Discrete Mathematics

Essentials and Applications

1st Edition - April 29, 2022
Author: Ali Grami
Language: English
Paperback ISBN:
9 7 8 - 0 - 1 2 - 8 2 0 6 5 6 - 0
eBook ISBN:
9 7 8 - 0 - 1 2 - 8 2 0 9 0 9 - 7

Discrete Mathematics: Essentials and Applications offers a comprehensive survey of the area, particularly concentrating on the basic principles and applications of Discrete Mathem… Read more

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Discrete Mathematics: Essentials and Applications offers a comprehensive survey of the area, particularly concentrating on the basic principles and applications of Discrete Mathematics. This up-to-date text provides proofs of significance, keeping the focus on numerous relevant examples and many pertinent applications. Written in a simple and clear tone, the title features insightful descriptions and intuitive explanations of all complex concepts and ensures a thorough understanding of the subject matter.

Cover image
Title page
Table of Contents
Copyright
Dedication
Preface
Acknowledgments
Chapter 1. Propositional Logic
1.1. Propositions
1.2. Basic Logical Operators
1.3. Conditional Statements
1.4. Propositional Equivalences
1.5. Logic Puzzles
Chapter 2. Predicate Logic
2.1. Predicates
2.2. Quantifiers
2.3. Negations of Quantified Statements
2.4. Nested Quantifiers
Chapter 3. Rules of Inference
3.1. Valid Arguments
3.2. Rules of Inference for Propositional Logic
3.3. Rules of Inference for Predicate Logic
3.4. Fallacies
Chapter 4. Proof Methods
4.1. Terminology
4.2. Proofs of Equivalence
4.3. Proof by Counterexample
4.4. Vacuous Proofs and Trivial Proofs
4.5. Direct Proofs
4.6. Proofs by Contraposition and Proofs by Contradiction
4.7. Proof by Cases and Proofs by Exhaustion
4.8. Existence Proofs: Constructive Proofs and Nonconstructive Proofs
4.9. Proof of a Disjunction
4.10. Uniqueness Proofs
Chapter 5. Sets
5.1. Definitions and Notation
5.2. Set Operations
5.3. Set Identities and Methods of Proof
5.4. Cardinality of Sets
5.5. Computer Representation of Sets
5.6. Multisets
5.7. Fuzzy Sets
5.8. Paradoxes in Set Theory
Chapter 6. Matrices
6.1. Definitions and Special Matrices
6.2. Matrix Addition and Scalar Multiplication
6.3. Matrix Multiplication
6.4. Matrix Inversion
6.5. Zero-One Matrix
6.6. Applications of Matrices
Chapter 7. Functions
7.1. Basic Definitions
7.2. Special Functions
7.3. One-to-One and Onto Functions
7.4. Compositions of Functions
Chapter 8. Boolean Algebra
8.1. Basic Definitions
8.2. Boolean Expressions and Boolean Functions
8.3. Identities of Boolean Algebra
8.4. Representing Boolean Functions
8.5. Functional Completeness
8.6. Logic Gates
8.7. Minimization of Combinational Circuits
Chapter 9. Relations
9.1. Relations on Sets
9.2. Properties of Relations
9.3. Representations of Relations
9.4. Operations on Relations
9.5. Closure Properties
9.6. Equivalence Relations
9.7. Partial Orderings
9.8. Relational Databases
Chapter 10. Number Theory
10.1. Numeral Systems
10.2. Divisibility
10.3. Prime Numbers
10.4. Greatest Common Divisors and Least Common Multiples
10.5. Divisibility Test
10.6. Congruences
10.7. Representations of Integers
10.8. Binary Operations
Chapter 11. Cryptography
11.1. Classical Cryptography
11.2. Modern Cryptography
11.3. Private-Key Cryptography
11.4. Public-Key Cryptography
11.5. The RSA Cryptosystem
Chapter 12. Algorithms
12.1. Algorithm Requirements
12.2. Algorithmic Paradigms
12.3. Complexity of Algorithms
12.4. Measuring Algorithm Efficiency
12.5. Sorting Algorithms
12.6. Search Algorithms
Chapter 13. Induction
13.1. Deductive Reasoning and Inductive Reasoning
13.2. Mathematical Induction
13.3. Applications of Mathematical Induction
13.4. Strong Induction
13.5. The Well-Ordering Principle
Chapter 14. Recursion
14.1. Sequences
14.2. Recursively Defined Functions
14.3. Recursive Algorithms
14.4. Solving Recurrence Relations by Iteration
14.5. Solving Linear Homogeneous Recurrence Relations with Constant Coefficients
14.6. Solving Linear Nonhomogeneous Recurrence Relations with Constant Coefficients
14.7. Solving Recurrence Relations Using Generating Functions
Chapter 15. Counting Methods
15.1. Basic Rules of Counting
15.2. The Pigeonhole Principle
15.3. Random Arrangements and Selections
15.4. Permutations and Combinations
15.5. Applications
Chapter 16. Discrete Probability
16.1. Basic Terminology
16.2. The Axioms of Probability
16.3. Joint Probability and Conditional Probability
16.4. Statistically Independent Events and Mutually Exclusive Events
16.5. Law of Total Probability and Bayes’ Theorem
16.6. Applications in Probability
Chapter 17. Discrete Random Variables
17.1. The Cumulative Distribution Function
17.2. The Probability Mass Function
17.3. Expected Values
17.4. Conditional Distributions
17.5. Upper Bounds on Probability
17.6. Special Random Variables and Their Applications
Chapter 18. Graphs
18.1. Basic Definitions and Terminology
18.2. Types of Graphs
18.3. Graph Representation and Isomorphism
18.4. Connectivity
18.5. Euler Circuits and Hamilton Circuits
18.6. Shortest-Path Problem
Chapter 19. Trees
19.1. Basic Definitions and Terminology
19.2. Tree Traversal
19.3. Spanning Trees
19.4. Minimum Spanning Trees
19.5. Applications of Trees
Chapter 20. Finite-State Machines
20.1. Types of Finite-State Machines
20.2. Finite-State Machines with No Output
20.3. Finite-State Machines with Output
List of Symbols
Glossary of Terms
Bibliography
Answers/Hints to Exercises
Index

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Ali Grami