内容

目次

Preface 1 The How, When, and Why of Mathematics Spotlight: GeorgePolya Tips on Doing Homework 2 Logically Speaking 3 Introducing theContrapositive and Converse 4 Set Notation and Quantifiers Tips onQuantification 5 Proof Techniques Tips on Definitions 6 Sets Spotlight:Paradoxes 7 Operations on Sets 8 More on Operations on Sets 9 The PowerSet and the Cartesian Product Tips on Writing Mathematics 10 RelationsTips on Reading Mathematics 11 Partitions Tips on Putting It All Together12 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 19Convergence of Sequences of Real Numbers 20 Equivalent Sets 21 FiniteSets and an Infinite Set 22 Countable and Uncountable Sets 23 MetricSpaces 24 Getting to Know Open and Closed Sets 25 Modular Arithmetic 26Fermat's Little Theorem Spotlight: Public and Secret Research 27 ProjectsTips on Talking about Mathematics 27.1 Picture Proofs 27.2 The BestNumber of All 27.3 Set Constructions 27.4 Rational and Irrational Numbers27.5 Irrationality of $e$ and $\pi $ 27.6 When does $fPascal's Triangle 27.8 The Cantor Set 27.9 TheCauchy-Bunyakovsky-Schwarz Inequality 27.10 Algebraic Numbers 27.11 TheRSA Code Spotlight: Hilbert's Seventh Problem 28 Appendix 28.1 AlgebraicProperties of $\@mathbb {R}$ 28.2 Order Properties of $\@mathbb {R}$ 28.3Polya's List References Index