Invitation to Discrete Mathematics

ISBN-10: 0198570422
ISBN-13: 9780198570424
Edition: 2nd 2008
List price: $71.00
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Description: 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  More...

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Book details

List price: $71.00
Edition: 2nd
Copyright year: 2008
Publisher: Oxford University Press, Incorporated
Publication date: 12/15/2008
Binding: Paperback
Pages: 456
Size: 6.00" wide x 9.00" long x 1.00" tall
Weight: 1.760
Language: English

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.

Jiri Matousek received his PhD in Mathematics from the Charles University in Prague in 1990 and is now Professor of Computer Science at Charles University Prague. He has held several visiting positions at universities in the U.S., Germany, Switzerland, Japan, and other countries. Humboldt Research Fellow in 1992 (Free University Berlin). Prize for Young Mathematicians of the 2nd European Congress of Mathematics in Budapest in 1996, speaker at the ICM 1998. Jaroslav Nesetril received his PhD from the Charles University in Prague in 1975 and is now Professor of Mathematics at Charles University Prague. He has held several visiting positions abroad (U.S.A., Canada, Germany). Currently he is the head of the Centre for Theoretical Computer Science (ITI) at Charles University and the director of the international center for Discrete Mathematics, Theoretical Computer Science and Their Applications (DIMATIA).

Introduction and basic concepts
An assortment of problems
Numbers and sets: notation
Mathematical induction and other proofs
Functions
Relations
Equivalences and other special types of relations
Orderings
Orderings and how they can be depicted
Orderings and linear orderings
Ordering by inclusion
Large implies tall or wide
Combinatorial counting
Functions and subsets
Permutations and factorials
Binomial coefficients
Estimates: an introduction
Estimates: the factorial function
Estimates: binomial coefficients
Inclusion-exclusion principle
The hatcheck lady & co.
Graphs: an introduction
The notion of a graph; isomorphism
Subgraphs, components, adjacency matrix
Graph score
Eulerian graphs
Eulerian directed graphs
2-connectivity
Triangle-free graphs: an extremal problem
Trees
Definition and characterizations of trees
Isomorphism of trees
Spanning trees of a graph
The minimum spanning tree problem
Jarn�k's algorithm and Borůvka's algorithm
Drawing graphs in the plane
Drawing in the plane and on other surfaces
Cycles in planar graphs
Euler's formula
Coloring maps: the four-color problem
Double-counting
Parity arguments
Sperner's theorem on independent systems
An extremal problem: forbidden four-cycles
The number of spanning trees
The result
A proof via score
A proof with vertebrates
A proof using the Pr�fer code
Proofs working with determinants
The simplest proof?
Finite projective planes
Definition and basic properties
Existence of finite projective planes
Orthogonal Latin squares
Combinatorial applications
Probability and probabilistic proofs
Proofs by counting
Finite probability spaces
Random variables and their expectation
Several applications
Order from disorder: Ramsey's theorem
A party of six
Ramsey's theorem for graphs
A lower bound for the Ramsey numbers
Generating functions
Combinatorial applications of polynomials
Calculation with power series
Fibonacci numbers and the golden section
Binary trees
On rolling the dice
Random walk
Integer partitions
Applications of linear algebra
Block designs
Fisher's inequality
Covering by complete bipartite graphs
Cycle space of a graph
Circulations and cuts: cycle space revisited
Probabilistic checking
Appendix: Prerequisites from algebra
Bibliography
Hints to selected exercises
Index

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