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TOPICS IN DISCRETE MATHEMATICS

A.F. PixleyHarvey Mudd College

July 21, 2010

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Contents

Preface v

1 Combinatorics 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Pigeonhole Principle . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Ramseys Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Counting Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.5 Permutations and combinations . . . . . . . . . . . . . . . . . . . . . 191.6 Permutations and combinations with repetitions . . . . . . . . . . . . 281.7 The binomial coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 381.8 The principle of inclusion and exclusion . . . . . . . . . . . . . . . . . 45

2 The Integers 532.1 Divisibility and Primes . . . . . . . . . . . . . . . . . . . . . . . . . . 532.2 GCD and LCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.3 The Division Algorithm and the Euclidean Algorithm . . . . . . . . . 622.4 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.5 Counting; Eulers -function . . . . . . . . . . . . . . . . . . . . . . . 692.6 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.7 Classical theorems about congruences . . . . . . . . . . . . . . . . . . 792.8 The complexity of arithmetical computation . . . . . . . . . . . . . . 85

3 The Discrete Calculus 933.1 The calculus of finite differences . . . . . . . . . . . . . . . . . . . . . 933.2 The summation calculus . . . . . . . . . . . . . . . . . . . . . . . . . 1023.3 Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083.4 Application: the complexity of the Euclidean algorithm . . . . . . . . 114

4 Order and Algebra 1174.1 Ordered sets and lattices . . . . . . . . . . . . . . . . . . . . . . . . . 1174.2 Isomorphism and duality . . . . . . . . . . . . . . . . . . . . . . . . . 1194.3 Lattices as algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.4 Modular and distributive lattices . . . . . . . . . . . . . . . . . . . . 125

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iv CONTENTS

4.5 Boolean algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.6 The representation of Boolean algebras . . . . . . . . . . . . . . . . . 137

5 Finite State Machines 1455.1 Machines-introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1455.2 Semigroups and monoids . . . . . . . . . . . . . . . . . . . . . . . . . 1465.3 Machines - formal theory . . . . . . . . . . . . . . . . . . . . . . . . . 1485.4 The theorems of Myhill and Nerode . . . . . . . . . . . . . . . . . . . 152

6 Appendix: Induction 161

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Preface

This text is intended as an introduction to a selection of topics in discrete mathemat-ics. The choice of topics in most such introductory texts is usually governed by thesupposed needs of students intending to emphasize computer science in their subse-quent studies. Our intended audience is somewhat larger and is intended to includeany student seriously interested in any of the mathematical sciences. For this reasonthe choice of each topic is to a large extent governed by its intrinsic mathematicalimportance. Also, for each topic introduced an attempt has been made to developthe topic in sufficient depth so that at least one reasonably nontrivial theorem can beproved, and so that the student can appreciate the existence of new and unexploredmathematical territory.

For reasons that are not entirely clear, at least to me, discrete mathematics seemsto be not as amenable to the intuitive sort of development so much enjoyed in thestudy of beginning calculus. Perhaps one reason for this is the fortuitous notation usedfor derivatives and integrals which makes such topics as the chain rule for derivativesand the change of variable theorems for integrals so easy to understand. But, forexample, in the discrete calculus, (presented in Chapter 3 of this book), despite manyefforts, the notation is not quite so natural and suggestive. It may also just be thecase that human intuition is, by nature, better adapted to the study of the continuousworld than to the discrete one. In any case, even in beginning discrete mathematics,the role of proper mathematical reasoning and hence the role of careful proofs seemsto be more essential than in beginning continuous mathematics. Hence we place agreat deal of emphasis on careful mathematical reasoning throughout the text.

Because of this, the prerequisites I have had in my mind in writing the text, beyondrigorous courses in single variable and multivariable calculus, include linear algebraas well as elementary computer programming. While little specific information fromthese subjects is used, the expectation is that the reader has developed sufficientmathematical maturity to begin to engage in reasonably sophisticated mathematicalreasoning. We do assume familiarity with the meanings of elementary set and logicalnotation. Concerning sets this means the membership () and inclusion () relations,unions, intersections, complements, cartesian products, etc.. Concerning logic thismeans the propositional connectives (or, and, negation, and implication)and the meanings of the existential and universal quantifiers. We develop more ofthese topics as we need them.

Mathematical induction plays an important role in the topics studied and anappendix on this subject is included. In teaching from this text I like to begin thecourse with this appendix.

Chapters 1 (Combinatorics) and 2 (The Integers) are the longest and the mostimportant in the text. With the exception of the principle of inclusion and exclusion(Section 1.8) which is used in Section 2.5 to obtain Legendres formula for the Euler -function, and a little knowledge of the binomial coefficients, there is little dependence

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of Chapter 2 on Chapter 1. The remaining chapters depend on the first two in varyingamounts, but not at all on each other.

Chapter 1

Combinatorics

1.1 Introduction

Combinatorics is concerned with the possible arrangements or configurations of ob-jects in a set. Three main kinds of combinatorial problems occur: existential, enu-merative, and constructive. Existential combinatorics studies the existence or non-existence of certain configurations. The celebrated four color problem Is therea map of possible countries on the surface of a sphere which requires more thanfour colors to distinguish between countries? is probably the most famous exampleof existential combinatorics. Its negative solution by Appel and Haken in 1976required over 1000 hours of computer time and involved nearly 10 billion separatelogical decisions.

Enumerative combinatorics is concerned with counting the number of configura-tions of a specific kind. Examples abound in everyday life: how many ways can alegislative committee of five members be chosen from among ten Democrats and sixRepublicans so that the Republicans are denied a majority? In how many ways cansuch a committee, once chosen, be seated around a circular table? These and manyother simple counting problems come to mind.

Constructive combinatorics deals with methods for actually finding specific con-figurations, as opposed to simply demonstrating their existence. For example, LosAngeles County contains at least 10 million residents and by no means does any hu-man being have anywhere near that many hairs on his or her head. Consequently wemust conclude (by existential combinatorics) that at any instant at least two peoplein LA County have precisely the same number of hairs on their heads! This simpleassertion of existence is, however, a far cry from actually prescribing a method of find-ing such a pair of people which is not even a mathematical problem. Constructivecombinatorics, on the other hand, is primarily concerned with devising algorithms mechanical procedures for actually constructing a desired configuration.

In the following discussion we will examine some basic combinatorial ideas withemphasis on the mathematical principles underlying them. We shall be primarily

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2 Chapter 1 Combinatorics

concerned with enumerative combinatorics since this classical area has the most con-nections with other areas of mathematics. We shall not be much concerned at allwith constructive combinatorics and only in the following discussion of the Pigeon-hole principle and Ramseys theorem will we be studying a primary area of existentialcombinatorics.

1.2 The Pigeonhole Principle

If we put into pigeonholes more pigeons than we have pigeonholes then at least oneof the pigeonholes contains at least two pigeons. If n people are wearing n + 1 hats,then someone is wearing two hats. The purely mathematical content of either of theseassertions as well as of the LA County hair assertion above is the same:

Proposition 1.2.1 (Pigeonhole Principle) If a set of at least n + 1 objects is parti-tioned into n non-overlapping subsets, then one of the subsets contains at least twoobjects.

The proof of the proposition is simply the observation that if each of the n non-overlapping subsets contained at most 1 object, then altogether we would only accountfor at most n of the at least n+ 1 objects.

In order to see how to apply the Pigeonhole Principle some discussion of partitionsof finite sets is in order. A partition of a set S is a subdivision of S into non-emptysubsets which are disjoint and exhaustive, i.e.: each element of S must belong to oneand only one of the subsets. Thus = {A1, ..., An} is a partition of S if the followingconditions are met: each Ai 6= , Ai Aj = for i 6= j, and S = A1 An.The Ai are called the blocks or classes of the partition and the number of blocks nis called the ind