An invitation to discrete mathematics

✍ Scribed by Matousek J., Nesetril J.

• Publisher: OUP
• Year: 2008
• Tongue: English
• Leaves: 461
• Edition: 2ed
• Category: Library

✦ Synopsis

This book 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). Being more narrowly focused than many discrete mathematics textbooks and treating selected topics in an unusual depth and from several points of view, 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 a delightful reading both for beginners and for mathematical professionals.The main topics include: elementary counting problems, asymptotic estimates, partially ordered sets, basic graph theory and graph algorithms, finite projective planes, elementary probability and the probabilistic method, generating functions, Ramsey's theorem, and combinatorial applications of linear algebra. General mathematical notions going beyond the high-school level are thoroughly explained in the introductory chapter. An appendix summarizes the undergraduate algebra needed in some of the more advanced sections of the book.

✦ Table of Contents

Cover......Page 1
Invitation to Discrete Mathematics (2nd edition)......Page 4
Copyright......Page 5
Preface to the second edition......Page 6
Preface......Page 8
Contents......Page 16
1 Introduction and basic concepts......Page 19
1.1 An assortment of problems......Page 20
1.2 Numbers and sets: notation......Page 25
1.3 Mathematical induction and other proofs......Page 34
1.4 Functions......Page 43
1.5 Relations......Page 50
1.6 Equivalences and other special types of relations......Page 54
2.1 Orderings and how they can be depicted......Page 61
2.2 Orderings and linear orderings......Page 66
2.3 Ordering by inclusion......Page 70
2.4 Large implies tall or wide......Page 73
3.1 Functions and subsets......Page 77
3.2 Permutations and factorials......Page 82
3.3 Binomial coefficients......Page 85
3.4 Estimates: an introduction......Page 96
3.5 Estimates: the factorial function......Page 103
3.6 Estimates: binomial coefficients......Page 111
3.7 Inclusion–exclusion principle......Page 116
3.8 The hatcheck lady & co.......Page 121
4.1 The notion of a graph; isomorphism......Page 127
4.2 Subgraphs, components, adjacency matrix......Page 136
4.3 Graph score......Page 143
4.4 Eulerian graphs......Page 148
4.5 Eulerian directed graphs......Page 156
4.6 2-connectivity......Page 161
4.7 Triangle-free graphs: an extremal problem......Page 166
5.1 Definition and characterizations of trees......Page 171
5.2 Isomorphism of trees......Page 177
5.3 Spanning trees of a graph......Page 184
5.4 The minimum spanning tree problem......Page 188
5.5 Jarnik’s algorithm and Boruvka’s algorithm......Page 194
6.1 Drawing in the plane and on other surfaces......Page 200
6.2 Cycles in planar graphs......Page 208
6.3 Euler’s formula......Page 214
6.4 Coloring maps: the four-color problem......Page 224
7.1 Parity arguments......Page 235
7.2 Sperner’s theorem on independent systems......Page 244
7.3 An extremal problem: forbidden four-cycles......Page 251
8.1 The result......Page 257
8.2 A proof via score......Page 258
8.3 A proof with vertebrates......Page 260
8.4 A proof using the Pr¨ufer code......Page 263
8.5 Proofs working with determinants......Page 265
8.6 The simplest proof?......Page 276
9.1 Definition and basic properties......Page 279
9.2 Existence of finite projective planes......Page 289
9.3 Orthogonal Latin squares......Page 295
9.4 Combinatorial applications......Page 299
10.1 Proofs by counting......Page 302
10.2 Finite probability spaces......Page 309
10.3 Random variables and their expectation......Page 319
10.4 Several applications......Page 325
11 Order from disorder: Ramsey’s theorem......Page 335
11.1 A party of six......Page 336
11.2 Ramsey’s theorem for graphs......Page 337
11.3 A lower bound for the Ramsey numbers......Page 339
12.1 Combinatorial applications of polynomials......Page 343
12.2 Calculation with power series......Page 347
12.3 Fibonacci numbers and the golden section......Page 358
12.4 Binary trees......Page 366
12.5 On rolling the dice......Page 371
12.6 Random walk......Page 372
12.7 Integer partitions......Page 375
13.1 Block designs......Page 382
13.2 Fisher’s inequality......Page 387
13.3 Covering by complete bipartite graphs......Page 391
13.4 Cycle space of a graph......Page 394
13.5 Circulations and cuts: cycle space revisited......Page 398
13.6 Probabilistic checking......Page 402
Appendix: Prerequisites from algebra......Page 413
Bibliography......Page 420
Hints to selected exercises......Page 425
Index......Page 451