Contents 1 Propositional Logic: Proofs from Axioms and Inference Rules... 1 1.1 Introduction... 1 1.1.1 An Example Demonstrating the Use of Logic in Real Life... 2 1.2 The Pure Propositional Calculus... 4 1.2.1 Formulae, Axioms, Inference Rules, and Proofs... 5 1.3 The Pure Positive Implicational Propositional Calculus... 9 1.3.1 Examples of Proofs in the Implicational Calculus... 9 1.3.2 Derived Rules: Implications Subject to Hypotheses... 11 1.3.3 A Guide for Proofs: an Implicational Deduction Theorem.. 14 1.3.4 Example: Law of Assertion from the Deduction Theorem.. 18 1.3.5 More Examples to Design Proofs of Implicational Theorems... 21 1.3.6 Another Guide for Proofs: Substitutivity of Equivalences.. 23 1.3.7 More Derived Rules of Inference... 25 1.3.8 The Laws of Commutation and of Assertion... 27 1.3.9 Exercises on the Classical Implicational Calculus... 28 1.3.10 Equivalent Implicational Axiom Systems... 29 1.3.11 Exercises on Kleene s Axioms... 30 1.3.12 Exercises on Tarski s Axioms... 31 1.4 Proofs by the Converse Law of Contraposition... 32 1.4.1 Examples of Proofs in the Full Propositional Calculus... 32 1.4.2 Guides for Proofs in the Propositional Calculus... 34 1.4.3 Proofs by Reductio ad Absurdum... 35 1.4.4 Proofs by Cases... 36 1.4.5 Exercises on Frege s and Church s Axioms... 37 1.5 Other Connectives... 38 1.5.1 Definitions of Other Connectives... 38 1.5.2 Examples of Proofs of Theorems with Conjunctions... 38 1.5.3 Examples of Proofs of Theorems with Equivalences... 41 1.5.4 Examples of Proofs of Theorems with Disjunctions... 44 vii
viii Contents 1.5.5 Examples of Proofs with Conjunctions and Disjunctions... 46 1.5.6 Exercises on Other Connectives... 47 1.6 Patterns of Deduction with Other Connectives... 48 1.6.1 Conjunctions of Implications... 48 1.6.2 Proofs by Cases or by Contradiction... 53 1.6.3 Exercises on Patterns of Deduction... 54 1.6.4 Equivalent Classical Axiom Systems... 55 1.6.5 Exercises on Kleene s, Rosser s, and Tarski s Axioms... 56 1.7 Completeness, Decidability, Independence, Provability, and Soundness... 56 1.7.1 Multi-Valued Fuzzy Logics... 56 1.7.2 Sound Multi-Valued Fuzzy Logics... 57 1.7.3 Independence and Unprovability... 59 1.7.4 Complete Multi-Valued Fuzzy Logics... 61 1.7.5 Peirce s Law as a Denial of the Antecedent... 62 1.7.6 Exercises on Church s and Łukasiewicz s Triadic Systems... 62 1.8 Boolean Logic... 63 1.8.1 The Truth Table of the Logical Implication... 63 1.8.2 Boolean Logic on Earth and in Space... 65 1.9 Automated Theorem Proving... 67 1.9.1 The Provability Theorem... 67 1.9.2 The Completeness Theorem... 69 1.9.3 Example: Peirce s Law from the Completeness Theorem... 69 1.9.4 Exercises on the Deduction Theorem... 72 2 First-Order Logic: Proofs with Quantifiers... 75 2.1 Introduction... 75 2.2 The Pure Predicate Calculus of First Order... 75 2.2.1 Logical Predicates... 75 2.2.2 Variables, Quantifiers, and Formulae... 77 2.2.3 Proper Substitutions of Free or Bound Variables... 78 2.2.4 Axioms and Rules for the Pure Predicate Calculus... 80 2.2.5 Exercises on Quantifiers... 82 2.2.6 Examples with Implicational and Predicate Calculi... 82 2.2.7 Examples with Pure Propositional and Predicate Calculi... 86 2.2.8 Other Axiomatic Systems for the Pure Predicate Calculus.. 87 2.2.9 Exercises on Kleene s, Margaris s, and Rosser s Axioms... 89 2.3 Methods of Proof for the Pure Predicate Calculus... 90 2.3.1 Substituting Equivalent Formulae... 90 2.3.2 Discharging Hypotheses... 91 2.3.3 Prenex Normal Form... 95 2.3.4 Proofs with More than One Quantifier... 96 2.3.5 Exercises on the Substitutivity of Equivalence... 97
Contents ix 2.4 Predicate Calculus with Other Connectives... 98 2.4.1 Universal Quantifiers and Conjunctions or Disjunctions... 98 2.4.2 Existential Quantifiers and Conjunctions or Disjunctions... 100 2.4.3 Exercises on Quantifiers with Other Connectives... 101 2.5 Equality-Predicates... 101 2.5.1 First-Order Predicate Calculi with an Equality-Predicate... 102 2.5.2 Simple Applied Predicate Calculi with an Equality-Predicate... 103 2.5.3 Other Axiom Systems for the Equality-Predicate... 106 2.5.4 Defined Ranking-Predicates... 107 2.5.5 Exercises on Equality-Predicates... 107 3 Set Theory: Proofs by Detachment, Contraposition, and Contradiction... 109 3.1 Introduction... 109 3.2 Sets and Subsets... 110 3.2.1 Equality and Extensionality... 110 3.2.2 The Empty Set... 114 3.2.3 Subsets and Supersets... 114 3.2.4 Exercises on Sets and Subsets... 118 3.3 Pairing, Power, and Separation... 119 3.3.1 Pairing... 119 3.3.2 Power Sets... 122 3.3.3 Separation of Sets... 124 3.3.4 Exercises on Pairing, Power, and Separation of Sets... 126 3.4 Unions and Intersections of Sets... 127 3.4.1 Unions of Sets... 127 3.4.2 Intersections of Sets... 132 3.4.3 Unions and Intersections of Sets... 135 3.4.4 Exercises on Unions and Intersections of Sets... 139 3.5 Cartesian Products and Relations... 142 3.5.1 Cartesian Products of Sets... 142 3.5.2 Cartesian Products of Unions and Intersections... 147 3.5.3 Mathematical Relations and Directed Graphs... 149 3.5.4 Exercises on Cartesian Products of Sets... 153 3.6 Mathematical Functions... 154 3.6.1 Mathematical Functions... 154 3.6.2 Images and Inverse Images of Sets by Functions... 159 3.6.3 Exercises on Mathematical Functions... 162 3.7 Composite and Inverse Functions... 164 3.7.1 Compositions of Functions... 164 3.7.2 Injective, Surjective, Bijective, and Inverse Functions... 166 3.7.3 The Set of all Functions from a Set to a Set... 171 3.7.4 Exercises on Injective, Surjective, and Inverse Functions... 173
x Contents 3.8 Equivalence Relations... 174 3.8.1 Reflexive, Symmetric, Transitive, or Anti-Symmetric Relations... 174 3.8.2 Partitions and Equivalence Relations... 175 3.8.3 Exercises on Equivalence Relations... 179 3.9 Ordering Relations... 180 3.9.1 Preorders and Partial Orders... 180 3.9.2 Total Orders and Well-Orderings... 182 3.9.3 Exercises on Ordering Relations... 185 4 Mathematical Induction: Definitions and Proofs by Induction... 189 4.1 Introduction... 189 4.2 Mathematical Induction... 190 4.2.1 The Axiom of Infinity... 190 4.2.2 The Principle of Mathematical Induction... 193 4.2.3 Definitions by Mathematical Induction... 195 4.2.4 Exercises on Mathematical Induction... 197 4.3 Arithmetic with Natural Numbers... 198 4.3.1 Addition with Natural Numbers... 198 4.3.2 Multiplication with Natural Numbers... 200 4.3.3 Exercises on Arithmetic by Induction... 203 4.4 Orders and Cancellations... 205 4.4.1 Orders on the Natural Numbers... 205 4.4.2 Laws of Arithmetic Cancellations... 210 4.4.3 Exercises on Orders and Cancellations... 213 4.5 Integers... 214 4.5.1 Negative Integers... 214 4.5.2 Arithmetic with Integers... 217 4.5.3 Order on the Integers... 220 4.5.4 Nonnegative Integral Powers of Integers... 226 4.5.5 Exercises on Integers with Induction... 228 4.6 Rational Numbers... 229 4.6.1 Definition of Rational Numbers... 229 4.6.2 Arithmetic with Rational Numbers... 231 4.6.3 Notation for Sums and Products... 237 4.6.4 Order on the Rational Numbers... 241 4.6.5 Exercises on Rational Numbers... 243 4.7 Finite Cardinality... 244 4.7.1 Equal Cardinalities... 244 4.7.2 Finite Sets... 248 4.7.3 Exercises on Finite Sets... 252 4.8 Infinite Cardinality... 252 4.8.1 Infinite Sets... 252 4.8.2 Denumerable Sets... 254 4.8.3 The Bernstein Cantor Schröder Theorem... 258
Contents xi 4.8.4 Denumerability of all Finite Sequences of Natural Numbers... 260 4.8.5 Other Infinite Sets... 262 4.8.6 Further Issues in Cardinality... 263 4.8.7 Exercises on Infinite Sets... 265 5 Well-Formed Sets: Proofs by Transfinite Induction with Already Well-Ordered Sets... 267 5.1 Introduction... 267 5.2 Transfinite Methods... 267 5.2.1 Transfinite Induction... 267 5.2.2 Transfinite Construction... 269 5.2.3 Exercises on Transfinite Methods... 271 5.3 Transfinite Sets and Ordinals... 271 5.3.1 Transitive Sets... 271 5.3.2 Ordinals... 272 5.3.3 Well-Ordered Sets of Ordinals... 274 5.3.4 Unions and Intersections of Sets of Ordinals... 275 5.3.5 Exercises on Ordinals... 276 5.4 Regularity of Well-Formed Sets... 277 5.4.1 Well-Formed Sets... 277 5.4.2 Regularity... 279 5.4.3 Exercises on Well-Formed Sets... 281 6 The Axiom of Choice: Proofs by Transfinite Induction... 283 6.1 Introduction... 283 6.2 The Choice Principle... 283 6.2.1 The Choice-Function Principle... 284 6.2.2 The Choice-Set Principle... 286 6.2.3 Exercises on Choice Principles... 288 6.3 Maximality and Well-Ordering Principles... 288 6.3.1 Zermelo s Well-Ordering Principle... 288 6.3.2 Zorn s Maximal-Element Principle... 290 6.3.3 Exercises on Maximality and Well-Orderings... 292 6.4 Unions, Intersections, and Products of Families of Sets... 292 6.4.1 The Multiplicative Principle... 292 6.4.2 The Distributive Principle... 293 6.4.3 Exercises on the Distributive and Multiplicative Principles... 295 6.5 Equivalence of the Choice, Zorn s, and Zermelo s Principles... 295 6.5.1 Towers of Sets... 296 6.5.2 Zorn s Maximality from the Choice Principle... 297 6.5.3 Exercises on Towers of Sets... 299 6.6 Yet Other Principles Related to the Axiom of Choice... 299 6.6.1 Yet Other Principles Equivalent to the Axiom of Choice... 299
xii Contents 6.6.2 Consequences of the Axiom of Choice... 300 6.6.3 Exercises on Related Principles... 301 7 Applications: Nobel-Prize Winning Applications of Sets, Functions, and Relations... 303 7.1 Introduction... 303 7.2 Game Theory... 304 7.2.1 Introduction... 304 7.2.2 Mathematical Models for The Prisoner s Dilemma... 305 7.2.3 Dominant Strategies... 307 7.2.4 Mixed Strategies... 309 7.2.5 Existence of Nash Equilibria for Two Players with Two Mixed Strategies... 310 7.2.6 Exercises on Mathematical Games... 313 7.3 Match Making... 315 7.3.1 Introduction... 315 7.3.2 A Mathematical Model for Optimal Match Making... 316 7.3.3 An Algorithm for Optimal Match Making with a Match Maker... 317 7.3.4 An Algorithm for Optimal Match Making Without a Match Maker... 319 7.3.5 Exercises on Gale & Shapley s Algorithms... 320 7.3.6 Projects... 321 7.4 Arrow s Impossibility Theorem... 321 7.4.1 Introduction... 321 7.4.2 A Mathematical Model for Arrow s Impossibility Theorem... 324 7.4.3 Statement and Proof of Arrow s Impossibility Theorem... 326 7.4.4 Exercises on Arrow s Impossibility Theorem... 330 Solutions to Some Odd-Numbered Exercises... 331 References... 373 Index... 381
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