Discrete Structures | by Satinder Bal Gupta | 2010 | ISBN: 9788131805800. Principle of Mathematical Induction. Permutations and Combinations. Matrix Algebra. Propositional Calculus.

January 11th, 2011 | by kutenk |

Discrete Structures
by Satinder Bal Gupta
2010 (646 pages)
ISBN:9788131805800
Including solved questions in each chapter, this book will help students learn the techniques to solve the mathematical and computational problems independently.
Discrete Structures
Preface to the Second Edition
Syllabus
Uttar Pardesh Technical University, Lucknow
Punjab Technical University, Jalandhar Discrete Structures (CS-203)
Punjab Technical University, Jalandhar Computer Mathematical Foundation (MCA-104 (N2)

Chapter 1 – Sets
1.1. Introduction
1.2. Set Formation
1.3. Standard Notations
1.4. Kinds of Sets
1.5. Operations on Sets
1.6. Algebra of Sets
1.7. (a) Principle of Duality for Sets
1.7. (b) Cardinality of a Set
1.8. Cartesian Product of Two Sets
1.9. Partitions of Sets and Venn-Diagrams
1.10. Venn-Diagrams
1.11. Cross Partition
1.12. Countable and Uncountable Sets
1.13. Countable Set
1.14. Uncountable Set
1.15. (a) Minsets or Minterms
1.15. (b) Minset Normal Form
1.16. Maxsets or Maxterms

Chapter 2 – Relations
2.1. Introduction
2.2. Ordered Pair
2.3. Cartesian Product of Sets
2.4. Relation (or Binary Relation)
2.5. Total Number of Relations
2.6. Domain and Range of a Relation
2.7. Inverse Relation
2.8. Types of Relations
2.9. Symmetric Relation
2.10. Transitive Relation
2.11. Antisymmetric Relation
2.12. Equivalence Relation
2.13. Partial Order Relation
2.14. (a) Diagonal Relation
2.14. (b) Ternary Relation
2.15. Closure Properties of Relations
2.16. Composition of Relations
2.17. Directed Graph or Digraph of a Relation
2.18. Representation of Relations
2.19. Matrix Representation of a Relation R
2.20. Paths in Relations
2.21. Computation of MR2 Using MR
2.22. Computation of the Matrix of Rn
2.23. Computation of the Matrix of R8
2.24. Computation of the Matrix of R*
2.25. Composition of Paths
2.26. R8 Connective Relation for R
2.27. Warshall’s Algorithm to Find Transitive Closure
2.28. Equivalence Class

Chapter 3 – Functions
3.1. Function
3.2. Function as a Set
3.3. Domain of a Function
3.4. Co-Domain of a Function
3.5. Image of an Element
3.6. Range of a Function
3.7. Representation of a Function
3.8. (a) Everywhere Defined Function
3.8. (b) Graph of a Function
3.8. (c) Function as a Relation
3.9. Types of Functions
3.10. Equal Functions
3.11. (a) Identity Functions
3.11. (b) Constant Map
3.12. Invertible (Inverse) Functions
3.13. Logarithmic Functions
3.14. Permutation Functions
3.15. Inverse of a Permutation
3.16. Composition (Product) of Two Permutations
3.17. Cyclic Permutation
3.18. Disjoint Cycles
3.19. Even and Odd Permutations
3.20. Even Permutation
3.21. Odd Permutation
3.22. Composition of Functions
3.23. Identity Function
3.24. Inverse Function
3.25. Method to Find the Inverse of a Bijection

Chapter 4A – Mathematical Induction
4.1. Principle of Mathematical Induction
4.2. Working Rule
4.3. Peano’s Axioms

Chapter 4B – Basic Counting Principles
4.4. Introduction
4.5. Basic Counting Principles
4.6. Sum Rule
4.7. Product Rule

Chapter 5 – Permutations and Combinations
5.1. Define Factorial n
5.2. Permutation
5.3. Permutation with Restrictions
5.4. Permutations When All of the Objects are Not Distinct
5.5. Permutations with Repeated Objects
5.6. Circular Permutations
5.7. Combination
5.8. Pigeonhole Principle
5.9. Extended Pigeonhole Principle

Chapter 6 – Inclusion-Exclusion Principle
Chapter 7 – Recurrence Relations and Generating Functions
7.1. Introduction
7.2. Recurrence Relations
7.3. Order of a Recurrence Relation
7.4. (a) Degree of Recurrence Relation
7.4. (b) Linear Recurrence Relation
7.5. Formation of Recurrence Relations
7.6. Linear Recurrence Relation of Order n with Constant Coefficients
7.7. Homogeneous Linear Recurrence Relation of Order n.
7.8. Characteristic Equation
7.9. Algorithm for Solving Homogeneous Linear Recurrence Relation of Order n with Constant Coefficients
7.10. Non-Homogeneous Linear Recurrence Relation of Order n with Constant Coefficients
7.11. Algorithm for Solving Non-Homogeneous Linear Recurrence Relation of Order n with Constant Coefficients
7.12. Generating Functions or Numeric Functions
7.13. Generating Functions for Some Standard Sequences
7.14. Solution of Recurrence Relation by the Method of Generating Functions

Chapter 8 – Groups
8.1. Introduction
8.2. Algebraic Structure
8.3. Binary Operation
8.4. Tables of Operation
8.5. Properties of Binary Operations
8.6. Semi-Group
8.7. Subsemi-Group
8.8. Free Semi-Group
8.9. Congruence Relation
8.10. Monoid
8.11. Submonoid
8.12. Group
8.13. Zm the Integers Modulo m
8.14. Finite and Infinite Group
8.15. Order of Group
8.16. Subgroup
8.17. Abelian Group
8.18. (a) Cosets
8.18. (b) Coset Representative System for H in G
8.19. Index of a Subgroup
8.20. Normal Subgroup
8.21. Quotient Group
8.22. Cyclic Group
8.23. (a) Group Homomorphism
8.23. (b) Group Isomorphism
8.24. Kernel f
8.25. Image f
8.26. Elementary Applications of Groups in Coding Theory
8.27. Message
8.28. Word
8.29. Encoding Function
8.30. Weight
8.31. Parity Check Code
8.32. Hamming Distance
8.33. Minimum Distance
8.34. Group Codes
8.35. Applications of Groups in Coding Theory

Chapter 9 – Rings
9.1. Ring
9.2. Commutative Ring
9.3. Ring with Unity
9.4. Finite and Infinite Ring
9.5. Ring with Zero Divisors
9.6. Ring without Zero Divisors
9.7. Direct Product of Rings
9.8. Ring Isomorphism
9.9. Subring
9.10. Units
9.11. Integral Domain
9.12. Field
9.13. Gaussian Integers
9.14. Ideals
9.15. Sum of Ideals
9.16. Quotient Ring
9.17. Principal Ideal
9.18. Principal Ideal Domain (P.I.D.)
9.19. Euclidean Domain
9.20. Associate

Chapter 10A – Posets
10.1. Partially Ordered Relation
10.2. Comparable Elements
10.3. Non-Comparable Elements
10.4. Linearly Ordered Set or Totally Ordered Set
10.5. Hasse Diagrams
10.6. Elements of Poset
10.7. Greatest Element
10.8. Least Element
10.9. Upper Bound
10.10. Lower Bound
10.11. Least Upper Bound (Supremum)
10.12. Greatest Lower Bound (Infimum)

Chapter 10B – Lattices
10.13. Lattice
10.14. Join
10.15. Meet
10.16. Bounded Lattices
10.17. Sublattices
10.18. Isomorphic Lattices
10.19. Distributive Lattice
10.20. Join-Irreducible
10.21. Meet-Irreducible
10.22. Atom
10.23. Antiatom
10.24. Complemented Lattices
10.25. Modular Lattice
10.26. Direct Product of Lattices

Chapter 10C – Boolean Algebra
10.27. Introduction
10.28. (a) Unary Operation
10.28. (b) Binary Operation
10.28. (c) Boolean Algebra
10.28. (d) Alternate Definition of Boolean Algebra
10.29. Sub-Algebra
10.30. Atoms of a Boolean Algebra
10.31. Isomorphic Boolean Algebras
10.32. Laws of Boolean Algebra
10.33. Principle of Duality
10.34. Boolean Expression or Boolean Function
10.35. Literal
10.36. Fundamental Product
10.37. Sum-of-Products Form or Sop Form
10.38. Complete Sum-of-Products Form
10.39. Minterm
10.40. Boolean Expansion Theorem
10.41. Disjunctive Normal Form or Sum-of-Products (or Sop) Form
10.42. Conjunctive Normal Form or Products of Sums (or Pos) Form
10.43. (a) Obtaining a Disjunctive Normal Form
10.43. (b) Obtaining a Conjunctive Normal Form
10.44. Prime Implicants
10.45. Karnaugh Map
10.46. Adjacent Fundamental Products
10.47. Karnaugh Map for Two Variables
10.48. Karnaugh Map for Three Variables
10.49. Karnaugh Map for Four Variables
10.50. Looping
10.51. Looping Groups of Two
10.52. Looping Groups of Four
10.53. Looping Groups of Eigths
10.54. Karnaugh Map Method for Finding Prime Implicants and Minimal Form for a Boolean Expression
10.55. Basic Rectangle for a Three Variable K-Map
10.56. Applications of Boolean Algebra to Switching Circuits
10.57. Truth Table for the Switches Connected in Parallel
10.58. Truth Table for the Switches Connected in Series
10.59. Application of Boolean Algebra to Logic Circuits
10.60. Conversion of Boolean Expression to Logic Circuit
10.61. Equivalent Logical Circuits

Chapter 11 – Graphs
11.1. Introduction
11.2. Basic Terminology
11.3. Directed Graph
11.4. (a) Undirected Graphs
11.4. (b) Mixed Graph
11.4. (c) Finite Graph
11.4. (d) Linear Graph
11.4. (e) Discrete or Null Graph
11.5. Simple Graph
11.6. Complement Graph
11.7. (a) Degree
11.7. (b) Indegree and Outdegree
11.8. Source and Sink
11.9. Even and Odd Vertex
11.10. Adjacent Vertices
11.11. Path in a Graph
11.12. Undirected Complete Graph
11.13. Connected Graph
11.14. Disconnected Graph
11.15. Connected Component
11.16. Subgraph
11.17. (a) Spanning Subgraph
11.17. (b) Complement of a Graph
11.17. (c) Complement of a Subgraph
11.18. (a) Cut Set
11.18. (b) Cut Points or Cut Vertices
11.19. Edge Connectivity
11.20. Bridge (Cut Edges)
11.21. Isomorphic Graphs
11.22. Order and Size of Graph
11.23. Homeomorphic Graphs
11.24. Weakly Connected
11.25. Unilaterally Connected Digraph
11.26. Strongly Connected Digraph
11.27. Disconnected Digraph
11.28. Directed Complete Graph
11.29. Labelled Graphs
11.30. Weighted Graphs
11.31. Multiple Edges
11.32. Multigraph
11.33. Traversable Multigraphs
11.34. Representation of Graphs
11.35. Other Important Graphs
11.36. Euler Path
11.37. Euler Circuit
11.38. Euler Graph
11.39. Fleury’s Algorithm
11.40. Hamiltonian Path
11.41. Hamiltonian Circuit
11.42. Hamiltonian Graph
11.43. Rules for Constructing Hamilton Paths and Hamilton Circuits in a Graph
11.44. Regular Graph
11.45. Planar Graph
11.46. Region of a Graph
11.47. Properties of Planar Graphs
11.48. State and Prove Euler’s Theorem
11.49. Non Planar Graphs
11.50. Properties of Non Planar Graphs
11.51. Graph Colouring
11.52. Chromatic Number of G
11.53. Shortest Path in Weighted Graphs
11.54. Dijkstra’s Algorithm

Chapter 12 – Trees
12.1. Introduction
12.2. Tree
12.3. Directed Trees
12.4. Ordered Trees
12.5. Rooted Trees
12.6. Path Length of a Vertex
12.7. Forest
12.8. Binary Tree
12.9. Basic Terminology
12.10. Binary Expression Trees
12.11. Complete Binary Tree
12.12. Full Binary Tree
12.13. Traversing Binary Trees
12.14. Algorithms
12.15. Binary Search Trees
12.16. Inserting into a Binary Search Tree
12.17. Spanning Tree
12.18. Minimum Spanning Tree
12.19. Kruskal’s Algorithm to Find Minimum Spanning Tree

Chapter 13 – Propositional Calculus
13.1: Basic Logic Operations
13.2: Statement
13.3. Proposition
13.4. Propositional Variables
13.5. Truth Table
13.6. Combination of Propositions
13.7. (a) Laws of the Algegbra of Propositions
13.7. (b) Variations in Conditional Statement
13.8. Principle of Duality
13.9. Logical Implication
13.10. Logically Equivalence of Propositions
13.11. Tautologies
13.12. Contradiction
13.13. Contingency
13.14. Functionally Complete Sets of Connectives
13.15. Argument
13.16. Proof of Validity
13.17. Quantifiers
13.18. Existential Quantifier
13.19. Universal Quantifier
13.20. Negation of Quantified Propositions
13.21. Propositions with Multiple Quantifiers

Chapter 14 – Matrix Algebra
14.1. (a) Matrix
14.1. (b) Kinds of Matrices
14.2. Addition of Matrices
14.3. Multiplication of a Matrix by a Scalar
14.4. Properties of Matrix Addition
14.5. Matrix Multiplication
14.6. Properties of Matrix Multiplication
14.7. Transpose of a Matrix
14.8. Properties of Transpose of a Matrix
14.9. Symmetric Matrix
14.10. Skew-Symmetric Matrix (or Anti-Symmetric Matrix)
14.11. Every Square Matrix Can Uniquely be Expressed as the Sum of a Symmetric Matrix and a Skew-Symmetric Matrix
14.12. Orthogonal Matrix
14.13. For Any Two Orthogonal Matrices A and B, Show that AB is an Orthogonal Matrix
14.14. Adjoint of a Square Matrix
14.15. An Important Relation between a Square Matrix A and adj A
14.16. Singular Matrices and Non-Singular Matrices
14.17. Inverse (or Reciprocal) of a Square Matrix
14.18. The Inverse of a Square Matrix, if it Exists, is Unique
14.19. The Necessary and Sufficient Condition for a Square Matrix A to Possess Inverse is that | A | ? 0 (i.e., A is Non-Singular)
14.20. If A is Invertible, then so is A-1 and (A-1)-1 = A
14.21. If A and B be Two Non-Singular Square Matrices of the Same Order, then (AB)-1 = B-1 A-1
14.22. If A is a Non-Singular Square Matrix, then so is A’ and (A’)-1 = (A-1)’
14.23. If A and B are Two Non-Singular Square Matrices of the Same Order, Then
14.24. Solution of Simultaneous Linear Equations by Matrix Inversion Method or Matrix Method
14.25. If A is Non-Singular Matrix, then the System of Equations AX = B Has a Unique Solution Given by X = A-1 B
14.26. Rank of a Matrix
14.27. To Determine the Rank of a Matrix A
14.28. Echelon form of a Matrix
14.29. Rank of a Matrix by Using Echelon Form
14.30. Solution of a System of Linear Equations (Rank Method) or (Gauss Jordan Method)
14.31. Theorem : If A is a Non-Singular Matrix, then the Matrix Equation AX = B has a Unique Solution
14.32. Gauss Elimination Method

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