Book:Steven G. Krantz/Discrete Mathematics Demystified

Subject Matter

 * Discrete Mathematics

Contents

 * Preface


 * CHAPTER 1 Logic
 * 1.1 Sentential Logic
 * 1.2 "And" and "Or"
 * 1.3 "Not"
 * 1.4 "If-Then"
 * 1.5 Contrapositive, Converse, and "Iff"
 * 1.6 Quantifiers
 * Exercises


 * CHAPTER 2 Methods of Mathematical Proof
 * 2.1 What Is a Proof?
 * 2.2 Direct Proof
 * 2.3 Proof by Contradiction
 * 2.4 Proof by Induction
 * 2.5 Other Methods of Proof
 * Exercises


 * CHAPTER 3 Set Theory
 * 3.1 Rudiments
 * 3.2 Elements of Set Theory
 * 3.3 Venn Diagrams
 * 3.4 Further Ideas in Elementary Set Theory
 * Exercises


 * CHAPTER 4 Functions and Relations
 * 4.1 A Word About Number Systems
 * 4.2 Relations and Functions
 * 4.3 Functions
 * 4.4 Combining Functions
 * 4.5 Types of Functions
 * Exercises


 * CHAPTER 5 Number Systems
 * 5.1 Preliminary Remarks
 * 5.2 The Natural Number System
 * 5.3 The Integers
 * 5.4 The Rational Numbers
 * 5.5 The Real Number System
 * 5.6 The Nonstandard Real Number System
 * 5.7 The Complex Numbers
 * 5.8 The Quaternions, the Cayley Numbers, and Beyond
 * Exercises


 * CHAPTER 6 Counting Arguments
 * 6.1 The Pigeonhole Principle
 * 6.2 Orders and Permutations
 * 6.3 Choosing and the Binomial Coefficients
 * 6.4 Other Counting Arguments
 * 6.5 Generating Functions
 * 6.6 A Few Words About Recursion Relations
 * 6.7 Probability
 * 6.8 Pascal's Triangle
 * 6.9 Ramsey Theory
 * Exercises


 * CHAPTER 7 Matrices
 * 7.1 What Is a Matrix?
 * 7.2 Fundamental Operations on Matrices
 * 7.3 Gaussian Elimination
 * 7.4 The Inverse of a Matrix
 * 7.5 Markov Chains
 * 7.6 Linear Programming
 * Exercises


 * CHAPTER 8 Graph Theory
 * 8.1 Introduction
 * 8.2 Fundamental Ideas of Graph Theory
 * 8.3 Application to the Königsberg Bridge Problem
 * 8.4 Coloring Problems
 * 8.5 The Traveling Salesman Problem
 * Exercises


 * CHAPTER 9 Number Theory
 * 9.1 Divisibility
 * 9.2 Primes
 * 9.3 Modular Arithmetic
 * 9.4 The Concept of a Group
 * 9.5 Some Theorems of Fermat
 * Exercises


 * CHAPTER 10 Cryptography
 * 10.1 Background on Alan Turing
 * 10.2 The Turing Machine
 * 10.3 More on the Life of Alan Turing
 * 10.4 What Is Cryptography?
 * 10.5 Encruption by Way of Affine Transformations
 * 10.6 Digraph Transformations
 * 10.7 RSA Encryption
 * Exercises


 * CHAPTER 11 Boolean Algebra
 * 11.1 Description of Boolean Algebra
 * 11.2 Axioms of Boolean Algebra
 * 11.3 Theorems in Boolean Algebra
 * 11.4 Illustration of the Use of Boolean Logic
 * Exercises


 * CHAPTER 12 Sequences
 * 12.1 Introductory Remarks
 * 12.2 Infinite Sequences of Real Numbers
 * 12.3 The Tail of a Sequence
 * 12.4 A Basic Theorem
 * 12.5 The Pinching Theorem
 * 12.6 Some Special Sequences
 * Exercises


 * CHAPTER 13 Series
 * 13.1 Fundamental Ideas
 * 13.2 Some Examples
 * 13.3 The Harmonic Series
 * 13.4 Series of Powers
 * 13.5 Repeating Decimals
 * 13.6 An Application
 * 13.7 A Basic Test for Convergence
 * 13.8 Basic Properties of Series
 * 13.9 Geometric Series
 * 13.10 Convergence of $p$-Series
 * 13.11 The Comparison Test
 * 13.12 A Test for Divergence
 * 13.13 The Ratio Test
 * 13.14 The Root Test
 * Exercises


 * Final Exam
 * Solutions to Exercises
 * Bibliography
 * Index