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Matters Mathematical: Second Edition
I. N. Herstein and I. Kaplansky
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AMS Chelsea Publishing
1978; 246 pp; hardcover
ISBN-10: 0-8284-0300-7
ISBN-13: 978-0-8284-0300-9
List Price: US$34
Member Price: US$30.60
Order Code: CHEL/300
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From the Preface: "This book is based on notes prepared for a course at the University of Chicago. The course was intended for nonmajors whose mathematical training was somewhat limited ... Mastery of the material requires nothing beyond algebra and geometry normally covered in high school ... [I]t could be used in courses designed for students who intend to teach mathematics ... We want the reader to see mathematics as a living subject in which new results are constantly being obtained."

Reviews

"It is refreshing to find a book composed of so many of the most enjoyable and interesting topics in mathematics, such as number theory, group theory, finite geometry, and game theory. The authors have succeeded in their attempt to impart to the reader "something of the flavor of modern mathematics." Even the standard discussion of set theory in the first chapter is expanded in the last chapter by a treatment of infinite sets. Each topic is dealt with thoroughly, including a complete discussion of the most current work and unsolved problems in each area ... This book is a must for schools that train secondary school teachers of mathematics and will be appreciated by all persons who have a strong interest in mathematics. Matters Mathematical is both an excellent text and reference book for all undergraduate and graduate libraries."

-- CHOICE

"A fascinating ... introduction to abstract mathematics."

-- American Mathematical Monthly

"The book originates from lectures for nonmajors in mathematics, which were expanded to give a text which could also be used as general introductory material for mathematics students, especially for future teachers ... the authors have succeeded very well in creating an adequate picture of mathematics for their audience, and ... the topics chosen are optimal for their intentions."

-- Mathematical Reviews

"The style is clear and should be easily understood by a non-specialist ... of interest even to a specialist ... very pleasant reading indeed."

-- Zentralblatt MATH

Table of Contents

Sets and Functions
  • 1 Sets
  • 2 Sets and counting
  • 3 Functions
Number Theory
  • 1 Prime numbers
  • 2 Some formal aspects
  • 3 Some formal consequences
  • 4 Some basic properties
  • 5 Equivalence relations
  • 6 Congruence
  • 7 Applications of the pigeon-hole principle
  • 8 Waring's problem
  • 9 Fermat and Mersenne primes
Permutations
  • 1 The set \(A(S)\)
  • 2 Cycle decomposition
  • 3 Even and odd
  • 4 The interlacing shuffle
  • 5 The Josephus permutation
  • Bibliography
Group Theory
  • 1 Definition and examples of groups
  • 2 Some beginning notions and results
  • 3 Subgroups
  • 4 Lagrange's theorem
  • 5 Isomorphism
Finite Geometry
  • 1 Introduction
  • 2 Affine planes
  • 3 Counting arguments
  • 4 Planes of low order
  • 5 Coordinate affine planes
  • 6 Parallelograms and midpoints
  • 7 The nonexistence of planes of order 6
  • Suggestions for further reading
Game Theory
  • 1 Probability
  • 2 Mathematical expectation
  • 3 Preliminary remarks on games
  • 4 Examples
  • 5 The general two-by-two game
  • 6 Simplified poker
  • 7 The game without a name
  • 8 Goofspiel
  • Suggestions for further reading
Infinite Sets
  • 1 Infinite numbers; countable sets
  • 2 Uncountable sets
  • Index
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