Modern Algebra with Applications - W.Gilbert, K.Nicholson, Wiley-Interscience, 2004.pdf

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MODERN ALGEBRA
WITH APPLICATIONS
PURE AND APPLIED MATHEMATICS
A Wiley-Interscience Series of Texts, Monograph, and Tracts
Founded by RICHARD COURANT
Editors: MYRON B. ALLEN III, DAVID A. COX, PETER LAX
Editors Emeriti: PETER HILTON, HARRY HOCHSTADT, JOHN TOLAND
A complete list of the titles in this series appears at the end of this volume.
MODERN ALGEBRA
WITH APPLICATIONS
Second Edition
WILLIAM J. GILBERT
University of Waterloo
Department of Pure Mathematics
Waterloo, Ontario, Canada
W. KEITH NICHOLSON
University of Calgary
Department of Mathematics and Statistics
Calgary, Alberta, Canada
A JOHN WILEY & SONS, INC., PUBLICATION
Cover:
Still image from the applet KaleidoHedron, Copyright
2000 by Greg Egan, from his
website http://www.netspace.net.au/∼gregegan/. The pattern has the symmetry of the icosahedral
group.
Copyright
2004 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
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Library of Congress Cataloging-in-Publication Data:
Gilbert, William J., 1941–
Modern algebra with applications / William J. Gilbert, W. Keith Nicholson.—2nd ed.
p. cm.—(Pure and applied mathematics)
Includes bibliographical references and index.
ISBN 0-471-41451-4 (cloth)
1. Algebra, Abstract. I. Nicholson, W. Keith. II. Title. III. Pure and applied
mathematics (John Wiley & Sons : Unnumbered)
QA162.G53 2003
512—dc21
2003049734
Printed in the United States of America.
10 9 8 7 6 5 4 3 2 1
CONTENTS
Preface to the First Edition
Preface to the Second Edition
List of Symbols
1
Introduction
Classical Algebra, 1
Modern Algebra, 2
Binary Operations, 2
Algebraic Structures, 4
Extending Number Systems,
2
Boolean Algebras
Algebra of Sets, 7
Number of Elements in a Set, 11
Boolean Algebras, 13
Propositional Logic, 16
Switching Circuits, 19
Divisors, 21
Posets and Lattices, 23
Normal Forms and Simplification of Circuits,
Transistor Gates, 36
Representation Theorem, 39
Exercises, 41
3
Groups
Groups and Symmetries,
Subgroups, 54
48
ix
xiii
xv
1
5
7
26
47
v

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