Viii 3.5.3 3.5.4 Programming and Primitive Recursion Minimalization 173 172 Induction and Recursion 121 3.1 Induction on Natural Numbers 1223.1.1 3.1.2 3.1.3 3.1.4 3.1.5 3.1.6 3.2 Sums 3.2.1 3.2.2 3.2.3 3.2.4 Introduction 122 Natural Numbers 123 Mathematical Induction 124 Induction for Proving Properties of Addition 128 Changing the Induction Base 130 Strong Induction 131 and Related Constructs 132 Introduction 132 Recursive Definitions of Sums and Products 133 Identities Involving Sums 135 Double Sums and Matrices 139ģ.3 Proof by Recursion 141 3.3.1 Introduction 141 3.3.2 Recursive Definitions 143 3.3.3 Descending Sequences 146 3.3.4 The Principle of Proofs by Recursion 147 3.3.5 Structural Induction 149 3.4 Applications of Recursion to Programming 154 3.4.1 Introduction 154 3.4.2 Programming as Function Composition 154 3.4.3 Recursion in Programs 158 3.4.4 Programs Involving Trees 163 3.5 Recursive Functions 166 3.5.1 Introduction 166 3.5.2 Primitive Recursive Functions Introduction 92 Basic Logical Equivalences 92 Other Important Equivalences 94Ģ.5 Equational Logic 96 2.5.1 Introduction 96 2.5.2 Equality 96 2.5.3 Equality and Uniqueness 99 2.5.4 Functions and Equational Logic 100 2.5.5 Function Compositions 103 2.5.6 Properties of Operators 105 2.5.7 Identity and Zero Elements 108 2.5.8 Derivations in Equational Logic 111 2.5.9 Equational Logic in Practice 113 2.5.10 Boolean Algebra 115 6 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 Introduction 60 The Universe of Discourse 60 Predicates 61 Variables and Instantiations 63 Quantifiers 65 Restrictions of Quantifiers to Certain Groups 68Ģ.2 Interpretations and ValidityIntroduction 70 Interpretations 71 Validity 74 Invalid Expressions 76 Proving Validity 78Ģ.3 Derivations 79 2.3.1 Introduction 79 2.3.2 Universal Instantiation 80 2.3.3 Universal Generalization 81 2.3.4 Deduction Theorem and Universal Generalization 2.3.5 Dropping the Universal Quantifiers 85 2.3.6 Existential Generalization 87 2.3.7 Existential Instantiation 88Ĭontents 2.4 Logical Equivalences2.4.1 2.4.2 2.4.3 Predicate Calculus 59 2.1 Syntactic Components of Predicate Calculus 60.1. Yi1.4.3 Tautologies and Sound Reasoning 26 1.4.4 Contradictions 26 1.4.5 Important Types of Tautologies 27 1.5 Logical Equivalences and Their Use 28 1.5.1 Introduction 28 1.5.2 Proving Logical Equivalences by Truth Tables 29 1.5.3 Statement Algebra 30 1.5.4 Removing Conditionals and Biconditionals 32 1.5.5 Essential Laws for Statement Algebra 33 1.5.6 Shortcuts for Manipulating Expressions 34 1.5.7 Normal Forms 36 1.5.8 Truth Tables and Disjunctive Normal Forms 38 1.5.9 Conjunctive Normal Forms and Complementation 40 1.6 Logical Implications and Derivations 42 1.6.1 Introduction 42 1.6.2 Logical Implications 43 1.6.3 Soundness Proofs through Truth Tables 44 1.6.4 Proofs 46 1.6.5 Systems for Derivations 49 1.6.6 The Deduction Theorem 52 Preface xv Propositional Calculus 11.1 Logical Arguments and Propositions 1 1.1.1 Introduction 1 1.1.2 Some Important Logical Arguments 2 1.1.3 Propositions 4 1.2 Logical Connectives 6 1.2.1 Introduction 6 1.2.2 Negation 6 1.2.3 Conjunction 7 1.2.4 Disjunction 8 1.2.5 Conditional 9 1.2.6 Biconditional 11 1.2.7 Further Remarks on Connectivesġ.3 Compound Propositions 13 1.3.1 Introduction 13 1.3.2 Logical Expressions 13 1.3.3 Analysis of Compound Propositions 15 1.3.4 Precedence Rules 18 1.3.5 Evaluation of Expressions and Truth Tables 1.3.6 Examples of Compound Propositions 21 1.4 Tautologies and Contradictions 1.4.1 Introduction 23 1.4.2 Tautologies 24 23 HALL, Upper Saddle River, New Jersey 07458 ![]() JEAN-PAUL TREMBLAYDepartment of Computer Science University of Saskatchewan WINFRIED KARL GRASSMANNDepartment of Computer Science University of Saskatchewan ![]() We highly recommend that you practice the skills that you will learn in this week by doing the puzzles at OUTCOMES: By the end of this week’s material you will be able to : understand the information conveyed by a truth-tableuse truth-tables to determine whether a deductive argument is valididentify quantifiers and categoriesbuild a Venn Diagram for any statement using quantifiers or categoriesOPTIONAL READING: If you want more examples or more detailed discussions of these topics, we recommend Understanding Arguments, Ninth Edition, Chapter 7.LOGIC AND DISCRETE MATHEMATICSA Computer Science Perspective It will also teach you to understand the functioning of these phrases using a device called a “Venn Diagram”, which shows how the truth or falsity of propositions that use these phrases depends upon the truth or falsity of other propositions that use these phrases. ![]() CONTENT: This week will teach you how such phrases as “all”, “some”, and “none” can work to guarantee the validity or invalidity of the deductive arguments in which they occur.
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