Elements of Advanced Mathematics, Third Edition: (Textbooks in Mathematics)

For many years, this classroom-tested, best-selling text has guided mathematics students to more advanced studies in topology, abstract algebra, and real analysis. Elements of Advanced Mathematics, Third Edition retains the content and character of previous editions while making the material more up-to-date and significant. This third edition adds four new chapters on point-set topology, theoretical computer science, the P/NP problem, and zero-knowledge proofs and RSA encryption. The topology chapter builds on the existing real analysis material. The computer science chapters connect basic set theory and logic with current hot topics in the technology sector. Presenting ideas at the cutting edge of modern cryptography and security analysis, the cryptography chapter shows students how mathematics is used in the real world and gives them the impetus for further exploration. This edition also includes more exercises sets in each chapter, expanded treatment of proofs, and new proof techniques. Continuing to bridge computationally oriented mathematics with more theoretically based mathematics, this text provides a path for students to understand the rigor, axiomatics, set theory, and proofs of mathematics. It gives them the background, tools, and skills needed in more advanced courses.

Table of Contents:
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.

About the Author :
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.

Review :
… one of the difficulties that students have with university mathematics is being able to relate it to what they’ve done at school. In this respect, the work on logic, sets, proof, relations and functions plays an essential bridging role. But another problem to be addressed is to re-present mathematics as a way of knowing—rather than a static body of formalised knowledge. In this book, Steven Krantz tackles this anomaly by including many open-ended problems in the rich collections of exercises. … The new chapters on theoretical computer science are concisely lucid, and I learned much by reading them. … this book engages the reader in really meaningful aspects of mathematics: it is well organized and is written with accuracy. … it is recommended as a possible course text for those who are planning to teach a foundation course. —P.N. Ruane, MAA Reviews, July 2012 Retains the content and character of previous editions while making the material more up-to-date and significant. … gives readers the background, tools, and skills necessary in more advanced mathematical work. — L'Enseignement Mathematique, 2012