Elements of Advanced Mathematics, Third Edition / Edition 3

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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.

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Editorial Reviews

From the Publisher
… 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

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Product Details

  • ISBN-13: 9781439898345
  • Publisher: Taylor & Francis
  • Publication date: 4/2/2012
  • Edition description: New Edition
  • Edition number: 3
  • Pages: 367
  • Product dimensions: 6.30 (w) x 9.30 (h) x 0.90 (d)

Meet 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.

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Table of Contents

Basic Logic
Principles of Logic
"And" and "Or"
Contrapositive, Converse, and "Iff"
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
Order Relations
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
Open Sets and Closed Sets
Compact Sets
The Cantor Set

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

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
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



Exercises appear at the end of each chapter.

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