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Note for Mathematical Foundations of Computer Science - mfcs By Digbijay Patil

  • Mathematical Foundations of Computer Science - mfcs
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q1 st ar t 0 q21 CS103 0 q0 0, 1 Mathematical 1Foundations of Computing ℒ(M) = Σ (00 ∪ 11) * q3 1 Preliminary Course Notes Keith Schwarz Fall 2015 q41

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Table of Contents Chapter 0 Introduction 7 0.1 How These Notes are Organized 7 0.2 Acknowledgements 8 Chapter 1 Sets and Cantor's Theorem 9 1.1 What is a Set? 9 1.2 Operations on Sets 11 1.3 Special Sets 14 1.4 Set-Builder Notation 16 Filtering Sets 16 Transforming Sets 18 1.5 Relations on Sets 19 Set Equality 19 Subsets and Supersets 20 The Empty Set and Vacuous Truths 21 1.6 The Power Set 22 1.7 Cardinality 24 What is Cardinality? 24 The Difficulty With Infinite Cardinalities 26 A Formal Definition of Cardinality 27 1.8 Cantor's Theorem 30 How Large is the Power Set? 30 Cantor's Diagonal Argument 31 Formalizing the Diagonal Argument 34 Proving Cantor's Theorem 36 1.9 Why Cantor's Theorem Matters 37 1.10 The Limits of Computation 38 What Does This Mean? 39 1.11 Chapter Summary 40 Chapter 2 Mathematical Proof 2.1 What is a Proof? 41 Transitioning to Proof-Based Mathematics 42 What Can We Assume? 42 2.2 Direct Proofs 43 Proof by Cases 45 A Quick Aside: Choosing Letters 47 Proofs about Sets 47 Lemmas 51 41

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Proofs with Vacuous Truths 55 2.3 Indirect Proofs 56 Logical Implication 56 Proof by Contradiction 58 Rational and Irrational Numbers 61 Proof by Contrapositive 63 2.4 Writing Elegant Proofs 66 Treat Proofs as Essays 66 Avoid Shorthand or Unnecessary Symbols 67 Write Multiple Drafts 68 Avoid “Clearly” and “Obviously” 68 2.5 Chapter Summary 69 2.6 Chapter Exercises 70 Chapter 3 Mathematical Induction 3.1 The Principle of Mathematical Induction 73 The Flipping Glasses Puzzle 74 3.2 Summations 78 Summation Notation 84 Summing Odd Numbers 86 Manipulating Summations 90 Telescoping Series 95 Products 100 3.3 Induction and Recursion 101 Monoids and Folds ★ 106 3.4 Variants on Induction 110 Starting Induction Later 110 Why We Can Start Later ★ 113 Fibonacci Induction ★ 116 Climbing Down Stairs 118 Computing Fibonacci Numbers 122 3.5 Strong Induction 127 The Unstacking Game 131 A Foray into Number Theory 140 3.6 The Well-Ordering Principle ★ 149 Proof by Infinite Descent 149 Proving the Well-Ordering Principle 152 An Incorrect Proof 152 A Correct Proof 154 3.7 Chapter Summary 155 3.8 Chapter Exercises 156 73

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Chapter 4 Graph Theory 161 4.1 Basic Definitions 161 Ordered and Unordered Pairs 161 A Formal Definition of Graphs 163 Navigating a Graph 164 4.2 Graph Connectivity 167 Connected Components 168 2-Edge-Connected Graphs ★ 175 Trees ★ 184 Properties of Trees 192 Directed Connectivity 201 4.3 DAGs 205 Topological Orderings 209 Condensations ★ 217 4.4 Matchings ★ 222 Proving Berge's Theorem 230 4.5 Chapter Summary 242 4.6 Chapter Exercises 243 Chapter 5 Relations 5.1 Basic Terminology 247 Tuples and the Cartesian Product 247 Cartesian Powers 250 A Formal Definition of Relations 251 Special Binary Relations 254 Binary Relations and Graphs 256 5.2 Equivalence Relations 262 Equivalence Classes 262 Equivalence Classes and Graph Connectivity 271 5.3 Order Relations 271 Strict Orders 271 Partial Orders 275 Hasse Diagrams 280 Preorders ★ 286 Properties of Preorders 287 Combining Orderings ★ 293 The Product Ordering 294 The Lexicographical Ordering 296 5.4 Well-Ordered and Well-Founded Sets ★ 301 Well-Orders 301 Properties of Well-Ordered Sets 303 Well-Founded Orders 305 247

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