Book:Steven G. Krantz/Discrete Mathematics Demystified

Steven G. Krantz: Discrete Mathematics Demystified

Published $\text {2009}$, McGraw-Hill

ISBN 978-0071549486

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