Fundamental approach to discrete mathematics

Cover
......Page 1
Preface
......Page 8
Contents......Page 10
1.1 Statement (Proposition)
......Page 16
1.2 Logical Connectives
......Page 17
1.3 Conditional
......Page 18
1.4 Bi-Conditional
......Page 19
1.7 Contra Positive
......Page 20
1.10 NOR
......Page 21
1.13 Satisfiable
......Page 22
1.16 Mathematical Induction
......Page 23
Solved Examples
......Page 24
Exercises
......Page 30
2.1 Sets
......Page 33
2.2 Types of Sets
......Page 34
2.3 Cardinality of a Set
......Page 35
2.4 Subset and Superset
......Page 36
2.7 Operations on Sets
......Page 37
2.9 Application of Set Theory
......Page 42
2.10 Product of Sets
......Page 43
Solved Examples
......Page 44
Exercises
......Page 53
3.1 Binary Relation......Page 55
3.2 Inverse Relation
......Page 56
3.3 Graph of Relation
......Page 57
3.4 Kind of Relation
......Page 58
3.7 Identity Relation
......Page 59
3.10 Composition of Relations
......Page 60
3.11 Types of Relations
......Page 62
3.12 Types of Relations and Relation Matrix
......Page 64
3.13 Equivalence Relation
......Page 66
3.14 Partial Order Relation
......Page 67
3.15 Total Order Relation
......Page 68
3.16 Closures of Relations
......Page 69
3.17 Equivalence Classes
......Page 70
3.18 Partitions......Page 71
Solved Examples
......Page 72
Exercises......Page 80
4.1 Function
......Page 84
4.3 Types of Function
......Page 86
4.4 Graph of Function
......Page 87
4.5 Composition of Functions
......Page 89
4.6 Inverse Function
......Page 91
4.7 Some Important Functions
......Page 92
4.8 Hash Function
......Page 94
Solved Examples
......Page 95
Exercises
......Page 104
5.2 Algebraic Structure
......Page 108
5.3 Group
......Page 109
5.4 Subgroup
......Page 113
5.5 Cyclic Group
......Page 116
5.7 Homomorphism
......Page 118
Solved Examples
......Page 119
Exercises
......Page 132
6.1 Terminologies
......Page 133
6.5 Distance Between the Code Words
......Page 134
6.6 Error Correction for Block Code
......Page 135
Solved Examples
......Page 136
Exercises
......Page 138
7.1 Ring
......Page 139
7.2 Special Types of Ring
......Page 141
7.5 Division Ring
......Page 143
7.6 Field
......Page 144
7.7 The Pigeonhole Principle
......Page 145
7.8 Characteristics of a Ring
......Page 146
7.9 Sub Ring
......Page 147
7.10 Homomorphism
......Page 148
7.11 Kernel of Homomorphism of Ring
......Page 149
Solved Examples
......Page 150
Exercises
......Page 159
8.1 Gates
......Page 161
8.2 More Logic Gates
......Page 163
8.3 Combinatorial Circuit
......Page 165
8.4 Boolean Expression
......Page 166
8.5 Equivalent Combinatorial Circuits
......Page 167
8.6 Boolean Algebra
......Page 168
8.7 Dual of a Statement
......Page 171
8.8 Boolean Function
......Page 172
8.9 Various Normal Forms
......Page 173
Solved Examples
......Page 174
Exercises
......Page 183
9.1 Lattices
......Page 188
9.2 Hasse Diagram
......Page 189
9.3 Principle of Duality
......Page 190
9.4 Distributive Lattice
......Page 192
9.5 Bounded Lattice
......Page 194
9.6 Complemented Lattice
......Page 195
Solved Examples
......Page 196
Exercises
......Page 199
10.1 Graph
......Page 201
10.2 Kinds of Graph
......Page 203
10.4 Weighted Graph
......Page 204
10.6 Path
......Page 205
10.9 Cycle
......Page 206
10.11 Acyclic Graph
......Page 207
10.12 Matrix Representation of Graphs
......Page 208
10.13 Connected Graph
......Page 211
10.15 Bipartite Graph
......Page 212
10.16 Subgraph
......Page 213
10.17 Walks
......Page 214
10.18 Operations on Graphs
......Page 215
10.19 Fusion of Graphs
......Page 217
Solved Examples
......Page 221
Exercises
......Page 235
11.1 Tree
......Page 241
11.2 Fundamental Terminologies
......Page 242
11.3 Binary Tree
......Page 243
11.4 Bridge
......Page 244
11.6 Eccentricity
......Page 245
11.9 Central Point and Centre
......Page 246
11.10 Spanning Tree
......Page 247
11.11 Searching Algorithms
......Page 249
11.12 Shortest Path Algorithms
......Page 253
11.13 Cut Vertices
......Page 255
11.14 Euler Graph
......Page 256
11.16 Closure of a Graph
......Page 258
11.17 Travelling Salesman Problem
......Page 259
Solved Examples
......Page 260
Exercises
......Page 275