Reading, Writing, and Proving by Daepp, Ulrich; Gorkin, Pamela;

A Closer Look at Mathematics

Sorozatcím: Undergraduate Texts in Mathematics;

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21 546 Ft

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A termék adatai:

Kiadó Springer
Megjelenés dátuma 2003. szeptember 1.
Kötetek száma Book

ISBN 9780387008349
Kötéstípus Keménykötés
Terjedelem408 oldal
Méret 235x155 mm
Súly 789 g
Nyelv angol
Illusztrációk 49 Illustrations, black & white

Kategóriák

A matematika általános kérdései A matematika alapjai, matematikai logika Számelmélet Analízis

Rövid leírás:

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.

Több

Hosszú leírás:

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.

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.

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)

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Tartalomjegyzék:

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