Synopses & Reviews
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.
Review
From the reviews:
U. Daepp and P. Gorkin
Reading, Writing, and Proving
A Closer Look at Mathematics
"Aids students in their transition from calculus (or precalculus) to higher-level mathematics . . . The authors have included a wide variety of examples, exercises with solutions, problems, and over 40 illustrations."
—L'ENSEIGNEMENT MATHEMATIQUE
"Daepp and Gorkin (both, Bucknell Univ.) offer another in the growing genre of books designed to teach mathematics students the rigor required to write valid proofs … . The book is well written and should be easy for a first- or second- year college mathematics student to read. There are many ‘tips’ offered throughout, along with many examples and exercises … . A book worthy of serious consideration for courses whose goal is to prepare students for upper-division mathematics courses. Summing Up: Highly recommended." (J.R. Burke, CHOICE, 2003)
"The book Reading, Writing, and Proving … provides a fresh, interesting, and readable approach to the often-dreaded ‘Introduction to Proof’ class. … RWP contains more than enough material for a one-semester course … . I was charmed by this book and found it quite enticing. … My students found the overall style, the abundance of solved exercises, and the wealth of additional historical information and advice in the book exceptionally useful. … well-conceived, solidly executed, and very useful textbook." (Maria G. Fung, MAA online, December, 2004)
"The book is intended for undergraduate students beginning their mathematical career or attending their first course in calculus. … Throughout the book … students are encouraged to 1) learn to understand the problem, 2) devise a plan to solve the problem, 3) carry out that plan, and 4) look back and check what the results told them. This concept is very valuable. … The book is written in an informal way, which will please the beginner and not offend the more experienced reader." (EMS Newsletter, December, 2005)
Synopsis
The reader of this book is probably about to teach or take a "first course in proof techniques." At this point, students have an intuitive sense of why things are true, but not the exposure to detailed and critical thinking necessary to survive in the mathematical world. The authors have written this book to bridge the gap. The authors' aim is to teach students to read, write and do mathematics independently, and to do it with clarity, precision, and care.
Synopsis
This book, which is based on Plya'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 Plya'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.
Synopsis
This book, based on Pólya'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 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.
Table of Contents
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