Invitation to Discrete Mathematics / Edition 1

Invitation to Discrete Mathematics / Edition 1

by Jiri Matousek, Jaroslav Nesetril
     
 

ISBN-10: 0198502079

ISBN-13: 9780198502074

Pub. Date: 01/28/1999

Publisher: Oxford University Press, USA

This second edition of Invitation to Discrete Mathematics is a clear and self-contained introduction to discrete mathematics. Aimed mainly at undergraduate and early graduate students of mathematics and computer science, it is written with the goal of stimulating interest in mathematics and an active, problem-solving approach to the presented material. The reader is

Overview

This second edition of Invitation to Discrete Mathematics is a clear and self-contained introduction to discrete mathematics. Aimed mainly at undergraduate and early graduate students of mathematics and computer science, it is written with the goal of stimulating interest in mathematics and an active, problem-solving approach to the presented material. The reader is led to an understanding of the basic principles and methods of actually doing mathematics (and having fun at that). By focussing on a more selective range of topics than many discrete mathematics textbooks, allowing greater depth of treatment using a number of different approaches, the book reflects the conviction of the authors, active and internationally renowned mathematicians, that the most important gain from studying mathematics is the cultivation of clear and logical thinking and habits useful for attacking new problems. More than 400 enclosed exercises with a wide range of difficulty, many of them accompanied by hints for solution, support this approach to teaching. The readers will appreciate the lively and informal style of the text accompanied by more than 200 drawings and diagrams. Specialists in various parts of science with a basic mathematical education wishing to apply discrete mathematics in their field can use the book as a useful source, and even experts in combinatorics may occasionally learn from pointers to research literature or from presentations of recent results. Invitation to Discrete Mathematics should make delightful reading both for beginners and for mathematical professionals.

Product Details

ISBN-13:
9780198502074
Publisher:
Oxford University Press, USA
Publication date:
01/28/1999
Edition description:
Older Edition
Pages:
426
Product dimensions:
9.10(w) x 6.10(h) x 1.10(d)

Table of Contents

1 Introduction and basic concepts 1

1.1 An assortment of problems 2

1.2 Numbers and sets: notation 7

1.3 Mathematical induction and other proofs 16

1.4 Functions 25

1.5 Relations 32

1.6 Equivalences and other special types of relations 36

2 Orderings 43

2.1 Orderings and how they can be depicted 43

2.2 Orderings and linear orderings 48

2.3 Ordering by inclusion 52

2.4 Large implies tall or wide 55

3 Combinatorial counting 59

3.1 Functions and subsets 59

3.2 Permutations and factorials 64

3.3 Binomial coefficients 67

3.4 Estimates: an introduction 78

3.5 Estimates: the factorial function 85

3.6 Estimates: binomial coefficients 93

3.7 Inclusion-exclusion principle 98

3.8 The hatcheck lady & co. 103

4 Graphs: an introduction 109

4.1 The notion of a graph; isomorphism 109

4.2 Subgraphs, components, adjacency matrix 118

4.3 Graph score 125

4.4 Eulerian graphs 130

4.5 Eulerian directed graphs 138

4.6 2-connectivity 143

4.7 Triangle-free graphs: an extremal problem 148

5 Trees 153

5.1 Definition and characterizations of trees 153

5.2 Isomorphism of trees 159

5.3 Spanning trees of a graph 166

5.4 The minimum spanning tree problem 170

5.5 Jarník's algorithm and Bor&uaa;vka's algorithm 176

6 Drawing graphs in the plane 182

6.1 Drawing in the plane and on other surfaces 182

6.2 Cycles in planar graphs 190

6.3 Euler's formula 196

6.4 Coloring maps: the four-color problem 206

7 Double-counting 217

7.1 Parity arguments 217

7.2 Sperner's theorem on independent systems 226

7.3 An extremal problem: forbidden four-cycles 233

8 The number of spanning trees 239

8.1 The result239

8.2 A proof via score 240

8.3 A proof with vertebrates 242

8.4 A proof using the Prüet;fer code 245

8.5 Proofs working with determinants 247

8.6 The simplest proof? 258

9 Finite projective planes 261

9.1 Definition and basic properties 261

9.2 Existence of finite projective planes 271

9.3 Orthogonal Latin squares 277

9.4 Combinatorial applications 281

10 Probability and probabilistic proofs 284

10.1 Proofs by counting 284

10.2 Finite probability spaces 291

10.3 Random variables and their expectation 301

10.4 Several applications 307

11 Order from disorder: Ramsey's theorem 317

11.1 A party of six 318

11.2 Ramsey's theorem for graphs 319

11.3 A lower bound for the Ramsey numbers 321

12 Generating functions 325

12.1 Combinatorial applications of polynomials 325

12.2 Calculation with power series 329

12.3 Fibonacci numbers and the golden section 340

12.4 Binary trees 348

12.5 On rolling the dice 353

12.6 Random walk 354

12.7 Integer partitions 357

13 Applications of linear algebra 364

13.1 Block designs 364

13.2 Fisher's inequality 369

13.3 Covering by complete bipartite graphs 373

13.4 Cycle space of a graph 376

13.5 Circulations and cuts: cycle space revisited 380

13.6 Probabilistic checking 384

Appendix: Prerequisites from algebra 395

Bibliography 402

Hints to selected exercises 407

Index 433

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