The Elements of Advanced Mathematics

Content: ""Halftitle Page""
 ""Title Page""
 ""Copyright""
 ""Table of Contents""
 ""Preface to the Fourth Edition""
 ""Preface to the Third Edition""
 ""Preface to the Second Edition""
 ""Preface to the First Edition""
 ""1. Basic Logic""
 ""1.1 Principles of Logic""
 ""1.2 Truth""
 ""1.3 â#x80
#x9C
Andâ#x80
#x9D
 and â#x80
#x9C
Orâ#x80
#x9D
""
 ""1.4 â#x80
#x9C
Notâ#x80
#x9D
""
 ""1.5 â#x80
#x9C
if-thenâ#x80
#x9D
""
 ""1.6 Contrapositive, Converse, and â#x80
#x9C
Iffâ#x80
#x9D
""
 ""1.7 Quantifiers""
 ""1.8 Truth and Provability""
 ""Exercises""
 ""2. Methods of Proof""
 ""2.1 What is a Proof?""
 ""2.2 Direct Proof""
 ""2.3 Proof by Contradiction"" ""2.4 Proof by Mathematical Induction""""2.5 Other Methods of Proof""
 ""2.5.1 Proof by Cases""
 ""2.5.2 Proof by Contraposition""
 ""2.5.3 Counting Arguments""
 ""Exercises""
 ""3. Set Theory""
 ""3.1 Undefinable Terms""
 ""3.2 Elements of Set Theory""
 ""3.3 Venn Diagrams""
 ""3.4 Further Ideas in Elementary Set Theory""
 ""3.5 Indexing and Extended Set Operations""
 ""Exercises""
 ""4. Relations and Functions""
 ""4.1 Relations""
 ""4.2 Order Relations""
 ""4.3 Functions""
 ""4.4 Combining Functions""
 ""4.5 Cantorâ#x80
#x99
s Notion of Cardinality""
 ""4.6 Hilbertâ#x80
#x99
s Hotel Infinity"" ""Exercises""""5. Axioms, Paradoxes, and Rigor""
 ""5.1 Axioms of Set Theory""
 ""5.2 The Axiom of Choice""
 ""5.2.1 Well Ordering""
 ""5.2.2 The Continuum Hypothesis""
 ""5.2.3 Zornâ#x80
#x99
s Lemma""
 ""5.2.4 The Hausdorff Maximality Principle""
 ""5.2.5 The Banachâ#x80
#x93
Tarski Paradox""
 ""5.3 Independence and Consistency""
 ""5.4 Set Theory and Arithmetic""
 ""Exercises""
 ""6. Number Systems""
 ""6.1 The Natural Number System""
 ""6.2 The Integers""
 ""6.3 The Rational Numbers""
 ""6.4 The Real Number System""
 ""6.5 OPTIONAL SECTION: Construction of the Real Number System"" ""6.6 The Nonstandard Real Number System""""6.6.1 The Need for Nonstandard Numbers""
 ""6.6.2 Filters and Ultrafilters""
 ""6.6.3 A Useful Measure""
 ""6.6.4 An Equivalence Relation""
 ""6.6.5 An Extension of the Real Number System""
 ""6.7 The Complex Numbers""
 ""6.8 The Quaternions, the Cayley Numbers, and Beyond""
 ""Exercises""
 ""7. More on the Real Number System""
 ""7.0 Introductory Remark""
 ""7.1 Sequences""
 ""7.2 Open Sets and Closed Sets""
 ""7.3 Compact Sets""
 ""7.4 The Cantor Set""
 ""7.4.1 Construction of a Remarkable Compact Set""
 ""Exercises""
 ""8. A Glimpse of Topology"" ""8.1 What Is Topology?""""8.2 First Definitions""
 ""8.3 Mappings""
 ""8.4 The Separation Axioms""
 ""8.5 Compactness""
 ""Exercises""
 ""9. Elementary Number Theory""
 ""9.1 Prime Numbers""
 ""9.2 Greatest Common Divisor""
 ""9.3 Modular Arithmetic""
 ""9.4 Wilsonâ#x80
#x99
s Theorem and the Chinese Remainder Theorem""
 ""9.5 The Eulerâ#x80
#x93
Fermat Theorem""
 ""9.6 Properties of Relatively Prime Integers""
 ""Exercises""
 ""10. Zero-Knowledge Proofs and Cryptography""
 ""10.1 Basics and Background""
 ""10.2 Preparation for RSA""
 ""10.3 The RSA System Enunciated""