Mathematical Foundations of Computer Science 1st Edition Satyanarayana 1000702715 9781032654195
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Mathematical Foundations of Computer Science 1st Edition Satyanarayana – Ebook Instant Download/Delivery ISBN(s): 9781032654195 , 1000702715
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Table of contents:
Cover
Half Title
Title Page
Copyright Page
Preface
Acknowledgement
Table of Contents
Chapter 1: Propositional Calculus
1.1 Statements and Notations
1.1.1 Subject and Predicate
1.1.2 Notation
1.2 Connectives and Truth Tables
1.2.1 Negation of a Statement
1.2.2 Conjunction (or meet)
1.2.3 Disjunction (or join)
1.2.4 Notation
1.2.5 Draw Truth Table for “p ↑ q”
1.2.6 Draw the Truth Table for “p ↓ q”
1.2.7 Statement Formulas and Truth Tables
1.2.8 Implication (or Conditional Statement)
1.2.9 Biconditional (or Double Implication)
1.2.10 Construction of the Truth Table for p ⇄ q
1.2.11 Well Formed Formulas
1.2.12 The Operation ⊕ or Δ
1.3 Tautology and Contradiction
1.3.1 Tautology
1.3.2 Contradiction (or Fallacy)
1.3.3 Contingency
1.4 Equivalence of Statements/ Formulas
1.4.1 Statements/Formulas
1.4.2 Equivalent Formulas
1.5 Duality Law and Tautological Implication
1.5.1 Duality Law
1.5.2 Tautological Implications
1.5.3 Converse
1.5.4 Inverse
1.5.5 Contrapositive
1.5.6 Some Implications
1.6 Normal Forms
1.6.1 Decision Problem
1.6.2 DNF
1.6.3 How to Find DNF
1.6.4 CNF
1.6.5 Principal Disjunctive Normal Form (PDNF)
1.6.6 Black Box Method (to find PDNF)
1.6.7 Principal Conjunctive Normal Form (PCNF)
1.6.8 How to Find PCNF (by using truth table or through PDNF)
1.7 The Theory of Inference for Statement Calculus
1.7.1 Tautology
1.7.2 Validity using Truth Table
1.7.3 Rules of Inference
1.7.4 Some Implications
1.7.5 Some Equivalences
1.8 Consistency of Premises and Indirect Method of Proof
1.8.1 Indirect Method of Proof
Exercises
Chapter 2: Predicate Calculus
2.1 Predicate Logic
2.1.1 Predicate
2.1.2 2-place Predicate
2.1.3 m-place Predicate
2.1.4 Connectives
2.2 Statement Functions, Variables and Quantifiers
2.2.1 Combined Statements (or Connectives)
2.2.2 Quantifiers
2.2.3 Universal Quantifier
2.2.4 Existential Quantifier
2.2.5 The Universe of Discourse
2.3 Free and Bound Variable
2.4 Inference Theory for the Predicate Calculus
2.4.1 Universal Specification (US)
2.4.2 Universal Generalization (UG)
2.4.3 Existential Specification (ES)
2.4.4 Existential Generalization (EG)
2.4.5 Formulas with More than One Quantifier
Exercises
Chapter 3: Number Theory
3.1 Properties of Integers
3.1.1 Properties of Integers
3.1.2 Z (the set of Integers)
3.2 Division Algorithm (or Division Theorem)
3.3 Greatest Common Divisor (GCD)
3.4 Euclidian Algorithm
3.5 Least Common Multiple
3.6 Testing for Prime Numbers
3.7 Fundamental Theorem of Arithmetic (or Unique Factorization Theorem)
3.8 Modular Arithmatic
3.9 Fermats Theorem
3.9.1 Corollary
Exercises
Chapter 4: Mathematical Induction
4.1 The Principle of Mathematical Induction
4.1.1 The Principle
Exercises
Chapter 5: Set Theory
5.1 Sets
5.1.1 Set
5.1.2 Sub Set
5.1.3 Equity of Sets
5.1.4 Powerset
5.1.5 Notations
5.2 Operations on Sets
5.2.1 Union and Intersection
5.2.2 Properties of Union and Intersection
5.2.3 Symmetric Difference
5.2.4 Set Complement
5.2.5 Cartesian Product
5.2.6 Disjoint Sets
5.2.7 Distributive Laws
5.2.8 Properties Related to Set Operations
5.2.9 Venn Diagrams
5.3 Principle of Inclusion and Exclusion
5.3.1 Principle
5.4 Relations, Binary Operations on Relations and Properties of Binary Relations
5.4.1 Domain and Range
5.4.2 Notation
5.4.3 Some Binary Operations on Relations
5.4.4 Properties of Relations
5.4.5 Composition of Relations
5.4.6 Notation
5.5 Relation Matrix and the Graph of a Relation
5.5.1 Some Properties of Relation Matrices
5.5.2 Pictorial Representation
5.5.3 Properties of the Graph of a Relation
5.6 Partition, Covering of a Set; and Equivalence Relations
5.6.1 Equivalence Relations
5.6.2 Equivalence Class
5.7 Transitive Closure
5.7.1 Properties of Transitive Closure
5.7.2 Representation of Closure Relations in the Form of Matrices
5.8 Compatibility Relations
5.8.1 Definition: (Maximal Compatibility Block)
5.8.2 Composition of Binary Relation
5.9 Partial Order Relations
5.9.1 Zorn’s Lemma
Exercises
Chapter 6: Functions
6.1 Definition and Examples of Functions
6.2 Types of Functions (including Bijective Functions)
6.2.1 Observation
6.2.2 Geometric Interpretation
6.2.3 Properties
6.2.4 Some Rules on Floor and Ceiling Functions
6.3 Composition of Functions and Inverse Functions
6.3.1 Observation
6.3.2 Notation
6.3.3 Definition
6.4 Permutation Functions
6.5 Recursive Functions
6.5.1 Theorem: (Recursion Theorem)
6.5.2 Observation
Exercises
Chapter 7: Graph Theory – I
7.1 Basic Concepts of Graphs
7.1.1 Example (Utilities Problem)
7.1.2 Observation
7.1.3 Example: Konigsberg Bridges (or Seven Bridges)
7.1.4 Properties of Konigsberg Bridges Graph
7.2 Subgraphs
7.2.1 Observations
7.3 Matrix Representation of Graphs
7.3.1 Incidence Matrices
7.4 Isomorphic Graphs
Exercises
Chapter 8: Graph Theory – II
8.1 Paths and Circuits
8.1.1 Corollary
8.2 Eularian Graphs
8.2.1 Corollary
8.3 Hamiltonian Graphs
8.3.1 Some Basic Rules for Constructing Hamiltonian Paths and Cycles
8.3.2 Corollary
8.3.3 Example (City-Route Puzzle)
8.3.4 Example (The Seating Arrangement Problem)
8.3.5 Travelling – Salesman Problem
8.4 Multiple Graphs
Exercises
Chapter 9: Graph Theory – III
9.1 Planar Graphs
9.1.1 Observation
9.2 Euler’s Formula
9.2.1 Corollary
9.3 Graph Colouring, Covering and Chromatic Numbers
9.3.1 The Scheduling Problem
9.3.2 Rules for Graph Coloring
Exercises
Chapter 10: Graph Theory – IV
10.1 Fundamental Concepts of Trees
10.1.1 Note (Formation of Components)
10.1.2 Some Properties of Trees
10.1.3 Corollary
10.2 Directed Trees
10.3 Binary Trees and Decision Trees
10.3.1 Lemma
10.4 Spanning Trees and Properties
10.5 Algorithms for Spanning Trees
10.5.1 Breadth-First Search Algorithm (BFS): (for a Spanning Tree)
10.5.2 Depth-First Search Algorithm (DFS): (for a Spanning Tree)
10.5.3 Minimal Spanning Trees
10.5.4 Kruskal Algorithm: (Finding Shortest Spanning Tree)
10.5.5 Prim’s Algorithm
Exercises
Chapter 11: Algebraic Structures
11.1 Some Algebraic Properties
11.2 Lattice as an Algebraic System
11.2.1 Properties of Lattices
11.3 Algebraic Systems with One Binary Operation
11.4 Properties of Binary Operations
11.4.1 Result
11.5 Semi Groups and Monoids
11.5.1 Observation
11.5.2 Result
11.5.3 Observations
11.6 Homomorphism of Semi Groups and Monoids
11.6.1 Definition
11.6.2 Result
11.6.3 Observation
11.6.4 Fundamental Theorem of Homomorphism
Exercises
Chapter 12: Algebraic Structures (Groups and Rings)
12.1 Fundamental Concepts in Group Theory
12.1.1 Result
12.2 Subgroups
12.2.1 Notation
12.3 Cosets
12.3.1 Remark
12.3.2 Theorem (Lagranges Theorem)
12.3.3 Remark
12.4 Rings
12.4.1 The Pigeon Hole Principle
Exercises
Chapter 13: Permutations and Combinations
13.1 Basic Counting Principles
13.1.1 Sum Rule
13.1.2 General Rule for Counting Event (or Sum Rule)
13.1.3 Product Rule
13.1.4 Indirect Counting
13.1.5 One to One Correspondence
13.1.6 Applications to Computer Science
13.2 Permutations
13.2.1 Fundamental Principles of Counting
13.2.2 Permutations of Distinct Things
13.2.3 Permutations with Repetitions
13.2.4 Dearrangements
13.3 Circular Permutations, Restricted Permutations
13.3.1 Circular Permutations
13.3.2 Clockwise and Anticlockwise Permutations
13.3.3 Restricted Permutations
13.4 Combinations and Restricted Combinations
13.5 The Pigeonhole Principle and its Applications
13.5.1 The Extended Pigeonhole Principle
Exercises
Chapter 14: Binomial Theorem
14.1 Binomial and Multinomial Coefficients
14.1.1 Pascal’s Triangle
14.1.2 Properties of Binomial Coefficients or Combinatorial Identities
14.1.3 Multinomial Coefficients
14.1.4 Multinomial Theorem
14.1.5 Multinomial Theorem – An Explanation
14.1.6 How to Find Number of Solutions of the Equation
14.2 Generating Functions of Permutations and Combinations
14.2.1 The Concept “Generating Function”
14.2.2 Properties of Generating Functions with Respect to Sum and Derivative
14.3 The Principle of Inclusion – Exclusion
14.3.1 Result
14.3.2 Principle of Inclusion-Exclusion for n-sets
14.3.3 Result
Exercises
Chapter 15: Recurrence Relations
15.1 Generating Function of Sequences
15.1.1 Generating Functions
15.2 Partial Fractions (Definition)
15.2.1 Partial Fractions
15.3 Calculating Coefficient of Generating Functions
15.3.1 Remark
15.3.2 Some Formulas (Special Cases of Binomial Theorem)
15.3.3 Geometric Series
15.3.4 A List of Formulas
15.4 Recurrence Relations and Formation of Recurrence Relation
15.4.1 Examples of Recurrence Relations
15.4.2 Some More Examples
15.4.3 Solutions of Recurrence Relations
Exercises
Chapter 16: Some Methods of Solving Recurrence Relations
16.1 Solving Linear Homogeneous Recurrence Relations by Substitution Method
16.1.1 Substitution Method
16.1.2 First Order Recurrence Relation
16.2 Generating Functions and the Method of Characteristic Roots
16.2.1 The Shifting Properties of Generating Functions
16.2.2 Sketch of the Method of Generating Functions
16.2.3 The Method of Characteristic Roots
16.3 Solving Inhomogeneous Recurrence Relations
16.3.1 Method
Exercises
Bibliography
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