Transition to Advanced Mathematics: Structure and Proof
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Transition to Advanced Mathematics: Structure and Proof

Transition to Advanced Mathematics: Structure and Proof

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About the Book

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.

Table of Contents:
Chapter 0. Introduction0.1. Why this book is0.2. What this book is0.3. What this book is not0.4. Advice to the Student0.5. Advice to the TeacherChapter 1. Preliminaries1.1. “And” “Or” “iff”1.2. Sets1.3. Functions1.4. Sequences1.5. Russell’s Paradox1.6. ExercisesChapter 2. Relations2.1. Orderings2.2. Equivalence Relations2.3. Modular Arithmetic2.4. ExercisesChapter 3. Proofs3.1. Mathematics and Proofs3.2. Propositional Logic3.3. Formulas3.4. Quantifiers3.5. ExercisesChapter 4. Principle of Induction4.1. Well-orderings4.2. Principle of Induction4.3. Polynomials4.4. Arithmetic-Geometric Inequality4.5. ExercisesChapter 5. Limits5.1. Limits5.2. Continuity5.3. Sequences of functions5.4. ExercisesChapter 6. Cardinality6.1. Cardinality6.2. Infinite Sets6.3. Uncountable Sets6.4. Countable Sets6.5. Functions and Computability6.6. ExercisesChapter 7. Divisibility7.1. Fundamental Theorem of Arithmetic7.2. The Division Algorithm7.3. Euclidean Algorithm7.4. Fermat’s Little Theorem7.5. Divisibility and Polynomials7.6. ExercisesChapter 8. The Real Numbers8.1. The Natural Numbers8.2. The Integers8.3. The Rational Numbers8.4. The Real Numbers8.5. The Least Upper Bound Principle8.6. Real Sequences8.7. Real Functions8.8. Cardinality of the Real Numbers8.9. ExercisesChapter 9. Complex Numbers9.1. Cubics9.2. Complex Numbers9.3. Tartaglia-Cardano Revisited9.4. Fundamental Theorem of Algebra9.5. Application to real polynomials9.6. Further remarks9.7. ExercisesAppendix A. The Greek AlphabetBibliographyIndex


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Product Details
  • ISBN-13: 9780071106474
  • Publisher: McGraw-Hill Education - Europe
  • Publisher Imprint: McGraw-Hill Professional
  • Height: 234 mm
  • Returnable: N
  • Weight: 390 gr
  • ISBN-10: 0071106472
  • Publisher Date: 16 May 2006
  • Binding: Paperback
  • Language: English
  • Spine Width: 1 mm
  • Width: 165 mm


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Transition to Advanced Mathematics: Structure and Proof
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