This book has enjoyed considerable use and appreciation during its first four editions. With hundreds of students having learned out of early editions, the author continues to find ways to modernize and maintain a unique presentation.
What sets the book apart is the excellent writing style, exposition, and unique and thorough sets of exercises. This edition offers a more instructive preface to assist instructors on developing the course they prefer. The prerequisites are more explicit and provide a roadmap for the course. Sample syllabi are included.
As would be expected in a fifth edition, the overall content and structure of the book are sound.
This new edition offers a more organized treatment of axiomatics. Throughout the book, there is a more careful and detailed treatment of the axioms of set theory. The rules of inference are more carefully elucidated.
Additional new features include:
- An emphasis on the art of proof.
- Enhanced number theory chapter presents some easily accessible but still-unsolved problems. These include the Goldbach conjecture, the twin prime conjecture, and so forth.
- The discussion of equivalence relations is revised to present reflexivity, symmetry, and transitivity before we define equivalence relations.
- The discussion of the RSA cryptosystem in Chapter 8 is expanded.
- The author introduces groups much earlier. Coverage of group theory, formerly in Chapter 11, has been moved up; this is an incisive example of an axiomatic theory.
Recognizing new ideas, the author has enhanced the overall presentation to create a fifth edition of this classic and widely-used textbook.
Author(s): Steven G. Krantz
Series: Textbooks in Mathematics
Edition: 5
Publisher: Chapman and Hall/CRC
Year: 2022
Language: English
Commentary: Publisher PDF
Pages: 312
City: Boca Raton, FL
Tags: Logic; Proof; Set Theory; Relations; Functions; Number System; Number Theory
Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
Author Bio
1. Basic Logic
1.1. Principles of Logic
1.2. Truth
1.3. “And” and “Or”
1.4. “Not”
1.5. “if-then”
1.6. Contrapositive, Converse, and “Iff”
1.7. Quantifiers
1.8. Truth and Provability
Exercises
2. Methods of Proof
2.1. What Is a Proof?
2.2. Where Do You Find a Proof?
2.3. What Is a Conjecture?
2.4. Direct Proof
2.5. Proof by Contradiction
2.6. Proof by Mathematical Induction
2.7. Other Methods of Proof
2.7.1. Proof by cases
2.7.2. Proof by contraposition
2.7.3. Counting arguments
Exercises
3. Set Theory
3.1. Undefinable Terms
3.2. Elements of Set Theory
3.3. Venn Diagrams
3.4. Further Ideas in Elementary Set Theory
3.5. Indexing and Extended Set Operations
Exercises
4. Relations and Functions
4.1. Relations
4.2. Order Relations
4.3. Functions
4.4. Combining Functions
4.5. Cantor’s Notion of Cardinality
4.6. Hilbert’s Hotel Infinity
Exercises
5. Number Systems
5.1. The Natural Number System
5.2. The Integers
5.3. The Rational Numbers
5.4. The Real Number System
5.5. Construction of the Real Numbers
5.6. The Nonstandard Real Number System
5.6.1. The need for nonstandard numbers
5.6.2. Filters and ultrafilters
5.6.3. A useful measure
5.6.4. An equivalence relation
5.6.5. An extension of the real number system
5.7. The Complex Numbers
5.8. The Quaternions, the Cayley Numbers, and Beyond
Exercises
6. More on the Real Number System
6.0. Introductory Remark
6.1. Sequences
6.2. Open Sets and Closed Sets
6.3. Compact Sets
6.4. The Cantor Set
6.4.1. Construction of a remarkable compact set
Exercises
7. Elementary Number Theory
7.1. Prime Numbers
7.2. Greatest Common Divisor
7.3. Modular Arithmetic
7.4. Theorems of Wilson and Chinese Remainder
7.5. The Euler–Fermat Theorem
7.6. Properties of Relatively Prime Integers
Exercises
8. Zero-Knowledge Proofs
8.1. Basics and Background
8.2. Preparation for RSA
8.2.1. Background ideas
8.2.2. Computational complexity
8.3. The RSA System Enunciated
8.3.1. Description of RSA
8.4. The RSA Encryption System Explicated
8.4.1. Explanation of RSA
8.5. Zero-Knowledge Proofs
8.5.1. How to keep a secret
8.6. Concluding Remarks
Exercises
9. Examples of Axiomatic Theories
9.1. Group Theory
9.2. Euclidean and Non-Euclidean Geometry
Exercises
APPENDIX: Axiomatics
A1. Axioms of Set Theory
A2. The Axiom of Choice
A2.1. Well ordering
A2.2. The continuum hypothesis
A2.3. Zorn’s lemma
A2.4. The Hausdorff maximality principle
A2.5. The Banach–Tarski paradox
A3. Independence and Consistency
A4. Set Theory and Arithmetic
Exercises
Solutions to Selected Exercises
Bibliography
Index
