Product Description
Wielding his widely recognized powers of explanation, Paul Krugman lays bare the hidden facts behind the $2 trillion tax cut.

With huge budget surpluses just ahead, the question of whether to cut taxes has shifted to when? and by how much? With Fuzzy Math, Paul Krugman dissects the Bush tax proposal and shows us who wins, who loses, and how quickly the tax cuts will consume the surplus. Always the equal-opportunity critic when it comes to faulty economics, Krugman also tucks into the Democratic alternatives to the Bush plan.

This little book packs a big wallop. Together with major media appearances, it puts Krugman's wisdom and steely-eyed analysis firmly at the center of the debate about how to spend upwards of $2 trillion. It may very well change the course of history. ... Read more

Customer Reviews (12)

Good overview of the Dubya tax cuts
While reading this book it is helpful to keep Krugman's (Nobel Prize winning economist) political stance in perspective (he himself admits this in the book). He is decidedly anti-conservative in his statements but seems to have taken on a more centrist approach lately.

The book itself is based on hard economic data and essentially shreds to pieces the Bush team's claims of the tax cuts being pro-poor and middle clsss. In fact the cuts disproportionately favored the rich and seriously rich families. Krugman lays out his case using numbers available to him in 2000 but the past 9 years seem to be bearing out his claims. The cuts did not favor a supply-side led boom in investment (which is the Republican stance) and in fact have helped pushed the federal deficit to historic highs. Add in the government led bailouts and spending programs being enacted by the current Democratic govt and the deficit is projected to reach 100% of the nation's GDP in the next decade.

Krugman details his claims as to why Bush favored tax cuts (smaller government) and why sometimes a case may be made for tax increases. The book does not go into much depth on this topic and very little economic justifications are provided for this point of view.

I cannot give this book more than 4 stars because it is too short (~120 pages) for a topic as deep as the federal deficit and tax reform. There are no references, notes or even an index which makes it seem as if Krugman has had the last word. However it is a good text that sheds light on how government fiscal policy may be analyzed by a curious citizen to separate fact from fiction.

prophetic?
In "Fuzzy Math," Paul Krugman debunks the deceptive hype deployed on behalf of the tax cut of 2001.Krugman points out how so extravagant a tax cut will force serious reductions in services - most likely, to social security.

Four years later, pundits and analysts told Americans of the dire threat to social security - a threat those same pundits and analysts dismissed when defending the cuts.As Krugman suggested they would.

Still, some might be disappointed to find that Krugman is less prophetic than simply an academic applying basic economic observations in a realistic manner.By clarifying processes of taxation, spending, and budgeting, Krugman succeeds in clearing away fog and myth, offering a healthy handbook for economics to all American citizens.

One of Krugman's best -- brief and informative
Every policy-maker and voter should read this book.After months of Krugman's anti-tac-cut NY Times Op-Eds, I was sick of hearing about this debate.But "Fuzzy Math" literally changed my mind in one night.It is not only a guide to the Bush tax cut but also a layman's guide to general tax policy, tax law, the federal budget, and distributional issues.Not only that, but Krugman provides a novel theory (at least to me) on why anti-big-government ideologues prefer tax cuts for the rich disproportionately over tax cuts for the bottom 99%.Krugman also exposes many statistical and other tricks that policy-makers play on the public in order to promote their programs.In short, this book does so much so thoroughly, and I am amazed that Krugman fit it all into so few pages.

Highly recommended
I recommend this book to anyone, even though the tax cuts Paul Krugman argues against have already come.Krugman, who is a New York Time od-ed writer and also a policy professor at Princeton, presents clear reasons why the Bush tax cuts are not a good idea.

Conservatives will find the book biased, which it is since Krugman is pretty democratic.Although conservatives might be able to argue the political philosophy of progressive versus regressive taxes, they will find it very difficult to challenge the numbers that Krugman presents.The end conclusion is that Bush has used "fuzzy math" to propose a tax cut and that the money is just not there for such a huge cut.Krugman is right.

Even though the cuts have already come, this book is a great (and quick) read because it gives a clear explanation of social security, medicare, and other issues related to the national budget.Clear, concise, and easy to understand.

This is important. Everybody should read this book.
This book needs to be read by every voting American, even those who support the Bush tax cut. Author Paul Krugman clearly explains the economic and political environments in which this tax plan takes place and concludes, first, that the tax cut is not only a bad idea but might have serious consequences as the Social Security/Medicare system becomes strapped and second, that "at every stage of the debate Bush and his people have tried to obscure what they were really proposing."
"Fuzzy Math" is a book written for intelligent lay people. I personally read it in two sittings (it's only 122 short pages), then, thinking that I must have missed smething, went back and read it again. It turns out I missed nothing. Krugman breaks down complex economic concepts and explains them with great lucidity and a little bit of wit. It's really an easy read.
Krugman begins by explaining how Bush arrived at his tax cut as the centerpiece of his campaign, first as an antidote to Steve Forbes' "Flat Tax" crusade and second, to secure the support of the far right elements of the Republican Party. He then describes the efficacy of tax cuts as an economic tool, particularly as they might be used to stimulate a sluggish economy (never an issue for Bush until the economy suddenly turned sour). He concludes that this is best left to the Federal Reserve Board's manipulation of interest rates. He further compares "demand side" tax reductions, aimed primarily at consumers, with "supply side" cuts which are directed toward potential producers and demonstrates that despite the Reagan rhetoric, the economic recovery of the early '80's was demand side driven and that a real supply side expansion occurred during the late '90's happened despite Bill Clinton's upper bracket tax increase.
Nexy Krugman explains the Federal Budget, beginning with where the money goes and then where it comes from. He explains that we've gone from being a "military state" to a "retirement state". He admittedly caricatures that, based on federal spending "the federal government has become a large retirement community that does some military stuff and a bit of humanitarian stuff on the side". He also explains that our national retirement program is not fully funded (as is a private pension plan). Instead the current group of retirees is living off the contributions of the current group of workers and that enormous problems will begin when the number of retirees begins to swell as the number of workers begins to shrink (about 2011). This is aleo why privatization of Social Security/Medicare is a bad idea: it will simply pull the rug out from under the feet of the current group of retirees. He discusses the origins of the recent budget surplus andhow it was tied to the recent economic boom.
He then breaks down the Bush tax cut, explaining who gets what. Using figures from the Center on Budget and Policy Priorities and Citizens for Tax Justice (stats from conservative think tanks are unavailable) he concludes that about 40% of American families will get nothing or very little while the top 1% will collect about 45% of the benefits. He analyzes the Treasury Department's statistics in light of this data and exposes the hucksterism involved in the official Bush line. Unfortunately this is the only piont at which Krugman cites sources although he uses statistics elsewhere in this book. More citations would have given the book a little more authority.
Finally he proposes an alternative, a "smaller, faster, cheaper, better" cut that will get money into the hands of consumers faster and will be "front loaded" (benefits sooner) as opposed to Bush's "back loaded" (most benefits arrive much later) and so will have an immediate effect on the economy.
Krugman concludes with a swipe at the "utter dishonesty of the sales campaign".
There is no reason why every American citizen should not read this book. It explains what's going on in the tax debate and does so clearly and simply. In fact, bookshelves in any participatory democracy should be full of books like this. ... Read more

Product Description
Presents the fundamentals of fuzzy systems and neural networks and then joins them synergistically - combining the feature extraction and modeling capabilities of the neural network with the representation capabilities of fuzzy systems. ... Read more

Customer Reviews (1)

a great tool for engineering research
I have been impressed by the clarity of the presented material and the intelligent choice of the various examples that are found in the text.One rarely finds such a well connected presentation of knowledge.This book gave me the capability to further expand my exploration efforts in this very interesting subject. ... Read more

Product Description
This digital document is an article from The Sporting News, published by Sporting News Publishing Co. on February 25, 2005. The length of the article is 1581 words. The page length shown above is based on a typical 300-word page. The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase. You can view it with any web browser.

Citation Details
Title: Vikings' Moss would be another team's gain: addition by substraction? That's fuzzy math in Minnesota.(NFL)(National Football League)
Author: Vinnie Iyer
Publication: The Sporting News (Magazine/Journal)
Date: February 25, 2005
Publisher: Sporting News Publishing Co.
Volume: 229Issue: 8Page: 71(2)

Distributed by Thomson Gale ... Read more

Product Description
This digital document is an article from National Right to Life News, published by National Right to Life Committee, Inc. on May 1, 2004. The length of the article is 1107 words. The page length shown above is based on a typical 300-word page. The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase. You can view it with any web browser.

Citation Details
Title: Abortion and Breast Cancer: Only Fuzzy Math Can Make the ABC Link Disappear.
Publication: National Right to Life News (Magazine/Journal)
Date: May 1, 2004
Publisher: National Right to Life Committee, Inc.
Volume: 31Issue: 5Page: NA

Distributed by Thomson Gale ... Read more

Product Description
Chinese edition of Paul Krugman's Fuzzy Math: The Essential Guide to the Bush Tax Plan. ... Read more

Product Description
This digital document is an article from Atlantic Economic Journal, published by Atlantic Economic Society on December 1, 2002. The length of the article is 495 words. The page length shown above is based on a typical 300-word page. The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase. You can view it with any web browser.

Citation Details
Title: The fuzzy math on butterfly ballots and the Buchanan vote. (Anthology).(Pat Buchanan and Palm Beach County, Florida's butterfly ballot in the November 7, 2000 presidential election)
Author: Fahim Ahmed
Publication: Atlantic Economic Journal (Refereed)
Date: December 1, 2002
Publisher: Atlantic Economic Society
Volume: 30Issue: 4Page: 442(1)

Distributed by Thomson Gale ... Read more

Product Description
This digital document is an article from The Register-Guard (Eugene, OR), published by The Register Guard on February 29, 2004. The length of the article is 681 words. The page length shown above is based on a typical 300-word page. The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase. You can view it with any web browser.

Citation Details
Title: More fuzzy math.(Editorials)(New overtime rules need additional analysis)(Editorial)
Publication: The Register-Guard (Eugene, OR) (Newspaper)
Date: February 29, 2004
Publisher: The Register Guard
Page: B1

Article Type: Editorial

Distributed by Thomson Gale ... Read more

Product Description

The book contains ten chapters as follows, Prepare Knowledge, Regression and Self-regression Models with Fuzzy Coefficients; Regression and Self-regression Models with Fuzzy Variables, Fuzzy Input/output Model, Fuzzy Cluster Analysis and Fuzzy Recognition, Fuzzy Linear Programming, Fuzzy Geometric Programming, Fuzzy Relative Equation and Its Optimizing, Interval and Fuzzy Differential Equations and Interval and Fuzzy Functional and Their Variation.

It can not only be used as teaching materials or reference books for under-graduates in higher education, master graduates and doctor graduates in the courses of applied mathematics, computer science, artificial intelligence, fuzzy information process and automation, operations research, system science and engineering, and the like, but also serves as a reference book for researchers in these fields, particularly, for researchers in soft science.

Product Description
The first edition of Fuzzy Logic with Engineering Applications (1995) was the first classroom text for undergraduates in the field. Now updated for the second time, this new edition features the latest advances in the field including material on expansion of the MLFE method using genetic algorithms, cognitive mapping, fuzzy agent-based models and total uncertainty. Redundant or obsolete topics have been removed, resulting in a more concise yet inclusive text that will ensure the book retains its broad appeal at the forefront of the literature.

Fuzzy Logic with Engineering Applications, 3rd Edition is oriented mainly towards methods and techniques. Every chapter has been revised, featuring new illustrations and examples throughout. Supporting MATLAB code is downloadable at www.wileyeurope.com/go/fuzzylogic. This will benefit student learning in all basic operations, the generation of membership functions, and the specialized applications in the latter chapters of the book, providing an invaluable tool for students as well as for self-study by practicing engineers.

Customer Reviews (7)

LOADED WITH ERRATA
I purchased the book for a graduate level engineering course.Perhaps the other raters are orders of magnitude smarter than I, because I found the book to be extremely difficult to understand, almost to the point of laughability.The narrative is unnecessarily complex in my opinion, requiring a serious student to read and reread multiple times until a concept can be grasped.And you better have a dictionary handy because you will likely come across terms that even a well educated person will not recognize.To make matters worse, the text is LOADED with errata.On several occasions I struggled for hours working through examples in various chapters, pulling my hair out and not getting it, only to discover later after confronting the professor that the example was wrong.Not only are there numerical errors laced throughout the text and in the answer appendix, but there are MANY instances where the concept/strategy is not fully explained in an example, leaving you to scratch your head as to why particular variables and their value were chosen (not calculated), why a certain step was performed or not performed, and why certain "tactical decisions" were made as you make your way through a procedure.Several of the sketches are sloppy too where the accompanying narrative does not always match with what the sketch is telling you.
For a rather complex subject, the last thing you need is to struggle with the text.There are so many errors, and so many instances where a complex idea or example is not fully explained that I would recommend purchasing other reference material; at least until the 3rd edition of this text comes out and corrects most of the errata.

Much better fuzzy logic books are out there.
With the exception of Ch. 1, the first part of the book is o.k. To see my problem with chapter 1, simply look at the article on bounded rationality he uses in his discussion of a chess game.First, several parts of the article are coppied word for word directly in his text;I have no problem with this, but they should appear in quotes.Second, I don't think his analysis of the article is accurate.
Ross does an excellent job of introducing binary logic and extending the results to fuzzy logic.However, other than this one chapter, the rest of the rest of the book is not worth the read.A much better book is by Yager and Filev.I went with Ross' text because it was published in 2004 (you know, latest is greatest) compared to the 1995 publishing of Yager's book.Don't make the same mistake, just get Yager's book; you will learn more and save yourself some money!

Excelent for a fisrt course in fuzzy logic and systems
The way the fuzzy paradigm is introduced and developed is very inspiring for those that come from more traditional ways of thinking. The author makes the transition from classical logic to fuzzy logic in a very clear way. The fields of aplication of fuzzy logic are very well represented in the book. I would stongly support the author to publish a second edition of the book.

excellent reference book
Very good book for a solid background on Fuzzy Logic.Highly reccomented.Ross is a very involved author and enthousiastic supporter of Fuzzy logic theory use in problem solving.Try to solve some of the problems in the book.They provide much inside on real applications of the fuzzy logic Theory

The least fuzzy book on fuzzy logic
If you decide to buy a book on fuzzy logic or applications of fuzzy logic, this is absolutely the best book for introductory level. In other words, this source gives a non-fuzzy and comprehensive knowledge to reader.
I strongly recommend this book to everybody for an exact comprehensive introduction to the theory of Zadeh. ... Read more

Product Description
Fuzzy Logic has gained increasing acceptance as a way to deal with complexity and uncertainty in many areas of science and engineering. This book is the first to address its practical applications to chemical systems. Ten distinguished authors discuss the role of fuzzy logic in the characterization of a variety of chemical concepts, including chirality, quantum systems, molecular engineering and design, and hierarchical classification methods. Fuzzy Logic in Chemistry will appeal to both students and professionals who are seeking to learn more about theory and applications in an area of growing importance to the physical sciences.

Key Features
* The first book on the applications of fuzzy logic in chemistry
* Covers a topic relevant to many disciplines, including molecular design
* Discusses applications of fuzzy logic to the physical sciences, a rapidly growing area
* Features chapters from highly distinguished authors in the physical sciences ... Read more

Customer Reviews (1)

First book to use fuzzy logic in chemistry
A surprising thing about this book is that it appears to be the first to use fuzzy logic in chemistry. Given that it was published in 1997, that seems rather late, since fuzzy logic was well established in computing and electric engineering in the 1980s.

Ah, but better late than never. Since most chemists have never dealt with fuzzy logic, the editor chose the wise step of having several research chemists write the chapters. Hopefully, this will ease acceptance of the method amongst chemists.

The subjects of the chapters tend to be hard, unsurprisingly. No softball simple textbook examples. Instead, we deal with issues like determining the shape of a large molecule, or the design of a drug. ... Read more

Product Description
Fuzzy sets and fuzzy logic are powerful mathematical tools for modelling and controlling uncertain systems in industry, humanity and nature; they are facilitators for approximate reasoning in decision making in the absence of complete and precise information. Their role is significant when applied to complex phenomena not easily described by traditional mathematics. The text provides an introductory course (with examples and problems), which brings in a systematic way fuzzy sets and fuzzy logic into the educational university and college system. It is designed to serve as a basic text for introducing engineers and scientists from various fields to the theory of fuzzy sets and fuzzy logic, thus enabling them to initiate projects and make applications. ... Read more

Customer Reviews (1)

Beautiful Introduction
I find the approach in this book to fuzzy logic interesting, useful, simple, and beautiful.Starting with something like interval numbers is almost certainly the best way to start a fuzzy logic text.It presents a simple example that is very easily understood and allows one to see what differentiates fuzzy logic from other multi-valued logics in an almost intuitive manner.
I have not worked through all of this book.But, while working through the truth table exercises in Chapter 8 I thought 'couldn't I do these on a spreadsheet like Microsoft Excel or Microsoft Works?'With Bojadziev's formulations how to produce truth tables on a spread sheet (I used Microsoft Works) came almost immediately.(The only real exception was that I had to treat fractional truth values, as functions.Expressly, in Works you type =1/2 in the cell.)To paraphrase Betrand Russell (correct this if I have the author wrong), 'a good notation has a certain suggestion to it.'
Since I did this, I noticed some textual errors.p. 164, (g) reads [~p->(q^~q)]->~p proof by contradiction.This is not a tautology.The tautological form, I think, Bojadziev is looking for is either [p->(q^~q)]->~p, or more likely [~p->(q^~q)]->p.Also, on p. 165 p^(p->q) should not read (1 0 1 0, which is q^(p->q) in context), but
1
0
0
0.Additionally, on p. 267 8.5 (a) for p->~q should read
(0 1/2 1 1/2 1 1 1 1 1).
Still, Bojadziev's notation is the most helpful I have seen for fuzzy logic texts.The notation helped me to think of simpler ways to deal with truth tables.This also allowed me to check many different forms of tautologies and see if they are tautologies in other than 2-value logics.For instance, modus tollens is only a quasi-tautology in greater than 2-valued logics (I have investigated up to 11-valued logics, and no doubt, this could be proven for all n-valued crisp logics).Modus ponens is a tautology for n-valued logics, where n is greater than 2 (at least up to an 11-valued logic, and no doubt this can be proven). This suggests, on a logical basis, that falsifiability is not as strong a criterion as verification.Without this book, such insights would have not happened or would have been much harder. ... Read more

Product Description

This is the first book presenting cardinality theory of fuzzy sets with triangular norms, including its scalar and "fuzzy" streams. This theory constitutes not only a powerful basis but also a useful tool for modelling and processing vague and imprecise quantitative information. The multiple application areas of the theory encompass computer science, soft computing, computing with words, and decision-making. Starting with a presentation of the fundamentals of triangular norms and fuzzy set theory, the book offers a self-contained, concise and systematic exposition of cardinalities of fuzzy sets that includes many examples.

Product Description
Neural networks and fuzzy systems are model free control design approaches that represent an advantage over classical control when dealing with complicated nonlinear actuator dynamics. Neuro-Fuzzy Control of Industrial Systems with Actuator Nonlinearities brings neural networks and fuzzy logic together with dynamical control systems. Each chapter presents powerful control approaches for the design of intelligent controllers to compensate for actuator nonlinearities such as time delay, friction, deadzone, and backlash that can be found in all industrial motion systems, plus a thorough development, rigorous stability proofs, and simulation examples for each design. In the final chapter, the authors develop a framework to implement intelligent control schemes on actual systems.

Rigorous stability proofs are further verified by computer simulations, and appendices contain the computer code needed to build intelligent controllers for real-time applications. Neural networks capture the parallel processing and learning capabilities of biological nervous systems, and fuzzy logic captures the decision-making capabilities of human linguistics and cognitive systems.

Audience
This book is written for students in a college curriculum, for practicing engineers in industry, and for university researchers. ... Read more

Product Description

First book that provides both theory and real world applications of fuzzy arithmetic in a comprehensive style.  

Provides a well-structured compendium that offers both a deeper knowledge about the theory of fuzzy arithmetic and an extensive view on its applications in the engineering sciences making it useful for graduate courses, researchers and engineers.

Presents the basic definitions and fundamental principles of fuzzy arithmetic, derived from fuzzy set theory.

Summarizes the state-of-the-art stage of fuzzy arithmetic, offers a comprehensive composition of different approaches including their benefits and drawbacks, and finally, and presents a completely new methodology of implementation of fuzzy arithmetic with particular emphasis on its subsequent application to real-world systems.

Concentrates on the application of fuzzy arithmetic to the simulation, analysis and identification of systems with uncertain model parameters, as they appear in various disciplines of engineering science.

Focuses on mechanical engineering, geotechnical engineering, biomedical engineering, and control engineering.

Product Description
The theory of cognitive maps was developed in 1976. Its main aim was the representation of (causal) relationships among “concepts” also known as “factors” or “nodes”.Concepts could be assigned values. Causal relationships between two concepts could be of three types: positive, negative or neutral. Increase in the value of a concept would yield a corresponding positive or negative increase at the concepts connected to it via relationships.In 1986 Bart Kosko introduced the notion of fuzziness to cognitive maps and created the theory ofFuzzy Cognitive Maps (FCMs).The relationship between two concepts in (FCMs) can take a value in the interval [-1,1]. This relationship value is called “weight”. For the last twenty years extensive research in the theory of FCMs has been performed that provided major improvements and enhancements in its theoretical underpinning. New methodologies and approaches have been developed. FCMs have also been applied to many different sectors. New software tools have been developed that automate FCM creation and management. The aim of this book is to present recent advances and state of the art in FCM theory,methodologies, applications and tools that exist to date scattered in journal papers, in a concrete and integrated manner. ... Read more

Product Description
This book provides the background information necessary to apply fuzzy set theory in various areas, including engineering fuzzy logic and decision making. The exercises at the end of each chapter deepen the understanding of the concepts and test one's ability to make necessary calculations. ... Read more

Customer Reviews (4)

Excellent introduction Fuzzy Logic theory
What I really liked in this book is that it is not an applications book. There is one chapter on practical applications, but the rest of the book goes through lots of definitions and theorems.So if you are interested in Fuzzy Logic as Logic (and not as a tool you an use) then you will like this book.

prof. review
thisis the best intro to fuzzy ever writen....im a university prof on the subject of AI so...Ive read tons of books on the subject and this is by far the best intro to fuzzy even for those who ll be learning on their own... very simple put and yet very abording on the matter!

A First Course in Fuzzy Logic
If you are looking for a book on Fuzzy Logic Theory, this is a good book. It explains the mathematical basis of fuzzy logic, the mathmatical symbols used, provides proofs for theorems, etc. The chapters and sections areclearly labeled in the table of contents, making it easy to zero in on atopic. It is a College Text Book, with problems to solve at the end of eachchapter. Selected problems have answers in the back of the book.Unfortunately, that wasn't the type of book that I expected or needed. Iwas looking more for the hands on, how to, type of book. That is the onlyreason it did not get 5 stars.

Outstanding, for people who are interested in this area
This is the clearest explanation and application of fuzzy logic that has been published. It is thorough, without being arcane or pedantic. ... Read more

Product Description
Fuzzy Control Synthesis and Analysis Edited by Shehu S. Farinwata Ford Motor Company, Research Laboratory, Dearborn, Michigan, USA Dimitar Filev Ford Motor Company, AMTDC, Redford, Michigan, USA Reza Langari Texas A & M University, College Station, Texas, USA Fuzzy techniques are used to cope with imprecision in the basic elements of a process under control. Written by an international team of researchers this edited volume covers the modeling, analysis and synthesis of fuzzy control systems. Features include:
? Comprehensive coverage of fuzzy dynamical systems, robustness, stability and sensitivity -- giving the reader a good grasp of the fundamentals of fuzzy control
? Focus on the analytical structures of new fuzzy modeling approaches based on the Takagi-Sugeno-Kang (TSK) or Takagi-Sugeno (TS) model
? Applications of fuzzy control to aircraft systems, rocket engines and automotive engines
? Problems and examples illustrating how fuzzy approaches may be applied to the modeling, analysis and synthesis of closed-loop systems
Design and control engineers will value the advanced control techniques and new design and analysis tools presented. Postgraduates studying fuzzy control will find this book a useful reference on synthesis, systems analysis and advanced nonlinear control methods. ... Read more

Customer Reviews (1)

Good, but only for experienced
Quite good book, however it is unsuitable for people how wants to learn the basis of control. I think it is readable only for people with experience in control field. ... Read more

Product Description
The book deals with fuzzy graph theory, fuzzy topology, fuzzy geometry, and fuzzy abstract algebra. The purpose of the book is to present the concepts of fuzzy mathematics which have applications to engineering, computer science, and mathematics. Some specific application areas are cluster analysis, digital image processing, fractal compression, chaotic fuzzy mappings, coding theory, automata theory, and nonlinear systems of fuzzy equations. The book is written with engineers and computer scientists in mind. Some mathematical sophistication is required of the reader. However the style is geared to a general audience. Most theorems are stated without proof and many examples are presented. Crisp results of mathematics are reviewed as needed. ... Read more