Problems and Exercises in Discrete Mathematics

Problems and Exercises in Discrete Mathematics

Kluwer Texts in the Mathematical Sciences VOLUME 14 A Graduate-Level Book Series The titles published in this series are listed at the end o/this volume.

Problems and Exercises in Discrete Mathematics by G. P. Gavrilov and A. A. Sapozhenko Department o/computational Mathematics and Cybernetics, Moscow State University, Moscow, Russia... " SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-4702-1 DOI 10.1007/978-94-017-2770-9 ISBN 978-94-017-2770-9 (ebook) This is a completely revised and updated edition of Selected Problems in Discrete Mathematics by the same authors. MIR, Moscow, 1989 Printed on acid-free paper A1l Rights Reserved 1996 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1996 Softcover reprint ofthe hardcover lst edition 1996 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, clectronic or mcchanical, including photocopying, rccording or by any information storage and retricval system, without writtcn pcrmission from the copyright owncr.

Contents Preface I PROBLEMS 1 1 Representations of Boolean Functions 3 1.1 Tabular Representations........ 3 1.2 Formulas... 8 1.3 Disjunctive and Conjunctive Normal Forms. 20 1.4 Polynomials... 25 1.5 Essential and Unessential Variables..... 30 2 Closed Classes and Completeness in Boolean Algebra 38 2.1 Closure Operation 38 2.2 Self-Dual Functions......... 43 2.3 Linear Functions... 47 2.4 Functions Preserving the Constants 52 2.5 Monotone Functions....55 2.6 Completeness and Closed Classes 60 3 Many-Valued Logics 66 3.1 Formulas of k-valued Logics............. 66 3.2 Closed Classes and Completeness in k-valued logic. 71 4 Graphs and Networks 81 4.1 Basic Concepts of Graph Theory... 81 4.2 Planarity, Connectivity, and Numerical Characteristics of Graphs 88 4.3 Directed Graphs....... 93 4.4 Trees and Bipolar Networks..................... 98 5 Elements of Coding Theory 110.5.1 Hamming's Distance. 110 5.2 Hamming's Codes...... 113.5.3 Linear Codes.... 118 5.4 Alphabetic Code Divisibility. 122 5.5 Optimal Codes.... 127 6 Finite Automata 134 6.1 Determined and Boundedly Determined Functions.. 134 6.2 Diagrams, Equations, and Circuits.... 148 ix v

VI CONTENTS 6.3 Closed Classes and Completeness in Automata. 7 Elements of Algorithm Theory 7.1 Turing Machines.. 7.2 Recursive Functions. 7.3 Computability. 8 Combinatorics 8.1 Permutations and Combinations. 8.2 Inclusion and Exclusion Formula 8.3 Recurrences and Generating Functions 8.4 Polya's Theory.... 8.5 Asymptotics and Inequalities. 8.6 Estimates in Graph Theory 9 Boolean Minimization 9.1 Faces of the n-cube. Covers and tests for tables 9.2 Constructing of the Reduced Disjunctive Normal Form Methods 9.3 Irredundant, Minimal, and Shortest DNFs.... 10 Logical Design 10.1 Circuits of Logical Elements 10.2 Contact Circuits.... 182 188 188 202.209 214 214 224 228 236.240.248 254.254 260.266 273 273.280 II ANSWERS, HINTS, SOLUTIONS 1 Representations of Boolean Functions 1.1 Tabular Representations........ 1.2 Formulas... 1.3 Disjunctive and Conjunctive Normal Forms. 1.4 Polynomials... 1.5 Essential and Unessential Variables... 2 Closed Classes and Completeness in Boolean Algebra 2.1 Closure Operation 2.2 Self-Dual Functions.... 2.3 Linear Functions......... 2.4 Functions Preserving the Constants 2.5 Monotone Functions.... 2.6 Completeness and Closed Classes 3 Many-Valued Logics 3.1 Formulas of k-valued Logics.... 3.2 Closed Classes and Completeness in k-valued logic 295 297 297 297 299.299 300 303.303 305 306 307.309 311 313 313 315

CONTENTS Vll 4 Graphs and Networks 4.1 Basic Concepts of Graph Theory.... 4.2 Planarity, Connectivity, and Numerical Characteristics of Graphs 4.3 Directed Graphs.... 4.4 Trees and Bipolar Networks 5 Elements of Coding Theory 5.1 Hamming's Distance 5.2 Hamming's Codes.... 5.3 Linear Codes.... 5.4 Alphabetic Code Divisibility 5.5 Optimal Codes.... 6 Finite Automata 6.1 Determined and Boundedly Determined Functions. 6.2 Diagrams, Equations, and Circuits.... 6.3 Closed Classes and Completeness in Automata. 7 Elements of Algorithm Theory 7.1 Turing Machines.. 7.2 Recursive Functions. 7.3 Computability. 8 Combinatorics 8.1 Permutations and Combinations. 8.2 Inclusion and Exclusion Formula 8.3 Recurrences and Generating Functions 8.4 Polya's Theory.... 8.5 Asymptotics and Inequalities. 8.6 Estimates in Graph Theory 9 Boolean Minimization 9.1 Faces of the n-cube. Covers and tests for tables 9.2 Constructing of the Reduced DNF Methods 9.3 Irredundant, Minimal, and Shortest DNFs 10 Logical Design 10.1 Circuits of Logical Elements 10.2 Contact Circuits.... Bibliography Index 325 325 326 327 328 329 329 331 333 335 337 339 339 341 345 348 348.349 351 353 353 363 368 379 383 401 403 403 406.408 409 409 410 415 419

Preface This book of problems is mainly intended for undergraduates. It can also be useful for postgraduates and researchers who apply methods of Discrete l\iathematics in their study and investigations. Lecturers can use this material for exercises during seminars. The contents are based on a course of lectures and seminars on Discrete Mathematics carried out by the authors and their colleagues over a number of years at the Department of Computational Mathematics and Cybernetics of Moscow State University. The Russian reader can use "Introduction to Discrete Mathematics" by S. V. Yablonsky as the theoretical guide while solving the problems in this collection. In the translation of the book, the authors essentially extended the theoretical introductions to the chapters, so the Western reader can use this book without any additional theoretical guidance. The study of mathematics is impossible without experience in solving tasks. There are many excellent textbooks on the classical fields of mathematics: analysis, algebra, differential equalities, etc. The situation is different in such modern fields as Discrete Mathematics and Theoretical Computer Sciences. To date a common notion on the subject of Discrete Mathematics has hardly been formed, although it usually includes Boolean algebra, k-valued logics, coding theory, automata theory, algorithm theory, combinatorics, graph theory, and logical design. Sometimes, several parts of logics, set theory and algebra, such as propositional calculus, and relations, are assigned to Discrete Mathematics. There are textbooks [4], [21], [22], [29], [31], [45], and also manuals and problem books on separate parts of Discrete Mathematics: Boolean algebra [16], [46], automata theory [20], [23], [24], [39], [41], algorithm theory [1], [2], [25], [27], graph theory [3], [6], [19], [32], [40], [43], [47], [30], [3.5], combinatorics [S], [17], [IS], [26], [34], [36], [37], [3S], [42], coding theory [28]. [33]. This problem book stands out because all the main parts of Discrete Mathematics are represented in it. Another peculiarity of the book is the functional approach to the subject. This approach is typical of the Moscow School of Discrete Mathematics. It was initiated by the well-known Yablonsky paper [45] and includes considering the objects of Discrete Mathematics as being generated from the elements of some basis by means of some relevant operations. It assumes an investigation into classes of objects, closed with respect to some sets of operations, and problems of expressibility and completeness. This approach is very productive in Boolean algebra, many-valued logics and automata, where it allows one to set a lot of varied problems. It is less successful in other parts, such as in coding theory and combinatorics. The writing of the book was begun in 1971 when the authors started their work ix

x PREFACE at the Computational Mathematics and Cybernetics Department of the Moscow State University. Our first experience was a small problem book of two parts [10] published by the M.S.U. Publishing House in 1974. Then, in 1977, the book [11] was published by Nauka Publishers. It was recommended as a manual for universities by the Education Ministry of the U.S.S.R. In 1980, 1981, and 1989, respectively, the Spanish [13], Hungarian [14] and English [15] translations of [11] appeared. In 1992, a second Russian edition [12] appeared. This manual is an essentially revised translation of our textbook [12] and consists of exercises (about 60%), tasks of intermediate difficulty (about 25%), and some difficult problems (about 15%). The most difficult problems are marked with an asterisk. In all, the book contains over 3000 problems and exercises. The book is divided into two parts. The first part contains problems and the second consists of answers, hints, and solutions. Each part has 10 chapters. The first two chapters of Part I are devoted to Boolean algebra which forms the basis of Discrete Mathematics. About a quarter of the total teaching time during lectures and seminars is devoted to Boolean algebra. The first chapter acquaints the reader with the various tools representing of a discrete function: tables, formulas, normal forms, polynomials, geometrical representations using the n-cube, etc. In the second chapter, the reader is introduced to the concepts of discrete functions, composition, functionally complete sets and closure. Some methods for testing the completeness and closure of sets of functions are also considered. The third chapter is devoted to k-valued logic. The problems presented here are intended to familiarize the reader with the canonical expansions of k-valued functions, equivalent transformations of formulas, closed classes of k-valued functions, and methods for testing the completeness and closure of sets of functions. Several problems in the chapter are intended to demonstrate the difference between k-valued logic (k > 2) and Boolean algebra. The fourth chapter contains problems in graph and network theory. The basic concepts (isomorphism, planarity, coloring, cover, etc.) and the methods of the theory are illustrated by exercises and problems. The fifth chapter deals with the elements of coding theory. Three topics are studied: the uniqueness of decoding problems in alphabetic coding, the design of optimal codes, the design of self-correcting and linear codes. The sixth chapter contains problems in automata theory. The problems collected here help in acquainting the reader with the notion of a discrete deterministic transformer of information (automaton), various tools of performance of automata (diagrams, tables, canonical equations and circuits), and operations over automata. The seventh chapter deals with the elements of algorithm theory and is intended to make the reader familiar with two models of algorithms: Turing machines and recursive functions. The eighth chapter is devoted to combinatorics. Here the properties of binomial coefficients, factorials and others combinatorial objects are studied. The chapter also contains sections devoted to the inclusion-exclusion formula, recurrent sequences, Polya's theory, asymptotic estimations in combinatorics and graph theory. The ninth chapter deals with Boolean minimization. The structure of the faces

PREFACE xi of the n-cube, covers, tests for tables, as well as the design methods of minimal, irredundant and reduced disjunctive normal forms are considered. The tenth chapter deals with logical design. The circuits of logical elements and the contact circuits are studied. The problems and exercises in this book have various origins. A considerable number of them are taken from mathematical folklore and are well-known to specialists in Discrete Mathematics. Most of the problems were conceived by the authors while preparing the material for seminars and examinations. Some of the problems are simple assertions from scientific articles. Sometimes, but far from always, we point out the names of the authors. Some of the problems were kindly supplied by our colleagues: O. B. Lupanov, V. K. Leontiev, V. B. Alekseev, S. V. Yablonsky, and G. Burosch. We express our thanks to them. The authors are deeply indebted to Serge Kostyukovich, Serge Levit, Andrew Sapozhenko, and Natalia Sumkina for their help in the preparation of the typescript and V. M. Khrapchenko for useful discussions on terminology. All reproaches on defects of design, misprints, linguistic errors, have to be addressed to Al. A. Sapozhenko who took on the task of translating the book and the preparation of the typescript. G. P. Gavrilov edited a considerable part of the typescript. G. P. Gavrilov, A. A. Sapozhenko