This text helps bridge computationally oriented mathematics with more theoretically oriented mathematics, preparing readers for more advanced courses that require understanding proofs. It covers logic, set theory, axiomatics, number systems, and reading, evaluating, and creating proofs. This third edition includes some of the more modern topics from theoretical computer science, such as the P/NP problem, Boolean algebra, and Church's thesis. Along with new problems and examples, it also illustrates logic in action and discusses topics from the real number system and topology.
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
