The Elements of Advanced Mathematics, Fourth Edition

✍ Scribed by Steven G. Krantz

• Publisher: Chapman and Hall/CRC
• Year: 2017
• Tongue: English
• Leaves: 414
• Series: Textbooks in Mathematics
• Edition: 4th ed
• Category: Library

✦ Synopsis

The Elements of Advanced Mathematics, Fourth Edition is the latest edition of the author’s bestselling series of texts. Expanding on previous editions, the new Edition continues to provide students with a better understanding of proofs, a core concept for higher level mathematics. To meet the needs of instructors, the text is aligned directly with course requirements.

The author connects computationally and theoretically based mathematics, helping students develop a foundation for higher level mathematics. To make the book more pertinent, the author removed obscure topics and included a chapter on elementary number theory. Students gain the momentum to further explore mathematics in the real world through an introduction to cryptography. These additions, along with new exercises and proof techniques, will provide readers with a strong and relevant command of mathematics.

✦ Table of Contents

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""