Elements of Logic via Numbers and SetsSpringer Science & Business Media, 25 sept. 1998 - 188 pages In mathematics we are interested in why a particular formula is true. Intuition and statistical evidence are insufficient, so we need to construct a formal logical proof. The purpose of this book is to describe why such proofs are important, what they are made of, how to recognize valid ones, how to distinguish different kinds, and how to construct them. This book is written for 1st year students with no previous experience of formulating proofs. Dave Johnson has drawn from his considerable experience to provide a text that concentrates on the most important elements of the subject using clear, simple explanations that require no background knowledge of logic. It gives many useful examples and problems, many with fully-worked solutions at the end of the book. In addition to a comprehensive index, there is also a useful `Dramatis Personae` an index to the many symbols introduced in the text, most of which will be new to students and which will be used throughout their degree programme. |
Table des matières
Numbers | 1 |
12 Proof by Contradiction | 5 |
13 Proof by Contraposition | 8 |
14 Proof by Induction | 10 |
15 Inductive Definition | 19 |
16 The Wellordering Principle | 27 |
Logic | 35 |
22 Truth Tables | 39 |
43 Number Systems | 79 |
44 Orderings | 85 |
Maps | 89 |
52 Examples | 94 |
53 Injections Surjections and Bijections | 99 |
54 Peanos Axioms | 105 |
Cardinal Numbers | 113 |
61 Cardinal Arithmetic | 114 |
23 Syllogisms | 43 |
24 Quantifiers | 48 |
Sets | 53 |
31 Introduction | 54 |
32 Operations | 58 |
33 Laws | 62 |
34 The Power Set | 65 |
Relations | 71 |
41 Equivalence Relations | 72 |
42 Congruences | 75 |
62 The CantorSchroederBernstein theorem | 118 |
63 Countable Sets | 121 |
64 Uncountable Sets | 126 |
Solutions to Exercises | 131 |
Guide to the Literature | 163 |
Bibliography | 165 |
Dramatis Personae in approximate order of appearance | 167 |
171 | |
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Expressions et termes fréquents
a₁ addition and multiplication algebra arithmetic assertion associative law assume P(m Axiom of Choice axioms bijection binary operation called cardinal Cauchy sequence Chapter codomain columns commutative law complex numbers contraposition coprime corresponding counter-example deduce define definition denote described disjoint distributive law divisor domain element equation equivalence classes equivalence relation example expression false finite set follows formula given set inclusion inductive base inductive step injection least logic lowest terms Map(A Map(B mathematics modulo multiplicative inverse natural map negation non-empty notation number systems pair partial ordering partition positive integers premise previous exercise prime propositions quantified r₁ rational numbers real numbers residue classes right-hand side set theory statement subsets surjective syllogistic symbols Theorem 6.3 topology true truth table truth-values Un+1 values variables Vx P(x whence write Z/nZ zero