Elements of advanced mathematics /
Steven G, Krantz
- 5
- Boca Raton CRC Press 2022
- 295p.
This book has enjoyed considerable use and appreciation during its first four editions. With hundreds of students learning from 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 iin developing their preferred course. The prerequisites are more explicit and provide a roadmap for the system. Sample syllabi are included.
As expected in the 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.
The enhanced number theory chapter presents some easily accessible but still-unsolved problems. These include the Goldbach conjecture, the twin prime conjecture, etc.
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 intelligent example of an axiomatic approach.
Recognizing new ideas, the author has enhanced the overall presentation to create a fifth edition of this classic and widely used textbook.