The elements of advanced mathematics / Steven G. Krantz.

Includes bibliographical references (pages 209-210) and index.

Machine generated contents note: 1 Basic Logic 1 -- 1.1 Principles of Logic1 -- 1.2 Truth2 -- 1.3 "And" and "Or" 3 3 -- 1.4 "Not"6 -- 1.5 "If- Then"7 -- 1.6 Contrapositive, Converse, and "Iff"10 -- 1.7 Quantifiers13 -- 1.8 Truth and Provability17 -- Exercises21 --2 Methods of Proof 27 -- 2.1 What is a Proof?27 -- 2.2 Direct Proof28 -- 2.3 Proof by Contradiction33 -- 2.4 Proof by Induction36 -- 2.5 Other Methods of Proof40 -- Exercises43 --3 Set Theory 47 -- 3.1 Undefinable Terms47 -- 3.2 Elements of Set Theory48 -- 3.3 Venn Diagrams52 -- 3.4 Further Ideas in Elementary Set Theory53 -- 3.5 Indexing and Extended Set Operations55 -- Exercises57 -- 4 Relations and Functions 61 -- 4.1 Relations61 -- 4.2 Order Relations 64 -- 4.3 Functions 66 -- 4.4 Combining Functions 69 -- 4.5 Cantor's Notion of Cardinality 73 -- Exercises 85 --5 Axioms of Set Theory, Paradoxes, and Rigor 91 -- 5.1 Axioms of Set Theory91 -- 5.2 The Axiom of Choice 94 -- 5.2.1 Well Ordering94 -- 5.2.2 The Continuum Hypothesis95 -- 5.2.3 Zorn's Lemma95 -- 5.2.4 The Hausdorff Maximality Principle96 -- 5.2.5 The Banach-Tarski Paradox97 -- 5.3 Independence and Consistency97 -- 5.4 Set Theory and Arithmetic 99 -- Exercises101 --6 Number Systems 105 -- 6.0 Preliminary Remarks105 -- 6.1 The Natural Number System106 -- 6.2 The Integers111 -- 6.3 The Rational Numbers117 -- 6.4 The Real Number System124 -- 6.5 The Non-Standard Real Number System133 -- 6.5.1 The Need for Non-Standard Numbers133 -- 6.5.2 Filters and Ultrafilters133 -- 6.5.3 A Useful Measure133 -- 6.5.4 An Equivalence Relation134 -- 6.5.5 An Extension of the Real Number System134 -- 6.6 The Complex Numbers135 -- 6.7 The Quaternions, the Cayley Numbers, and Beyond139 -- Exercises141 --7 More on the Real Number System 147 -- 7.0 Introduction 147 -- 7.1 Sequences147 -- 7.2 Open Sets and Closed Sets149 -- 7.3 Compact Sets151 -- 7.4 The Cantor Set152 -- 7.4.1 Construction of a Remarkable Compact Set153 -- Exercises156 -- 8 Examples of Axiomatic Theories 159 -- 8.0 Introductory Remarks159 -- 8.1 Group Theory159 -- 8.2 Euclidean and Non-Euclidean Geometry168 -- Exercises180.