Foundations of Logic and Mathematics

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1 Yves Nievergelt Foundations of Logic and Mathematics Applications to Computer Science and Cryptography Birkhäuser Boston Basel Berlin

2 Contents Preface Outline xiii xv A Theory 1 0 Boolean Algebraic Logic INTRODUCTION LOGICAL FORMULAE TheEmptySet An Alphabet for Logic Well-Formed Formulae Exercises LOGICAL TRUTH AND CONNECTIVES Logical Truth Logical Connectives Truth Tables Exercises TAUTOLOGIES AND CONTRADICTIONS Examples of Tautologies Contradictions Exercises OTHER METHODS OF PROOF Proofs by Tautologies Proofs by Contradictions Exercises SYNTHESIS OF LOGICAL FORMULAE Design of Logical Formulae with Specified Truth Values Simplification by Distributivity and Excluded Middle Exercises OTHER CONNECTIVES AND APPLICATIONS Other Logical Connectives Logical Connectives in Binary Arithmetic Exercises SYNTHESIS BY KARNAUGH TABLES Karnaugh Tables Exercises AN APPLICATION TO CIRCUITS 52

3 vi CONTENTS 0.9 PROJECTS 53 1 Logic and Deductive Reasoning INTRODUCTION PROPOSITIONAL CALCULUS Formulae, Axioms, Rules, and Proofs Examples of Proofs with Axioms PI and P Exercises CLASSICALIMPLICATIONAL CALCULUS Derived Rules: Implications Subject to Hypotheses Examples of Proofs of Implicational Theorems Exercises PROOFS BY CONTRAPOSITION Examples of Proofs with Axiom P Proofs by Reductio ad Absurdum Exercises 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 Examples of Proofs with Conjunctions and Disjunctions Exercises OTHER FORMS OF DEDUCTIVE REASONING Conjunctions of Implications Proofs by Cases or by Contradiction Exercises PREDICATE CALCULUS Predicates, Quantifiers, Free or Bound Variables Axioms and Rules for the Predicate Calculus Examples of Proofs with the Predicate Calculus Exercises INFERENCE WITH QUANTIFIERS Proofs with Quantifiers Proofs With Universal Quantifiers and Other Connectives Proofs with Quantifiers and Other Connectives Proofs with Restrictions On a Quantified Variable Proofs with More than One Quantified Variable Exercises FURTHER ISSUES IN LOGIC PROJECTS 96 2 Set Theory INTRODUCTION SETS AND SUBSETS Equality and Extensionality TheEmptySet Subsets and Supersets Exercises PAIRING, POWER, AND SEPARATION Pairing 104

4 CONTENTS vn Power Sets Separation of Sets Exercises UNIONS AND INTERSECTIONS OF SETS UnionsofSets Intersections of Sets Unions and Intersections of Sets Exercises CARTESIAN PRODUCTS AND RELATIONS Cartesian Products of Sets Cartesian Products of Unions and Intersections Mathematical Relations Exercises MATHEMATICAL FUNCTIONS Mathematical Functions Images and Inverse Images of Sets by Functions Exercises COMPOSITE AND INVERSE FUNCTIONS Compositions of Functions Injective, Surjective, Bijective, and Inverse Functions Exercises EQUIVALENCE RELATIONS Reflexive, Symmetrie, Transitive, or Anti-Symmetric Relations Partitions and Equivalence Relations Exercises ORDERING RELATIONS Preorders and Partial Orders Total Orders and Well-Orderings Exercises PROJECTS Induction, Recursion, Arithmetic, Cardinality INTRODUCTION MATHEMATICAL INDUCTION The Axiom oflnfinity The Principleof Mathematical Induction Definitions by Mathematical Induction or Recursion Exercises ARITHMETIC WITH NATURAL NUMBERS Addition with Natural Numbers Multiplication with Natural Numbers Exercises ORDERS AND CANCELLATIONS Orders on the Natural Numbers Laws of Arithmetic Cancellations Exercises INTEGERS Negative Integers Arithmetic with Integers Order on the Integers Non-Negative Integral Powers of Integers. 187

5 viii CONTENTS Exercises RATIONAL NUMBERS Definition of Rational Numbers Arithmetic with Rational Numbers Notation for Sums and Products Order on the Rational Numbers Exercises FINITE CARDINALITY Equal Cardinalities Finite Sets Exercises INFINITE CARDINALITY Infinite Sets Denumerable Sets The Bernstein-Cantor-Schröder Theorem Other Infinite Sets Further Issues in Cardinality Exercises ARITHMETIC IN FINANCE Introduction Percentages and Rates Sales and Income Taxes Compounded Interest Loans, Mortgages, and Savings Plans Perpetuities Exercises PROJECTS Decidability and Completeness INTRODUCTION LOGICS FOR SCIENTIFIC REASONING Scientific Reasoning Hypothesis Testing Brouwer & Heyting's Intuitionistic Logic Kolmogorov & Johansson's Minimal Propositional Calculus Exercises INCOMPLETENESS Tautologies and Theorems Incompleteness of the Implicational Calculus Exercises LOGICS NOT AMENABLE TO TRUTH TABLES Logics with Any Number of Values Practical Logics Not Amenable to Truth Tables Exercises AUTOMATED THEOREM PROVING The Deduction Theorem Example: Law of Assertion from the Deduction Theorem The Provability Theorem The Completeness Theorem Example: Peirce's Law from the Completeness Theorem Exercises 248

6 CONTENTS ix 4.5 TRANSFINITE METHODS Transfinite Induction Transfinite Construction Exercises TRANSITIVE SETS AND ORDINALS Transitive Sets Ordinals Well-Ordered Sets of Ordinals Unions and Intersections of Sets of Ordinals Exercises REGULARITY OF WELL-FORMED SETS Well-Formed Sets Regularity Exercises FURTHER ISSUES IN DECIDABHJTY PROJECTS 262 B Applications Number Theory and Codes INTRODUCTION THEEUCLIDEAN ALGORITHM Division With Integers Greatest Common Divisors Exercises DIGITAL EXPANSION AND ARITHMETIC Expansion of Integers in Powers of an Integral Base Digital Integer Arithmetic Exercises PRIME NUMBERS PrimeNumbers Prime-Number Factorization Primality Testing Exercises MODULAR ARITHMETIC Modular Integers Modular Addition Modular Multiplication Modular Division Divisibility Tests Exercises MODULAR CODES The International Standard Book Number (ISBN) Code The Universal Product Code (UPC) The Bank Identification Code The U.S. postal nine-digit ZIP Code Exercises EUCLID, EULER, &FERMATS THEOREMS Fermat's Little Theorem The Euler-Fermat Theorem 295

7 x CONTENTS Exercises RIVEST-SHAMIR-ADLEMAN (RSA) CODES The Rivest-Shamir-Adleman (RSA) Public-Key ProofoftheCorrectnessoftheRSAAlgorithm Exercises FURTHER ISSUES IN NUMBER THEORY PROJECTS Ciphers, Combinatorics, and Probabilities INTRODUCTION ALGEBRA Definitions and Examples of Mathematical Groups Existence and Uniqueness of Identities and Inverses Bijections of Groups Exercises PERMUTATIONS Permutations and Orbits Transpositions Exercises CYCLIC PERMUTATIONS Cycles Factorization of Permutations by Cycles Exercises SIGNATURESOF PERMUTATIONS Even and Odd Permutations Signatures of Composite Permutations Exercises ARRANGEMENTS Arrangements Without Repetition Arrangements With Repetitions Exercises COMBINATIONS Combinations Without Repetition Combinations With Repetition Permutations With Repetition Exercises PROBABILITY Probability Spaces The Exclusion-Inclusion Principle Conditional Probabilities Applications of Binomial Probabilities Exercises THE ENIGMA MACHINES Design and Operation of the ENIGMA Machines Breaking the ENIGMA Ciphers Reconstructing the ENIGMA Wiring Exercises PROJECTS 359

8 CONTENTS xi 7 Graph Theory INTRODUCTION MATHEMATICAL GRAPHS Directed Graphs Undirected Graphs Degrees of Vertices Exercises PATH-CONNECTED GRAPHS Walks in Graphs Path-Connected Components of Graphs Longer Walks Longest Paths Exercises EULER AND HAMILTONIAN CIRCUITS Circuits Hamiltonian Paths and Circuits Eulerian Paths and Circuits Exercises MATHEMATICAL TREES Trees Spanning Trees Directed Trees and Star-Shaped Graphs Exercises WEIGHTED GRAPHS AND MULTIGRAPHS Weighted Graphs Shortest Path and Minimum-Weight Paths or Trees Maximal Flows in Transportation Graphs Exercises BIPARTITE GRAPHS AND MATCHINGS Bipartite Graphs Maximum Matching Exercises GRAPH THEORY IN SCIENCE TheShapeofMolecules TheShapeofHydrocarbons Sequences of Radioactive Decays Exercises FURTHER ISSUES: PLANAR GRAPHS PROJECTS 398 Bibliography 399 Index 405

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