Elements of Advanced Mathematics, Third Edition

Steven G. Krantz is a professor of mathematics at Washington University in St. Louis, Missouri. He has published over 150 papers and nearly 70 books and has been an editor of several journals. He earned a Ph.D. in mathematics from Princeton University. His research interests include complex variables, harmonic analysis, partial differential equations, geometry, interpolation of operators, and real analysis.

Basic Logic
Principles of Logic
Truth
"And" and "Or"
"Not"
"If-Then"
Contrapositive, Converse, and "Iff"
Quantifiers
Truth and Provability





Methods of Proof
What Is a Proof?
Direct Proof
Proof by Contradiction
Proof by Induction
Other Methods of Proof





Set Theory
Undefinable Terms
Elements of Set Theory
Venn Diagrams
Further Ideas in Elementary Set Theory
Indexing and Extended Set Operations





Relations and Functions
Relations
Order Relations
Functions
Combining Functions
Cantor’s Notion of Cardinality





Axioms of Set Theory, Paradoxes, and Rigor
Axioms of Set Theory
The Axiom of Choice
Independence and Consistency
Set Theory and Arithmetic





Number Systems
The Natural Number System
The Integers
The Rational Numbers
The Real Number System
The Nonstandard Real Number System
The Complex Numbers
The Quaternions, the Cayley Numbers, and Beyond





More on the Real Number System
Introductory Remark
Sequences
Open Sets and Closed Sets
Compact Sets
The Cantor Set





A Glimpse of Topology
What Is Topology?
First Definitions
Mappings
The Separation Axioms
Compactness





Theoretical Computer Science
Introductory Remarks
Primitive Recursive Functions
General Recursive Functions
Description of Boolean Algebra
Axioms of Boolean Algebra
Theorems in Boolean Algebra
Illustration of the Use of Boolean Logic
The Robbins Conjecture





The P/NP Problem
Introduction
The Complexity of a Problem
Comparing Polynomial and Exponential Complexity
Polynomial Complexity
Assertions That Can Be Verified in Polynomial Time
Nondeterministic Turing Machines
Foundations of NP-Completeness
Polynomial Equivalence
Definition of NP-Completeness





Examples of Axiomatic Theories
Group Theory
Euclidean and Non-Euclidean Geometry





Zero-Knowledge Proofs
Basics and Background
Preparation for RSA
The RSA System Enunciated
The RSA Encryption System Explicated
Zero-Knowledge Proofs





Solutions to Selected Exercises


Bibliography


Index





Exercises appear at the end of each chapter.