Discrete Mathematical Structures: A Succinct Foundation

✍ Scribed by B. V. Senthil Kumar; Hemen Dutta

• Publisher: CRC Press
• Year: 2020
• Tongue: English
• Leaves: 275
• Series: Mathematics and Its Applications: Modelling, Engineering, and Social Sciences
• Category: Library

✦ Synopsis

This book contains fundamental concepts on discrete mathematical structures in an easy to understand style so that the reader can grasp the contents and explanation easily. The concepts of discrete mathematical structures have application to computer science, engineering and information technology including in coding techniques, switching circuits, pointers and linked allocation, error corrections, as well as in data networking, Chemistry, Biology and many other scientific areas. The book is for undergraduate and graduate levels learners and educators associated with various courses and progammes in Mathematics, Computer Science, Engineering and Information Technology. The book should serve as a text and reference guide to many undergraduate and graduate programmes offered by many institutions including colleges and universities. Readers will find solved examples and end of chapter exercises to enhance reader comprehension.

Features

Offers comprehensive coverage of basic ideas of Logic, Mathematical Induction, Graph Theory, Algebraic Structures and Lattices and Boolean Algebra Provides end of chapter solved examples and practice problems Delivers materials on valid arguments and rules of inference with illustrations Focuses on algebraic structures to enable the reader to work with discrete structures

✦ Table of Contents

Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Preface
Authors
1 Logics and Proofs
1.1 Introduction
1.2 Proposition
1.3 Compound Propositions
1.4 Truth Table
1.5 Logical Operators
1.5.1 Negation
1.5.2 Conjunction
1.5.3 Disjunction
1.5.4 Molecular Statements
1.5.5 Conditional Statement [If then] [ → ]
1.5.6 Biconditional [If and only if or iff] [↔ or ⇌]
1.5.7 Solved Problems
1.5.8 Tautology
1.5.9 Contradiction
1.5.10 Contingency
1.5.11 Equivalence Formulas
1.5.12 Equivalent Formulas
1.5.13 Duality Law
1.5.14 Tautological Implication
1.5.15 Some More Equivalence Formulas
1.5.16 Solved Problems
1.6 Normal Forms
1.6.1 Principal Disjunctive Normal Form or Sum of Products Canonical Form
1.6.2 Principal Conjunctive Normal Form or Product of Sum Canonical Form
1.6.3 Solved Problems
1.7 Inference Theory
1.7.1 Rules of Inference
1.7.2 Solved Problems
1.8 Indirect Method of Proof
1.8.1 Method of Contradiction
1.8.2 Solved Problems
1.9 Method of Contrapositive
1.9.1 Solved Problems
1.10 Various Methods of Proof
1.10.1 Trivial Proof
1.10.2 Vacuous Proof
1.10.3 Direct Proof
1.11 Predicate Calculus
1.11.1 Quantifiers
1.11.2 Universe of Discourse, Free and Bound Variables
1.11.3 Solved Problems
1.11.4 Inference Theory for Predicate Calculus
1.11.5 Solved Problems
1.12 Additional Solved Problems
2 Combinatorics
2.1 Introduction
2.2 Mathematical Induction
2.2.1 Principle of Mathematical Induction
2.2.2 Procedure to Prove that a Statement P(n) is True for all Natural Numbers
2.2.3 Solved Problems
2.2.4 Problems for Practice
2.2.5 Strong Induction
2.2.6 Well-Ordering Property
2.3 Pigeonhole Principle
2.3.1 Generalized Pigeonhole Principle
2.3.2 Solved Problems
2.3.3 Another Form of Generalized Pigeonhole Principle
2.3.4 Solved Problems
2.3.5 Problems for Practice
2.4 Permutation
2.4.1 Permutations with Repetitions
2.4.2 Solved Problems
2.4.3 Problems for Practice
2.5 Combination
2.5.1 Solved Problems
2.5.2 Problems for Practice
2.5.3 Recurrence Relation
2.5.4 Solved Problems
2.5.5 Linear Recurrence Relation
2.5.6 Homogenous Recurrence Relation
2.5.7 Recurrence Relations Obtained from Solutions
2.6 Solving Linear Homogenous Recurrence Relations
2.6.1 Characteristic Equation
2.6.2 Algorithm for Solving k[sup(th)]-order Homogenous Linear Recurrence Relations
2.6.3 Solved Problems
2.7 Solving Linear Non-homogenous Recurrence Relations
2.7.1 Solved Problems
2.7.2 Problems for Practice
2.8 Generating Functions
2.8.1 Solved Problems
2.8.2 Solution of Recurrence Relations Using Generating Function
2.8.3 Solved Problems
2.8.4 Problems for Practice
2.9 Inclusion—Exclusion Principle
2.9.1 Solved Problems
2.9.2 Problems for Practice
3 Graphs
3.1 Introduction
3.2 Graphs and Graph Models
3.3 Graph Terminology and Special Types of Graphs
3.3.1 Solved Problems
3.3.2 Graph Colouring
3.3.3 Solved Problems
3.4 Representing Graphs and Graph Isomorphism
3.4.1 Solved Problems
3.4.2 Problems for Practice
3.5 Connectivity
3.5.1 Connected and Disconnected Graphs
3.6 Eulerian and Hamiltonian Paths
3.6.1 Hamiltonian Path and Hamiltonian Circuits
3.6.2 Solved Problems
3.6.3 Problems for Practice
3.6.4 Additional Problems for Practice
4 Algebraic Structures
4.1 Introduction
4.2 Algebraic Systems
4.2.1 Semigroups and Monoids
4.2.2 Solved Problems
4.2.3 Groups
4.2.4 Solved Problems
4.2.5 Subgroups
4.2.6 Cyclic Groups
4.2.7 Homomorphisms
4.2.8 Cosets and Normal Subgroups
4.2.9 Solved Problems
4.2.10 Permutation Functions
4.2.11 Solved Problems
4.2.12 Problems for Practice
4.2.13 Rings and Fields
4.2.14 Solved Problems
4.2.15 Problems for Practice
5 Lattices and Boolean Algebra
5.1 Introduction
5.2 Partial Ordering and Posets
5.2.1 Representation of a Poset by Hasse Diagram
5.2.2 Solved Problems
5.2.3 Problems for Practice
5.3 Lattices, Sublattices, Direct Product, Homomorphism of Lattices
5.3.1 Properties of Lattices
5.3.2 Theorems on Lattices
5.3.3 Solved Problems
5.3.4 Problem for Practice
5.4 Special Lattices
5.4.1 Solved Problems
5.4.2 Problems for Practice
5.5 Boolean Algebra
5.5.1 Solved Problems
5.5.2 Problems for Practice
Bibliography
Index