This book, which is based on Pólya's method of problem solving, aids students in their transition from calculus (or precalculus) to higher-level mathematics. The book begins by providing a great deal of guidance on how to approach definitions, examples, and theorems in mathematics. It ends by providing projects for independent study.

Students will follow Pólya's four step process: learn to understand the problem; devise a plan to solve the problem; carry out that plan; and look back and check what the results told them. Special emphasis is placed on reading carefully and writing well. The authors have included a wide variety of examples, exercises with solutions, problems, and over 40 illustrations, chosen to emphasize these goals. Historical connections are made throughout the text, and students are encouraged to use the rather extensive bibliography to begin making connections of their own. While standard texts in this area prepare students for future courses in algebra, this book also includes chapters on sequences, convergence, and metric spaces for those wanting to bridge the gap between the standard course in calculus and one in analysis.

Inhalt

Preface 1 The How, When, and Why of Mathematics Spotlight: George Polya Tips on Doing Homework 2 Logically Speaking 3 Introducing the Contrapositive and Converse 4 Set Notation and Quantifiers Tips on Quantification 5 Proof Techniques Tips on Definitions 6 Sets Spotlight: Paradoxes 7 Operations on Sets 8 More on Operations on Sets 9 The Power Set and the Cartesian Product Tips on Writing Mathematics 10 Relations Tips on Reading Mathematics 11 Partitions Tips on Putting It All Together 12 Order in the Reals Tips: You Solved it. Now What? 13 Functions, Domain, and Range Spotlight: The Definition of Function 14 Functions, One-to-one, and Onto 15 Inverses 16 Images and Inverse Images Spotlight: Minimum or Infimum 17 Mathematical Induction 18 Sequences 19 Convergence of Sequences of Real Numbers 20 Equivalent Sets 21 Finite Sets and an Infinite Set 22 Countable and Uncountable Sets 23 Metric Spaces 24 Getting to Know Open and Closed Sets 25 Modular Arithmetic 26 Fermat's Little Theorem Spotlight: Public and Secret Research 27 Projects Tips on Talking about Mathematics 27.1 Picture Proofs 27.2 The Best Number of All 27.3 Set Constructions 27.4 Rational and Irrational Numbers 27.5 Irrationality of $e$ and $\pi $ 27.6 When does $f^{-1} = 1/f$? 27.7 Pascal's Triangle 27.8 The Cantor Set 27.9 The Cauchy-Bunyakovsky-Schwarz Inequality 27.10 Algebraic Numbers 27.11 The RSA Code Spotlight: Hilbert's Seventh Problem 28 Appendix 28.1 Algebraic Properties of $@mathbb {R}$ 28.2 Order Properties of $@mathbb {R}$ 28.3 Polya's List References Index