Preface
0 Set Theory and Logic
0.1 Introduction to Set Theory
0.2 Functions and Relations
0.3 Inductive Proofs and Recursive Definitions
0.4 The Language of Logic
0.5 Notes and References
0.6 Exercises
1 Combinatorics
1.1 Two Basic Counting Rules
1.2 Permutations
1.3 Combinations
1.4 More on Permutations and Combinations
1.5 The Pigeonhole Principle
1.6 The Inclusion-Exclusion Principle
1.7 Summary of Results in Combinatorics
1.8 Notes and References
1.9 Exercises
2 Generating Functions
2.1 Introduction
2.2 Ordinary Generating Functions
2.3 Exponential Generating Functions
2.4 Notes and References
2.5 Exercises
3 Recurrence Relations
3.1 Introduction
3.2 Homogeneous Recurrence Relations
3.3 Inhomogeneous Recurrence Relations
3.4 Recurrence Relations and Generating Functions
3.5 Analysis of Alogorithms
3.6 Notes and References
3.7 Exercises
4 Graphs and Digraphs
4.1 Introduction
4.2 Adjacency Matrices and Incidence Matrices
4.3 Joining in Graphs
4.4 Reaching in Digraphs
4.5 Testing Connectedness
4.6 Strong Orientation of Graphs
4.7 Notes and References
4.8 Exercises
5 More on Graphs and Digraphs
5.1 Eulerian Paths and Eulerian Circuits
5.2 Coding and de Bruijn Digraphs
5.3 Hamiltonian Paths and Hamiltonian Cycles
5.4 Applications of Hamiltonian Cycles
5.5 Vertex Coloring and Planarity of Graphs
5.6 Notes and References
5.7 Exercises
6 Trees and Their Applications
6.1 Definitions and Properties
6.2 Spanning Trees
6.3 Binary Trees
6.4 Notes and References
6.5 Exercises
7 Spanning Tree Problems
7.1 More on Spanning Trees
7.2 Kruskal's Greedy Algorithm
7.3 Prim's Greedy Algorithm
7.4 Comparison of the Two Algorithms
7.5 Notes and References
7.6 Exercises
8 Shortest Path Problems
8.1 Introduction
8.2 Dijkstra's Algorithm
8.3 Floyd-Warshall Algorithm
8.4 Comparison of the Two Algorithms
8.5 Notes and References
8.6 Exercises
Appendix What is NP-Completeness?
A.1 Problems and Their Instances
A.2 The Size of an Instance
A.3 Algorithm to Solve a Problem
A.4 Complexity of an Algorithm
A.5 "The "Big Oh" or the O(·) Notation"
A.6 Easy Problems and Difficult Problems
A.7 The Class P and the Class NP
A.8 Polynomial Transformations and NP-Completeness
A.9 Coping with Hard Problems
Bibliography
Answers to Selected Exercises
Index

This concise text offers an introduction to discrete mathematics for undergraduate students in computer science and mathematics. Mathematics educators consider it vital that their students be exposed to a course in discrete methods that introduces them to combinatorial mathematics and to algebraic and logical structures focusing on the interplay between computer science and mathematics. The present volume emphasizes combinatorics, graph theory with applications to some stand network optimization problems, and algorithms to solve these problems.
Chapters 0?3 cover fundamental operations involving sets and the principle of mathematical induction, and standard combinatorial topics: basic counting principles, permutations, combinations, the inclusion-exclusion principle, generating functions, recurrence relations, and an introduction to the analysis of algorithms. Applications are emphasized wherever possible and more than 200 exercises at the ends of these chapters help students test their grasp of the material.
Chapters 4 and 5 survey graphs and digraphs, including their connectedness properties, applications of graph coloring, and more, with stress on applications to coding and other related problems. Two important problems in network optimization ― the minimal spanning tree problem and the shortest distance problem ― are covered in the last two chapters. A very brief nontechnical exposition of the theory of computational complexity and NP-completeness is outlined in the appendix.