9780387955858

Discrete Mathematics : Elementary and Beyond

by ; ;
  • ISBN13:

    9780387955858

  • ISBN10:

    0387955852

  • Format: Paperback
  • Copyright: 2/1/2003
  • Publisher: Springer Verlag

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Supplemental Materials

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Summary

Discrete mathematics is quickly becoming one of the most important areas of mathematical research, with applications to cryptography, linear programming, coding theory and the theory of computing. This book is aimed at undergraduate mathematics and computer science students interested in developing a feeling for what mathematics is all about, where mathematics can be helpful, and what kinds of questions mathematicians work on. The authors discuss a number of selected results and methods of discrete mathematics, mostly from the areas of combinatorics and graph theory, with a little number theory, probability, and combinatorial geometry. Wherever possible, the authors use proofs and problem solving to help students understand the solutions to problems. In addition, there are numerous examples, figures and exercises spread throughout the book. L?!szl?? Lov?!sz is a Senior Researcher in the Theory Group at Microsoft Corporation. He is a recipient of the 1999 Wolf Prize and the G??del Prize for the top paper in Computer Science. J??zsef Pelik?!n is Professor of Mathematics in the Department of Algebra and Number Theory at E??tv??s Lor?!nd University, Hungary. In 2002, he was elected Chairman of the Advisory Board of the International Mathematical Olympiad. Katalin Vesztergombi is Senior Lecturer in the Department of Mathematics at the University of Washington.

Author Biography

 L+íszl+¦ Lov+ísz is a Senior Researcher in the Theory Group at Microsoft Corporation. He is a recipient of the 1999 Wolf Prize and the G+¦del Prize for the top paper in Computer Science. J+¦zsef Pelik+ín is Professor of Mathematics in the Department of Algebra and Number Theory at E+¦tv+¦s Lor+índ University, Hungary. In 2002, he was elected Chairman of the Advisory Board of the International Mathematical Olympiad. Katalin Vesztergombi is Senior Lecturer in the Department of Mathematics at the University of Washington.

Table of Contents

Preface v
Let's Count!
1(24)
A Party
1(3)
Sets and the Like
4(5)
The Number of Subsets
9(5)
The Approximate Number of Subsets
14(1)
Sequences
15(2)
Permutations
17(2)
The Number of Ordered Subsets
19(1)
The Number of Subsets of a Given Size
20(5)
Combinatorial Tools
25(18)
Induction
25(5)
Comparing and Estimating Numbers
30(2)
Inclusion-Exclusion
32(2)
Pigeonholes
34(3)
The Twin Paradox and the Good Old Logarithm
37(6)
Binomial Coefficients and Pascal's Triangle
43(22)
The Binomial Theorem
43(2)
Distributing Presents
45(1)
Anagrams
46(2)
Distributing Money
48(1)
Pascal's Triangle
49(1)
Identities in Pascal's Triangle
50(4)
A Bird's-Eye View of Pascal's Triangle
54(3)
An Eagle's-Eye View: Fine Details
57(8)
Fibonacci Numbers
65(12)
Fibonacci's Exercise
65(3)
Lots of Identities
68(3)
A Formula for the Fibonacci Numbers
71(6)
Combinatorial Probability
77(10)
Events and Probabilities
77(2)
Independent Repetition of an Experiment
79(1)
The Law of Large Numbers
80(3)
The Law of Small Numbers and the Law of Very Large Numbers
83(4)
Integers, Divisors, and Primes
87(38)
Divisibility of Integers
87(1)
Primes and Their History
88(2)
Factorization into Primes
90(3)
On the Set of Primes
93(4)
Fermat's ``Little'' Theorem
97(2)
The Euclidean Algorithm
99(6)
Congruences
105(2)
Strange Numbers
107(7)
Number Theory and Combinatorics
114(3)
How to Test Whether a Number is a Prime?
117(8)
Graphs
125(16)
Even and Odd Degrees
125(5)
Paths, Cycles, and Connectivity
130(5)
Eulerian Walks and Hamiltonian Cycles
135(6)
Trees
141(16)
How to Define Trees
141(2)
How to Grow Trees
143(3)
How to Count Trees?
146(2)
How to Store Trees
148(5)
The Number of Unlabeled Trees
153(4)
Finding the Optimum
157(8)
Finding the Best Tree
157(4)
The Traveling Salesman Problem
161(4)
Matchings in Graphs
165(14)
A Dancing Problem
165(2)
Another matching problem
167(2)
The Main Theorem
169(2)
How to Find a Perfect Matching
171(8)
Combinatorics in Geometry
179(10)
Intersections of Diagonals
179(2)
Counting regions
181(3)
Convex Polygons
184(5)
Euler's Formula
189(8)
A Planet Under Attack
189(3)
Planar Graphs
192(2)
Euler's Formula for Polyhedra
194(3)
Coloring Maps and Graphs
197(14)
Coloring Regions with Two Colors
197(2)
Coloring Graphs with Two Colors
199(3)
Coloring graphs with many colors
202(2)
Map Coloring and the Four Color Theorem
204(7)
Finite Geometries, Codes, Latin Squares, and Other Pretty Creatures
211(28)
Small Exotic Worlds
211(6)
Finite Affine and Projective Planes
217(3)
Block Designs
220(4)
Steiner Systems
224(5)
Latin Squares
229(3)
Codes
232(7)
A Glimpse of Complexity and Cryptography
239(12)
A Connecticut Class in King Arthur's Court
239(3)
Classical Cryptography
242(2)
How to Save the Last Move in Chess
244(2)
How to Verify a Password---Without Learning it
246(1)
How to Find These Primes
246(1)
Public Key Cryptography
247(4)
Answers to Exercises
251(36)
Index 287

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