Fuzzy Sets and Fuzzy Logic: Theory and Applications [Facsimile] [Paperback]

A comprehensive and authoritative presentation of developments in the mathematics of fuzzy systems theory over the past thiry years. While the basic mathematics are presented, this book is not for the casual reader, but for those seriously interested in fuzzy systems theory. If the reader does not have a good mathematical background, he or she will find this book tough going. Coverage of theoretical fuzzy concepts is quite complete, including theory of fuzzy sets, fuzzy arithmetic, fuzzy relations, possiblity theory, fuzzy logic and uncertainty-based information.

The applications section presents theory which could be useful in applications rather than the applications themselves. References are given, but no distinction is made between theoretical work and real-world applications, and many of the references are old and out-of-date.

For a reference book on fuzzy mathematics, this book is superb; as a pointer to real-world applications, it leaves something to be desired.

George and Bo have been as thorough and lucid in preparing this book as well as George explicated systems thinking in the very first book of his I read, "An Approach to General Systems Theory." Here, as there, without compromising mathematical rigor, the goal of this book is to elaborate its subject matter in such a robust manner that it has multidisciplinary appeal. As always, the reader is given a flexible, almost interactive, access to the what, why and how of fuzzy thinking. Despite the exception taken by Professor Lotfi A. Zadeh, the "founder of fuzzy logic," the percipient reader will appreciate the authors' unusual association of "fuzzy measure," that is, the degree of belief that a particular element belongs to a crisp set, (not the degree of membership in the set), with Possibility Theory so as to clarify the differences between fuzzy set theory and probability theory. The illustrative applications are not only case studies that one may pick and choose from for examination and emulation but also constitute incontrovertible evidence of the successful and promising realization of the fuzzy paradigm. As a former professor of engineering at Rutgers University, I found the 79-page Instructor's manual helpful for self- or extended study and I assume it would be valuable for teaching. I have read many books on fuzzy logic and I judge this to be the most balanced to date, (early 1998), - not filled with C++ code or trying to sell a software package nor is it theoretically daunting - it is simply an inviting demonstration of how fuzzy logic clears up foggy modeling and analysis.

I would hesistate to give anything less than a 5 star review to anything on fuzzy set theory in the wide sense. Make no mistake reading this book is worth your time. Yet, some significant problems do exist with this text.

First off, read the proofs in this carefully and figure out if they do work. Klir and Yuan know that appealing to contradiction in theorem proving doesn't often work out in fuzzy theory. Yet, they go ahead and use it almost recklessly. One example is their proof on fuzzy numbers that says that they are all continuous on pages 99 to 100. After about a full, condensed page of mathematical reasoning they say that left fuzzy numbers are continuous from the left and that right fuzzy numbers are continuous from the right. After their supposed "proof" they claim that "The implication of Theorem 4.1 is that every fuzzy number be represented in the form of (4.1)." 4.1 shows a discontinuous fuzzy number. A jump discontinuity to speak more specifically. Consequently, their supposed "theorem" doesn't exactly work as a "theorem". Perhaps I misunderstand and they have some different idea of continuity. I don't get it though and neither does any other mathematician, as any break in a function whatsoever means discontinuity.

More interestingly, some of their axioms for fuzzy set don't hold. For instance, on page 62 Axiom i1 (i for intersection) says that i(a, 1)=a, which they label as the "boundary conidition." This does hold for drastic products. However, it doesn't hold for all fuzzy intersections. As Buckley and Eslami point out the axioms or necessary conditions for fuzzy intersections work out as "(1) 0<=a, b<=1 and i(a, b) is in

[0, 1]; (2) i(1, 1)=1; and (3) i(0, 1)=i(1, 0)=i(0, 0)=0." Consquently, (ab)/max{a, b, .5} qualifies as a fuzzy intersections. Here i(.6, .4)=.24/.6=24/60=2/5=.4

I don't exactly mean the above to significantly downgrade the work of Klir and Yuan. Their collection of papers of Zadeh does have signficant value, even if it costs a lot. The sheer enormity and very comprehensive nature of this quasi-encyclopedia makes it worth the read. The problems are interesting and challenging, if you choose to do them. I do appreciate the authors mentioning that the problems are meant to enchance the reader's understanding. That Klir and Yuan provide a comprehensive bibliography and consulted many, many original papers before and while writing their text alone indicates they do know something and did some thinking here. Their graphs do help to illustrate their ideas. So, I do advise that you read the book. Just read carefully.

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