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BookOfProof

# BookOfProof - Book of Proof Richard Hammack Virginia...

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Book of Proof Richard Hammack Virginia Commonwealth University

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To my students

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Contents Preface vii Introduction viii I Fundamentals 1. Sets 3 1.1. Introduction to Sets 3 1.2. The Cartesian Product 8 1.3. Subsets 11 1.4. Power Sets 14 1.5. Union, Intersection, Difference 17 1.6. Complement 19 1.7. Venn Diagrams 21 1.8. Indexed Sets 24 2. Logic 28 2.1. Statements 29 2.2. And, Or, Not 33 2.3. Conditional Statements 36 2.4. Biconditional Statements 39 2.5. Truth Tables for Statements 41 2.6. Logical Equivalence 44 2.7. Quantifiers 46 2.8. More on Conditional Statements 48 2.9. Translating English to Symbolic Logic 50 2.10. Negating Statements 52 2.11. Logical Inference 55 2.12. An Important Note 56 3. Counting 57 3.1. Counting Lists 57 3.2. Factorials 64 3.3. Counting Subsets 67 3.4. Pascal’s Triangle and the Binomial Theorem 72 3.5. Inclusion-Exclusion 75
v II How to Prove Conditional Statements 4. Direct Proof 81 4.1. Theorems 81 4.2. Definitions 83 4.3. Direct Proof 85 4.4. Using Cases 90 4.5. Treating Similar Cases 92 5. Contrapositive Proof 94 5.1. Contrapositive Proof 94 5.2. Congruence of Integers 97 5.3. Mathematical Writing 99 6. Proof by Contradiction 103 6.1. Proving Statements with Contradiction 104 6.2. Proving Conditional Statements by Contradiction 107 6.3. Combining Techniques 108 6.4. Some Words of Advice 109 III More on Proof 7. Proving Non-conditional Statements 113 7.1. If-And-Only-If Proof 113 7.2. Equivalent Statements 115 7.3. Existence Proofs 116 8. Proofs Involving Sets 119 8.1. How to Prove a A 119 8.2. How to Prove A B 121 8.3. How to Prove A = B 124 8.4. Examples: Perfect Numbers 127 9. Disproof 134 9.1. Counterexamples 136 9.2. Disproving Existence Statements 138 9.3. Disproof by Contradiction 139 10. Mathematical Induction 142 10.1. Proof by Strong Induction 148 10.2. Proof by Smallest Counterexample 152 10.3. Fibonacci Numbers 153

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vi IV Relations, Functions and Cardinality 11. Relations 161 11.1. Properties of Relations 164 11.2. Equivalence Relations 169 11.3. Equivalence Classes and Partitions 173 11.4. The Integers Modulo n 176 11.5. Relations Between Sets 179 12. Functions 181 12.1. Functions 181 12.2. Injective and Surjective Functions 186 12.3. The Pigeonhole Principle 190 12.4. Composition 193 12.5. Inverse Functions 196 12.6. Image and Preimage 199 13. Cardinality of Sets 202 13.1. Sets With Equal Cardinalities 202 13.2. Countable and Uncountable Sets 206 13.3. Comparing Cardinalities 211 Conclusion 215 Solutions 216 Index 274
Preface I n writing this book I have been motivated by the desire to create a high-quality textbook that costs almost nothing.

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BookOfProof - Book of Proof Richard Hammack Virginia...

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