Introductory logic and sets for Computer scientists

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1 Introductory logic and sets for Computer scientists Nimal Nissanke University of Reading ADDISON WESLEY LONGMAN Harlow, England II Reading, Massachusetts Menlo Park, California New York Don Mills, Ontario Amsterdam «* Bonn Sydney Singapore Tokyo Madrid II San Juan Milan Mexico City Seoul # Taipei

Contents Preface A note on the exercises XIII xvi 1 An overview of logic 1.1 1.2 1.3 1.4 1.5 1.6 1.7 What is logic?

2 Contents Preface A note on the exercises XIII xvi 1 An overview of logic What is logic? Logic in real arguments Arguments and deduction The logical Systems Scientific theories and specifications Mathematical modeis Limitations of our study Propositions and propositional connectives Propositions Connectives Ambiguity and imprecision 19 3 Propositional logic as a language About formal languages Grammar of propositional logic The semantics Truth tables Tautologies, contradictions and contingent propositions Notations: T vs true and F vs false Logical equivalence and logical implication An application: digital circuits 37 4 Transformational proofs Logical laws and how to establish them The logical laws listed 49

4.3 On the art of proofs 52 4.4 An application: programming 55 4.5 An application: digital circuits 57 5 Deductive proofs 60 5.1 Arguments 60 5.2 Validity 62 5.

3 4.3 On the art of proofs An application: programming An application: digital circuits 57 5 Deductive proofs Arguments Validity Demonstrating validity using truth tables Demonstrating invalidity Proving validity using deductive proofs Inference rules On the art of proofs An application: reasoning in theories 78 6 Predicates and quantifiers The need for predicate logic Predicates Unary predicates Predicate logic and propositional logic Quantifiers Elementary reasoning in predicate logic Predicates with higher arities An application: program specification 90 7 Further predicate logic Scope of quantifiers Bound and free variables Types of values Some Conventions Comparison with variable usage in programming Comparison with variable usage in mathematics Interpretation of formulae Interpretation of universal quantification Interpretation of existential quantification Theorems in predicate logic and proofs Useful theorems in predicate logic Proofs in predicate logic Recapitulation Formulae in propositional form Unknowns and genuine variables Notation for new inference rules What to guard against 119

Contents ix 9.6 Theorems in predicate logic 129 9.7 An application: program refinement 133 10 Proof by mathematical induction 141 10.1 What is mathematical induction? 141 10.2 Relationship with induction 142 10.

4 Contents ix 9.6 Theorems in predicate logic An application: program refinement Proof by mathematical induction What is mathematical induction? Relationship with induction The technique of mathematical induction An application: algorithm analysis Basic set theoretical concepts Whataresets? How to define sets? Setequality The empty set The universal set Subsets Sizeofsets Power set An extended notation for sets Operations on sets Binary set Operations Uses of sets Set theoretic laws listed Set theoretic proofs Generalized set Operations Disjointsets An application: formal System specification Relations: basic concepts Pairs and tuples Cartesian product Relations as sets Relations as predicates Graphical representation of relations Domain, ränge and field Inverse relation Identity relation An application: relational model of databases Advanced relational Operations Relational composition Iteration An application: relational image 216

14.4 Domain and ränge restriction and co-restriction 218 14.5 Relational overriding 220 15 Properties of binary relations 224 15.1 Properties of binary relations 225 15.2 Equivalence relations 227 15.

5 14.4 Domain and ränge restriction and co-restriction Relational overriding Properties of binary relations Properties of binary relations Equivalence relations Partitions and equivalence classes Relational closures Functions and their Classification Functions as relations Classification of functions Functions as expressions Function composition An application: more on formal System specification An application: functional programming Numbers Natural numbers Integers A tour through other number Systems Dense vs sparse sets Countability Numbers in Computers Sequencesand bags Sequences as sets Operations on sequences Bags Operations on bags Boolean algebra What is Boolean algebra? Theorems in Boolean algebra Proof of theorems Propositional Interpretation of Boolean algebra Set interpretation of Boolean algebra Significance of Boolean algebra Lambda abstraction of functions Curried functions Infix Operators Function names 319

Contents xi 20.4 Lambda abstraction 320 20.5 Lambda calculus 322 20.

6 Contents xi 20.4 Lambda abstraction Lambda calculus Modeiling mathematical objects 334 Appendix: Outline answers to exercises 336 References 374 Index 375

Logic and its Applications

Logic and its Applications Edmund Burke and Eric Foxley PRENTICE HALL London New York Toronto Sydney Tokyo Singapore Madrid Mexico City Munich Contents Preface xiii Propositional logic 1 1.1 Informal introduction