Contents Propositional Logic: Proofs from Axioms and Inference Rules

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1 Contents 1 Propositional Logic: Proofs from Axioms and Inference Rules Introduction An Example Demonstrating the Use of Logic in Real Life The Pure Propositional Calculus Formulae, Axioms, Inference Rules, and Proofs The Pure Positive Implicational Propositional Calculus Examples of Proofs in the Implicational Calculus Derived Rules: Implications Subject to Hypotheses A Guide for Proofs: an Implicational Deduction Theorem Example: Law of Assertion from the Deduction Theorem More Examples to Design Proofs of Implicational Theorems Another Guide for Proofs: Substitutivity of Equivalences More Derived Rules of Inference The Laws of Commutation and of Assertion Exercises on the Classical Implicational Calculus Equivalent Implicational Axiom Systems Exercises on Kleene s Axioms Exercises on Tarski s Axioms Proofs by the Converse Law of Contraposition Examples of Proofs in the Full Propositional Calculus Guides for Proofs in the Propositional Calculus Proofs by Reductio ad Absurdum Proofs by Cases Exercises on Frege s and Church s Axioms Other Connectives Definitions of Other Connectives Examples of Proofs of Theorems with Conjunctions Examples of Proofs of Theorems with Equivalences Examples of Proofs of Theorems with Disjunctions vii

2 viii Contents Examples of Proofs with Conjunctions and Disjunctions Exercises on Other Connectives Patterns of Deduction with Other Connectives Conjunctions of Implications Proofs by Cases or by Contradiction Exercises on Patterns of Deduction Equivalent Classical Axiom Systems Exercises on Kleene s, Rosser s, and Tarski s Axioms Completeness, Decidability, Independence, Provability, and Soundness Multi-Valued Fuzzy Logics Sound Multi-Valued Fuzzy Logics Independence and Unprovability Complete Multi-Valued Fuzzy Logics Peirce s Law as a Denial of the Antecedent Exercises on Church s and Łukasiewicz s Triadic Systems Boolean Logic The Truth Table of the Logical Implication Boolean Logic on Earth and in Space Automated Theorem Proving The Provability Theorem The Completeness Theorem Example: Peirce s Law from the Completeness Theorem Exercises on the Deduction Theorem First-Order Logic: Proofs with Quantifiers Introduction The Pure Predicate Calculus of First Order Logical Predicates Variables, Quantifiers, and Formulae Proper Substitutions of Free or Bound Variables Axioms and Rules for the Pure Predicate Calculus Exercises on Quantifiers Examples with Implicational and Predicate Calculi Examples with Pure Propositional and Predicate Calculi Other Axiomatic Systems for the Pure Predicate Calculus Exercises on Kleene s, Margaris s, and Rosser s Axioms Methods of Proof for the Pure Predicate Calculus Substituting Equivalent Formulae Discharging Hypotheses Prenex Normal Form Proofs with More than One Quantifier Exercises on the Substitutivity of Equivalence... 97

3 Contents ix 2.4 Predicate Calculus with Other Connectives Universal Quantifiers and Conjunctions or Disjunctions Existential Quantifiers and Conjunctions or Disjunctions Exercises on Quantifiers with Other Connectives Equality-Predicates First-Order Predicate Calculi with an Equality-Predicate Simple Applied Predicate Calculi with an Equality-Predicate Other Axiom Systems for the Equality-Predicate Defined Ranking-Predicates Exercises on Equality-Predicates Set Theory: Proofs by Detachment, Contraposition, and Contradiction Introduction Sets and Subsets Equality and Extensionality The Empty Set Subsets and Supersets Exercises on Sets and Subsets Pairing, Power, and Separation Pairing Power Sets Separation of Sets Exercises on Pairing, Power, and Separation of Sets Unions and Intersections of Sets Unions of Sets Intersections of Sets Unions and Intersections of Sets Exercises on Unions and Intersections of Sets Cartesian Products and Relations Cartesian Products of Sets Cartesian Products of Unions and Intersections Mathematical Relations and Directed Graphs Exercises on Cartesian Products of Sets Mathematical Functions Mathematical Functions Images and Inverse Images of Sets by Functions Exercises on Mathematical Functions Composite and Inverse Functions Compositions of Functions Injective, Surjective, Bijective, and Inverse Functions The Set of all Functions from a Set to a Set Exercises on Injective, Surjective, and Inverse Functions

4 x Contents 3.8 Equivalence Relations Reflexive, Symmetric, Transitive, or Anti-Symmetric Relations Partitions and Equivalence Relations Exercises on Equivalence Relations Ordering Relations Preorders and Partial Orders Total Orders and Well-Orderings Exercises on Ordering Relations Mathematical Induction: Definitions and Proofs by Induction Introduction Mathematical Induction The Axiom of Infinity The Principle of Mathematical Induction Definitions by Mathematical Induction Exercises on Mathematical Induction Arithmetic with Natural Numbers Addition with Natural Numbers Multiplication with Natural Numbers Exercises on Arithmetic by Induction Orders and Cancellations Orders on the Natural Numbers Laws of Arithmetic Cancellations Exercises on Orders and Cancellations Integers Negative Integers Arithmetic with Integers Order on the Integers Nonnegative Integral Powers of Integers Exercises on Integers with Induction Rational Numbers Definition of Rational Numbers Arithmetic with Rational Numbers Notation for Sums and Products Order on the Rational Numbers Exercises on Rational Numbers Finite Cardinality Equal Cardinalities Finite Sets Exercises on Finite Sets Infinite Cardinality Infinite Sets Denumerable Sets The Bernstein Cantor Schröder Theorem

5 Contents xi Denumerability of all Finite Sequences of Natural Numbers Other Infinite Sets Further Issues in Cardinality Exercises on Infinite Sets Well-Formed Sets: Proofs by Transfinite Induction with Already Well-Ordered Sets Introduction Transfinite Methods Transfinite Induction Transfinite Construction Exercises on Transfinite Methods Transfinite Sets and Ordinals Transitive Sets Ordinals Well-Ordered Sets of Ordinals Unions and Intersections of Sets of Ordinals Exercises on Ordinals Regularity of Well-Formed Sets Well-Formed Sets Regularity Exercises on Well-Formed Sets The Axiom of Choice: Proofs by Transfinite Induction Introduction The Choice Principle The Choice-Function Principle The Choice-Set Principle Exercises on Choice Principles Maximality and Well-Ordering Principles Zermelo s Well-Ordering Principle Zorn s Maximal-Element Principle Exercises on Maximality and Well-Orderings Unions, Intersections, and Products of Families of Sets The Multiplicative Principle The Distributive Principle Exercises on the Distributive and Multiplicative Principles Equivalence of the Choice, Zorn s, and Zermelo s Principles Towers of Sets Zorn s Maximality from the Choice Principle Exercises on Towers of Sets Yet Other Principles Related to the Axiom of Choice Yet Other Principles Equivalent to the Axiom of Choice

6 xii Contents Consequences of the Axiom of Choice Exercises on Related Principles Applications: Nobel-Prize Winning Applications of Sets, Functions, and Relations Introduction Game Theory Introduction Mathematical Models for The Prisoner s Dilemma Dominant Strategies Mixed Strategies Existence of Nash Equilibria for Two Players with Two Mixed Strategies Exercises on Mathematical Games Match Making Introduction A Mathematical Model for Optimal Match Making An Algorithm for Optimal Match Making with a Match Maker An Algorithm for Optimal Match Making Without a Match Maker Exercises on Gale & Shapley s Algorithms Projects Arrow s Impossibility Theorem Introduction A Mathematical Model for Arrow s Impossibility Theorem Statement and Proof of Arrow s Impossibility Theorem Exercises on Arrow s Impossibility Theorem Solutions to Some Odd-Numbered Exercises References Index

Logic, Mathematics, and Computer Science

Logic, Mathematics, and Computer Science Yves Nievergelt Logic, Mathematics, and Computer Science Modern Foundations with Practical Applications Second Edition 123 Yves Nievergelt Department of Mathematics