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Reading, Writing, and Proving

A Closer Look at Mathematics

Ulrich Daepp, Pamela Gorkin (Autoren)

Buch | Softcover

416 Seiten

2010 | 1st ed. Softcover of orig. ed. 2003
Springer-Verlag New York Inc.
978-1-4419-1840-6 (ISBN)

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This book, based on Polya's method of problem solving, aids students in their transition to higher-level mathematics. It begins by providing a great deal of guidance on how to approach definitions, examples, and theorems in mathematics and ends by providing projects for independent study. Students will follow Polya'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.

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

Erscheint lt. Verlag	29.11.2010
Reihe/Serie	Undergraduate Texts in Mathematics
Zusatzinfo	1, black & white illustrations
Verlagsort	New York, NY
Sprache	englisch
Maße	156 x 234 mm
Gewicht	581 g
Einbandart	Paperback
Themenwelt	Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika
	Mathematik / Informatik ► Mathematik ► Analysis
	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
ISBN-10	1-4419-1840-X / 144191840X
ISBN-13	978-1-4419-1840-6 / 9781441918406
Zustand	Neuware