Elements of advanced mathematics

著者

• Krantz, Steven G. (Steven George)

書誌事項

Elements of advanced mathematics

Steven G. Krantz

CRC Press, Taylor & Francis Group, c2012

3rd ed

• : hardback

注記

Includes bibliography (p. 341-343) and index

内容説明・目次

内容説明

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.

目次

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.

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詳細情報

• NII書誌ID(NCID): BB09105891
• ISBN:
  • 9781439898345
• 出版国コード: us
• タイトル言語コード: eng
• 本文言語コード: eng
• 出版地: Boca Raton, Fla.
• ページ数/冊数: xvi, 351 p.
• 大きさ: 25 cm
• 分類:
  • DC23 : 510
• 件名:
  • LCSH : Mathematics -- Textbooks