# Elements of Advanced Mathematics, Third Edition

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## Description

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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## Product details

- Hardback | 367 pages
- 154.94 x 236.22 x 22.86mm | 657.71g
- 20 Apr 2012
- Taylor & Francis Ltd
- Whittles Publishing
- London, United Kingdom
- English
- New edition
- 3rd New edition
- 3 Tables, black and white; 38 Illustrations, black and white
- 1439898340
- 9781439898345
- 1,265,098

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## Table of contents

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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## Review quote

-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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## About Steven G. Krantz

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