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Discrete Mathematics - Second Edition
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Chapter 1 Set Theory
1.1 Introduction
1.2 Sets and Elements
1.3 Universal Set and Empty Set
1.4 Subsets
1.5 Venn Diagrams
1.6 Set Operations
1.7 Algebra of Sets and Duality
1.8 Power Sets, Partitions
1.9 Mathematical Induction
1.10 Finite Sets , Counting Principle
Chapter 2 Relations
2.1 Introduction
2.2 Product Sets
2.3 Relations
2.4 Pictorial Representations of Relations
2.5 Composition of Relations
2.6 Types of Relations
2.7 Closure Properties
2.8 Equivalence Relations
2.9 Partial Ordering Relations
2.10 n-ary Relations
Chapter 3 Functions and Algorithms
3.1 Introduction
3.2 Functions
3.3 One-to-one, onto and Invertible functions
3.4 Mathematical Functions, Exponential and Logarithmic Functions
3.5 Sequences, Indexed class of sets
3.6 Recursively Defined Functions
3.7 Cardinality
3.8 Algorithms and Functions
3.9 Complexity of Algorithms
Chapter 4 LOGIC AND PROPOSITIONAL CALCULUS
4.1 Introduction
4.2 Propositions and Compound Propositions
4.3 Basic Logical Operations
4.4 Propositions and Truth Tables
4.5 Tautologies and Contradictions
4.6 Logical Equivalence
4.7 Algebra of Propositions
4.8 Conditional and Biconditional Statements
4.9 Arguments
4.10 Logical Implication
4.11 Propositional Functions, Quantifiers
4.12 Negation of Quantified Statements
Chapter 5 Vectors and Matrices
5.1 Introduction
5.2 Vectors
5.3 Matrices
5.4 Matrix Addition and Scalar Multiplication
5.5 Matrix Multiplication
5.6 Transpose
5.7 Square Matrices
5.8 Invertible (Non singular) Matrices
5.9 Determinants
5.10 Elementary Row Operations, Gaussian Elimination
5.11 Boolean Zero-one Matrices
Chapter 6 Counting
6.1 Introduction, Basic Counting Principles
6.2 Factorial Notation
6.3 Binomial Coefficients
6.4 Permutations
6.5 Combinations
6.6 The Pigeonhole Principle
6.7 The Inclusion-Exclusion Principle
6.8 Ordered and Unordered Partitions
Chapter 7 Probability Theory
7.1 Introduction
7.2 Sample Space and Events
7.3 Finite Probability Spaces
7.4 Conditional Probability
7.5 Independent Events
7.6 Independent Repeated Trials, Binomial Distribution
7.7 Random Variables
Chapter 8 Graph Theory
8.1 Introduction, Data Structures
8.2 Graphs and Multigraphs
8.3 Subgraphs, Isographs and Homeomorphic Isographs
8.4 Paths Connectivity
8.5 The Bridges of Konisberg, Traversible Multigraphs
8.6 Labeled and Weighted Graphs
8.7 Complete , Regular and Bipartite Graphs
8.8 Tree Graphs
8.9 Planar Graphs
8.10 Graph Colorings
8.11 Representing Graphs in Computer Memory
8.12 Graph Algorithms
Chapter 9 Directed Graphs
9.1 Introduction
9.2 Directed Graphs
9.3 Complete and Extended Binary Trees
9.4 Representing Binary Trees in Memory
9.5 Sequential Representation of Directed Graphs
9.6 Warshall's Algorothm Shortest Paths
9.7 Linked representation of Directed Graphs
9.8 Graph Algorithms Depth First and Breadth First Searches
9.9 Directed Cycle Free Graphs, Topological Sort
9.10 Pruning Algorithm for Shortest Path
Chapter 10 Binary Trees
10.1 Introduction
10.2 Binary Trees
10.3 Complete and Extended Binary Trees
10.4 Representing Binary Trees in Memory
10.5 Traversing Binary Trees
10.6 Binary Search Trees
10.7 Priority Queues, Heaps
10.8 Path Lengths, Huffman's Algorithm
10.9 General (Ordered Rooted) Trees Revisited
Chapter 11 Properties of the Integers
11.1 Introduction
11.2 Order and Inequalities, Absolute Value
11.3 Mathematical Induction
11.4 Division Algorithm
11.5 Divisibility Primes
11.6 Greatest Common Divisor, Euclidean Algorithm
11.7 Fundamental Theories of Arithmetic
11.8 Congruence Relation
11.9 Congruence Equations
Chapter 12 Algebraic Systems
12.1 Introduction
12.2 Operations
12.3 Semigroups
12.4 Groups
12.5 Subgroups, Normal Groups and Homomorphisms
12.6 Rings, Internal Domains and Fields
12.7 Polynomials Over a Field
Chapter 13 Languages, Grammars and Machines
13.1 Introduction
13.2 Alphabet, Words, Free Semigroup
13.3 Languages
13.4 Regular Expressions, Regular Languages
13.5 Finite State Automata
13.6 Grammars
13.7 Finite State Machines
13.8 Godel Numbers
13.9 Turing Machines
13.10 Computable Functions
Chapter 14 Ordered Sets and Lattices
14.1 Introduction
14.2 Ordered Sets
14.3 Hasse Diagrams of Partially Ordered Sets
14.4 Consistent Enumeration
14.5 Supremum and Infimum
14.6 Isomorphic (Similar) Ordered Sets
14.7 Well Ordered Sets
14.8 Lattices
14.9 Bounded Lattices
14.10 Distributive Lattices
14.11 Complements, Complemented Lattices
Chapter 15 Boolean Algebra
15.1 Introduction
15.2 Basic Definitions
15.3 Duality
15.4 Basic Theorems
15.5 Boolean Algebras as Lattices
15.6 Representation Theorem
15.7 Sum of Products Form of Sets
15.8 Sum of Products form for Boolean Algebras
15.9 Minimal Boolean Expressions, Prime Implicants
15.10 Logic Gates and Circuits
15.11 Truth Tables, Boolean Functions
15.12 Karnaugh Maps
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