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Notes on Discrete Mathematics James Aspnes 2017-08-29 13:34
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i Copyright c 2004–2017 by James Aspnes. Distributed under a Creative Com- mons Attribution-ShareAlike 4.0 International license: . org/licenses/by-sa/4.0/ .
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Contents Table of contents ii List of figures xvi List of tables xvii List of algorithms xviii Preface xix Syllabus xx Resources xxiii Internet resources xxiv Lecture schedule xxv 1 Introduction 1 1.1 So why do I need to learn all this nasty mathematics? . . . . 1 1.2 But isn’t math hard? . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Thinking about math with your heart . . . . . . . . . . . . . 3 1.4 What you should know about math . . . . . . . . . . . . . . . 3 1.4.1 Foundations and logic . . . . . . . . . . . . . . . . . . 4 1.4.2 Basic mathematics on the real numbers . . . . . . . . 4 1.4.3 Fundamental mathematical objects . . . . . . . . . . . 5 1.4.4 Modular arithmetic and polynomials . . . . . . . . . . 6 1.4.5 Linear algebra . . . . . . . . . . . . . . . . . . . . . . 6 1.4.6 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4.7 Counting . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4.8 Probability . . . . . . . . . . . . . . . . . . . . . . . . 7 ii
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CONTENTS iii 1.4.9 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Mathematical logic 9 2.1 The basic picture . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Axioms, models, and inference rules . . . . . . . . . . 9 2.1.2 Consistency . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.3 What can go wrong . . . . . . . . . . . . . . . . . . . 10 2.1.4 The language of logic . . . . . . . . . . . . . . . . . . 11 2.1.5 Standard axiom systems and models . . . . . . . . . . 11 2.2 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 Operations on propositions . . . . . . . . . . . . . . . 13 2.2.1.1 Precedence . . . . . . . . . . . . . . . . . . . 15 2.2.2 Truth tables . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.3 Tautologies and logical equivalence . . . . . . . . . . . 17 2.2.3.1 Inverses, converses, and contrapositives . . . 19 2.2.3.2 Equivalences involving true and false . . . . 21 Example . . . . . . . . . . . . . . . . . . . . . . 22 2.2.4 Normal forms . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 Predicate logic . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3.1 Variables and predicates . . . . . . . . . . . . . . . . . 25 2.3.2 Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.2.1 Universal quantifier . . . . . . . . . . . . . . 26 2.3.2.2 Existential quantifier . . . . . . . . . . . . . 26 2.3.2.3 Negation and quantifiers . . . . . . . . . . . 27 2.3.2.4 Restricting the scope of a quantifier . . . . . 27 2.3.2.5 Nested quantifiers . . . . . . . . . . . . . . . 28 2.3.2.6 Examples . . . . . . . . . . . . . . . . . . . . 30 2.3.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3.4 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3.4.1 Uniqueness . . . . . . . . . . . . . . . . . . . 32 2.3.5 Models . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.5.1 Examples . . . . . . . . . . . . . . . . . . . . 33 2.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.4.1 Inference Rules . . . . . . . . . . . . . . . . . . . . . . 35 2.4.2 Proofs, implication, and natural deduction . . . . . . . 36 2.4.2.1 The Deduction Theorem . . . . . . . . . . . 37 2.5 Natural deduction . . . . . . . . . . . . . . . . . . . . . . . . 38 2.5.1 Inference rules for equality . . . . . . . . . . . . . . . 38 2.5.2 Inference rules for quantified statements . . . . . . . . 40 2.6 Proof techniques . . . . . . . . . . . . . . . . . . . . . . . . . 41
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CONTENTS iv 3 Set theory 46 3.1 Naive set theory . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Operations on sets . . . . . . . . . . . . . . . . . . . . . . . . 47 3.3 Proving things about sets . . . . . . . . . . . . . . . . . . . . 49 3.4 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . 51 3.5 Cartesian products, relations, and functions . . . . . . . . . . 52 3.5.1 Examples of functions . . . . . . . . . . . . . . . . . . 54 3.5.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.5.3 Functions of more (or less) than one argument . . . . 55 3.5.4 Composition of functions . . . . . . . . . . . . . . . . 55 3.5.5 Functions with special properties . . . . . . . . . . . . 55 3.5.5.1 Surjections . . . . . . . . . . . . . . . . . . . 56 3.5.5.2 Injections . . . . . . . . . . . . . . . . . . . . 56 3.5.5.3 Bijections . . . . . . . . . . . . . . . . . . . . 56 3.5.5.4 Bijections and counting . . . . . . . . . . . . 56 3.6 Constructing the universe . . . . . . . . . . . . . . . . . . . . 57 3.7 Sizes and arithmetic . . . . . . . . . . . . . . . . . . . . . . . 59 3.7.1 Infinite sets . . . . . . . . . . . . . . . . . . . . . . . . 59 3.7.2 Countable sets . . . . . . . . . . . . . . . . . . . . . . 61 3.7.3 Uncountable sets . . . . . . . . . . . . . . . . . . . . . 61 3.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4 The real numbers 63 4.1 Field axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.1.1 Axioms for addition . . . . . . . . . . . . . . . . . . . 64 4.1.2 Axioms for multiplication . . . . . . . . . . . . . . . . 65 4.1.3 Axioms relating multiplication and addition . . . . . . 67 4.1.4 Other algebras satisfying the field axioms . . . . . . . 68 4.2 Order axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.3 Least upper bounds . . . . . . . . . . . . . . . . . . . . . . . 70 4.4 What’s missing: algebraic closure . . . . . . . . . . . . . . . . 72 4.5 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.6 Connection between the reals and other standard algebras . . 73 4.7 Extracting information from reals . . . . . . . . . . . . . . . . 74 5 Induction and recursion 76 5.1 Simple induction . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.2 Alternative base cases . . . . . . . . . . . . . . . . . . . . . . 78 5.3 Recursive definitions work . . . . . . . . . . . . . . . . . . . . 79 5.4 Other ways to think about induction . . . . . . . . . . . . . . 79
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CONTENTS v 5.5 Strong induction . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.6 Recursively-defined structures . . . . . . . . . . . . . . . . . . 82 5.6.1
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