These changes have fostered the development of many single-semester and multiple-semester courses where some of the following are introduced: 1. The technological advances of the last four decades have resulted in many changes in the undergraduate curriculum. I can only hope that this fifth edition will continue to enlighten and challenge all those who wish to learn about some of the many facets of the fascinating area of mathematics called discrete mathematics.
They have also been used in other nations such as Australia, Canada, England, Ireland, Japan, Mexico, the Netherlands, Scotland, Singapore, South Africa, and Sweden. The first four editions of this textbook have found their way into many colleges and universities here in the United States. Consequently, I continue to find myself simultaneously very humbled and very pleased with the way this textbook has been received by so many instructors and especially students.
At that time the idea of further editions never crossed my mind. Preface It has been more than twenty years since September 2, 1982, when I signed the contract to develop what turned into the first edition of this present textbook. NOTATION LOGIC SET THEORY PROBABILITY NUMBERS p>q ^p p/\q pVq p->q pq p*=>q To Fo Vx 3x x e A xiA °lt ACfi, 52 A AcB,BDA A?B A(A) AHB AUB AAB A A-B Ul€/A, ftielAi S AC5 Pr(A) Pr(A\B) X E(X) Var(Z) = o\ ox a\b alb gcd(#, b) lcm(a, b) ? /(Ai) /(A) /: A X A -> ? (C A) lA: A -> A /Ui 8° f f~X f~x Un€Z+An Un>o An: the Kleene closure of language A a finite state machine M with internal states 5, input alphabet if, output alphabet 0, next state function v: S X if -> S and output function o>: 5 X if -> 0 No part of this puj may be reproduced, stored in a retrieval system, or transmitted in any form or by electronic, mechanical, photocopying, recording, or otherwise, without the permission of the publisher. ed of: Discrete and combinatorial mathematics, cl999. A review of discrete and combinatorial mathematics / by Ralph P. Library of Congress Cataloging-in-Publication Data Grimaldi, Ralph P. Kruskal is reproduced courtesy of Leiden University. Huffman is reproduced courtesy of Manuel Enrique Bermudez of the Department of Computer and Information Science and Engi- Engineering at the University of Florida. The photographs of Andrei Nikolayevich Kolmogorov, Thomas Bayes, and Al-Khowarizmi are reproduc ed courtesy of the St. The photograph of Paul Erdos is reproduced courtesy of Christopher Barker. The photographs of Georg Cantor, Alan Mathison Turing, William Rowan Hamilton, and Leonardo Fibonacci are reproduced courtesy of The Granger Collection. The photographs of Andrew John Wiles and Rear Admiral Grace Murray Hopper are reproduced courtesy of AP/Wide World. Dijkstra is reproduced courtesy of the University of Texas at Austin. The photograph of Claude El wood Shannon is reproduced courtesy of the MIT Museum. Photographs of George Boole, Peter Gustav Lejeune Dirichlet, David Hilbert, Giuseppe Peano, James Joseph Sylvester, Sophie Germain, and Emmy Noether are reproduced courtesy of Historical Pictures/Stock Montage.
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Kelvin Photographs of Blaise Pascal, Aristotle, Lord Bertrand Arthur William Russell, Euclid, Au- Augusta Ada Byron (Countess of Lovelace), Gottfried Wilhelm Leibniz, Carl Friedrich Gauss, Leonhard Euler, Arthur Cayley, Pierre de Fermat, Niels Henrik Abel, and Evariste Galois are reproduced courtesy of the Bettman Archive (Corbis). Atkinson Dennis Schaefer Beth Anderson George V. Publisher: Senior Acquisitions Editor: Assistant Editor: Executive Marketing Manager: Senior Marketing Manager: Marketing Assistant: Managing Editor: Senior Production Supervisor: Senior Manufacturing Buyer: Composition and Technical Art Rendering: Production Services: Design Supervisor: Cover Designer: Photo Research and Design Specifications: Cover Illustration: Greg Tobin William Hoffman RoseAnne Johnson Yolanda Cossio Pamela Laskey Heather Peck Karen Guardino Peggy McMahon Hugh Crawford Techsetters, Inc. GRIMALDI Rose-Hulman Institute of Technology PEARSON Addison Wesley Boston San Francisco New York London Toronto Sydney Tokyo Singapore Madrid Mexico City Munich Paris Cape Town Hong Kong Montreaf DISCRETE AND COMBINATORIAL MATHEMATICS An Applied Introduction FIFTH EDITION RALPH P.
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