Book:Steven G. Krantz/Discrete Mathematics Demystified

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Steven G. Krantz: Discrete Mathematics Demystified

Published $2009$, McGraw-Hill

ISBN 978-0071549486.


Subject Matter


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