Elements of Advanced 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.

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

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

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

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

Axioms of Set Theory, Paradoxes, and RigorAxioms of Set Theory The Axiom of ChoiceIndependence and Consistency Set Theory and Arithmetic

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

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

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

Theoretical Computer ScienceIntroductory RemarksPrimitive Recursive FunctionsGeneral Recursive FunctionsDescription of Boolean AlgebraAxioms of Boolean AlgebraTheorems in Boolean AlgebraIllustration of the Use of Boolean LogicThe Robbins Conjecture

The P/NP ProblemIntroduction 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-CompletenessPolynomial Equivalence Definition of NP-Completeness

Examples of Axiomatic TheoriesGroup 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.