Bob; Mccarthy, John

Rilegato

ISBN 10: 007353353X ISBN 13: 9780073533537

Casa editrice: McGraw-Hill Education, 2006

Riassunto

This text is intended for the Foundations of Higher Math bridge course taken by prospective math majors following completion of the mainstream Calculus sequence, and is designed to help students develop the abstract mathematical thinking skills necessary for success in later upper-level majors math courses. As lower-level courses such as Calculus rely more exclusively on computational problems to service students in the sciences and engineering, math majors increasingly need clearer guidance and more rigorous practice in proof technique to adequately prepare themselves for the advanced math curriculum. With their friendly writing style Bob Dumas and John McCarthy teach students how to organize and structure their mathematical thoughts, how to read and manipulate abstract definitions, and how to prove or refute proofs by effectively evaluating them. Its wealth of exercises give students the practice they need, and its rich array of topics give instructors the flexibility they desire to cater coverage to the needs of their school’s majors curriculum.

This text is part of the Walter Rudin Student Series in Advanced Mathematics.

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Contenuti

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

Dettagli dell'edizione

• Editore: McGraw-Hill Education
• Data di pubblicazione: 2006
• Lingua: Inglese
• ISBN 10: 007353353X
• ISBN 13: 9780073533537
• Rilegatura: Copertina rigida
• Numero di pagine: 296
• Contatto del produttore: non disponibile
• Persona responsabile: Ufficio Anagrafiche Messaggerie; 0039 0245774342; ufficio.anagrafiche@meli.it; ; Via Verdi, 8.; Assago Milano; 20090; Italia