Reading, Writing, and Proving

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