Discrete Mathematics SECOND EDITION OXFORD UNIVERSITY PRESS. Norman L. Biggs. Professor of Mathematics London School of Economics University of London

Transcription

1 Discrete Mathematics SECOND EDITION Norman L. Biggs Professor of Mathematics London School of Economics University of London OXFORD UNIVERSITY PRESS

2 Contents PART I FOUNDATIONS Statements and proofs. 1 Some mathematical statements 3.2 How to do mathematics 4.3 Compound statements 6.4 Existential statements 7.5 Universal statements 8.6 Proof techniques 10.7 Miscellaneous Exercises Set notation 2.1 Sets of objects and numbers Subsets Union and intersection Miscellaneous Exercises The logical framework 3.1 The basic logical operations: not, or, and Logical equivalence if-then The converse statement The contrapositive statement Universal and existential quantifiers Miscellaneous Exercises Natural numbers 4.1 The'laws of algebra' Putting the natural numbers in order The principle of induction Summation formulae Recursive definitions Other forms of the principle of induction Greatest and least members How a conjecture becomes a theorem Miscellaneous Exercises 38

3 x Contents 5. Functions 5.1 The concept of a function Surjections, injections, bijections Composition of functions Bijections and inverse functions Miscellaneous Exercises How to count 6.1 Counting as a bijection The size of a set A counting problem Some applications of the pigeonhole principle Infinite sets Strange properties of infinite sets Miscellaneous Exercises Integers 7.1 Negative numbers Equivalence relations Classification Construction of the integers Properties of the integers Bounded subsets of Z Miscellaneous Exercises Divisibility and prime numbers 8.1 Divisibility Quotient and remainder Representation of integers The greatest common divisor Prime numbers Existence and uniqueness of prime factorization Miscellaneous Exercises Fractions and real numbers 9.1 Construction and properties of rational numbers Density of fractions Decimal representations of fractions Real numbers Approximations for real numbers The greatest lower bound property The real numbers are more plentiful than the rationals Miscellaneous Exercises 87

4 PART II TECHNIQUES 10. Principles of counting 10.1 The addition principle 10.2 Counting sets of pairs 10.3 Euler's function 10.4 Functions, words, and selections 10.5 Injections as ordered selections without repetition 10.6 Permutations 10.7 Miscellaneous Exercises 11. Subsets and designs 11.1 Binomial numbers 11.2 Unordered selections with repetition 11.3 The binomial theorem 11.4 The sieve principle 11.5 Some arithmetical applications 11.6 Designs 11.7 f-designs 11.8 Miscellaneous Exercises 12. Partition, classification, and distribution 12.1 Partitions of a set 12.2 Classification and equivalence relations 12.3 Distributions and the multinomial numbers 12.4 Partitions of a positive integer 12.5 Classification of permutations 12.6 Even and odd permutations 12.7 Miscellaneous Exercises 13. Modular arithmetic 13.1 Congruences 13.2 Z m and its arithmetic 13.3 Invertible elements of Z m 13.4 Cyclic constructions for designs 13.5 Latin squares 13.6 Miscellaneous Exercises PART III ALGORITHMS AND GRAPHS 14. Algorithms and their efficiency 14.1 What is an algorithm? 14.2 The language of programs 14.3 Algorithms and programs 14.4 Proving that an algorithm is correct

5 xii Contents Efficiency of algorithms Growth rates: the O notation Comparison of algorithms Introduction to sorting algorithms Miscellaneous Exercises Graphs Graphs and their representation Isomorphism of graphs Degree Paths and cycles Trees Colouring the vertices of a graph The greedy algorithm for vertex-colouring Miscellaneous Exercises Trees, sorting, and searching 16.1 Counting the leaves on a rooted tree Trees and sorting algorithms Spanning trees and the MST problem Depth-first search Breadth-first search The shortest path problem Miscellaneous Exercises Bipartite graphs and matching problems 17.1 Relations and bipartite graphs Edge colourings of graphs Application of edge colouring to latin squares Matchings Maximum matchings Transversals for families of finite sets Miscellaneous Exercises Digraphs, networks, and flows 18.1 Digraphs Networks and critical paths Flows and cuts The max-flow min-cut theorem The labelling algorithm for network flows Miscellaneous Exercises Recursive techniques 19.1 Generalities about recursion Linear recursion 241

6 Contents xiii 19.3 Recursive bisection Recursive optimization The framework of dynamic programming Examples of the dynamic programming method Miscellaneous Exercises 253 PART IV ALGEBRAIC METHODS 20. Groups 20.1 The axioms for a group Examples of groups Basic algebra in groups The order of a group element Isomorphism of groups Cyclic groups Subgroups Cosets and Lagrange's theorem Characterization of cyclic groups Miscellaneous Exercises Groups of permutations 21.1 Definitions and examples Orbits and stabilizers The size of an orbit The number of orbits Representation of groups by permutations Applications to group theory Miscellaneous Exercises Rings, fields, and polynomials 22.1 Rings Invertible elements of a ring Fields Polynomials The division algorithm for polynomials The Euclidean algorithm for polynomials Factorization of polynomials in theory Factorization of polynomials in practice Miscellaneous Exercises Finite fields and some applications 23.1 A field with nine elements The order of a finite field Construction of finite fields The primitive element theorem Finite fields and latin squares 322

7 xiv Contents 23.6 Finite geometry and designs Projective planes Squares in finite fields, Existence of finite fields Miscellaneous Exercises Error-correcting codes 24.1 Words, codes, and errors Linear codes Construction of linear codes Correcting errors in linear codes Cyclic codes Classification and properties of cyclic codes Miscellaneous Exercises Generating functions 25.1 Power series and their algebraic properties Partial fractions The binomial theorem for negative exponents Generating functions The homogeneous linear recursion Non-homogeneous linear recursions Miscellaneous Exercises Partitions of a positive integer 26.1 Partitions and diagrams Conjugate partitions Partitions and generating functions Generating functions for restricted partitions A mysterious identity The calculation of p(n) Miscellaneous Exercises Symmetry and counting 27.1 The cycle index of a group of permutations Cyclic and dihedral symmetry Symmetry in three dimensions The number of inequivalent colourings Sets of colourings and their generating functions Polya's theorem Miscellaneous Exercises 405 Answers to exercises 407 Index 421