By its nature, set theory does not depend on any previous mathematical knowl edge. Hence, an individual wanting to read this book can best find out if he is ready to do so by trying to read the first ten or twenty pages of Chapter 1. As a textbook, the book can serve for a course at the junior or senior level. If a course covers only some of the chapters, the author hopes that the student will read the rest himself in the next year or two. Set theory has always been a sub ject which people find pleasant to study at least partly by themselves. Chapters 1-7, or perhaps 1-8, present the core of the subject. (Chapter 8 is a short, easy discussion of the axiom of regularity). Even a hurried course should try to cover most of this core (of which more is said below). Chapter 9 presents the logic needed for a fully axiomatic set th ory and especially for independence or consistency results. Chapter 10 gives von Neumann's proof of the relative consistency of the regularity axiom and three similar related results. Von Neumann's 'inner model' proof is easy to grasp and yet it prepares one for the famous and more difficult work of GOdel and Cohen, which are the main topics of any book or course in set theory at the next level."

0 Basic Facts.- 1 Hey's Theorem and Consequences.- 2 Siegel-Weyl Reduction Theory.- 3 The Tamagawa Number and the Volume of G(?)/G(?).- 3.1 Statement of the main result.- 3.2 Proof of 3.1.- 3.3 The volume of G(?)/G(?).- 4 The Size of ?.- 4.1 Statement of results.- 4.2 Proofs.- 5 Margulis' Finiteness Theorem.- 5.1 The Result.- 5.2 Amenable groups.- 5.3 Kazhdan's property (T).- 5.4 Proof of 5.1; beginning.- 5.5 Interlude: parabolics and their opposites.- 5.6 Continuation of the proof.- 5.7 Contracting automorphisms and the Moore Ergodicity theorem.- 5.8 End of proof.- 5.9 Appendix on measure theory.- 6 A Zariski Dense and a Free Subgroup of ?.- 7 An Example.- 8 Problems.- 8.1 Generators.- 8.2 The congruence problem.- 8.3 Betti numbers.- References.

This ambitious exposition by Malik and Mordeson on the fuzzification of discrete structures not only supplies a solid basic text on this key topic, but also serves as a viable tool for learning basic fuzzy set concepts "from the ground up" due to its unusual lucidity of exposition. While the entire presentation of this book is in a completely traditional setting, with all propositions and theorems provided totally rigorous proofs, the readability of the presentation is not compromised in any way; in fact, the many ex cellently chosen examples illustrate the often tricky concepts the authors address. The book's specific topics - including fuzzy versions of decision trees, networks, graphs, automata, etc. - are so well presented, that it is clear that even those researchers not primarily interested in these topics will, after a cursory reading, choose to return to a more in-depth viewing of its pages. Naturally, when I come across such a well-written book, I not only think of how much better I could have written my co-authored monographs, but naturally, how this work, as distant as it seems to be from my own area of interest, could nevertheless connect with such. Before presenting the briefest of some ideas in this direction, let me state that my interest in fuzzy set theory (FST) has been, since about 1975, in connecting aspects of FST directly with corresponding probability concepts. One chief vehicle in carrying this out involves the concept of random sets."

Fuzzy sets were first proposed by Lotfi Zadeh in his seminal paper [366] in 1965, and ever since have been a center of many discussions, fervently admired and condemned. Both proponents and opponents consider the argu ments pointless because none of them would step back from their territory. And stiH, discussions burst out from a single sparkle like a conference pa per or a message on some fuzzy-mail newsgroup. Here is an excerpt from an e-mail messagepostedin1993tofuzzy-mail@vexpert. dbai. twvien. ac. at. by somebody who signed "Dave". , . . . Why then the "logic" in "fuzzy logic"? I don't think anyone has successfully used fuzzy sets for logical inference, nor do I think anyone wiH. In my admittedly neophyte opinion, "fuzzy logic" is a misnomer, an oxymoron. (1 would be delighted to be proven wrong on that. ) . . . I carne to the fuzzy literature with an open mind (and open wal let), high hopes and keen interest. I am very much disiHusioned with "fuzzy" per se, but I did happen across some extremely interesting things along the way. " Dave, thanks for the nice quote! Enthusiastic on the surface, are not many of us suspicious deep down? In some books and journals the word fuzzy is religiously avoided: fuzzy set theory is viewed as a second-hand cheap trick whose aim is nothing else but to devalue good classical theories and open up the way to lazy ignorants and newcomers.

This book comprises revised full versions of lectures given during the 9th European Summer School in Logic, Languages, and Information, ESSLLI'97, held in Aix-en-Provence, France, in August 1997. The six lectures presented introduce the reader to the state of the art in the area of generalized quantifiers and computation. Besides an introductory survey by the volume editor various aspects of generalized quantifiers are studied in depth.

Since the late 1980s, a large number of very user-friendly tools for fuzzy control, fuzzy expert systems, and fuzzy data analysis have emerged. This has changed the character of this area and started the area of `fuzzy technology'. The next large step in the development occurred in 1992 when almost independently in Europe, Japan and the USA, the three areas of fuzzy technology, artificial neural nets and genetic algorithms joined forces under the title of `computational intelligence' or `soft computing'. The synergies which were possible between these three areas have been exploited very successfully. Practical Applications of Fuzzy Sets focuses on model and real applications of fuzzy sets, and is structured into four major parts: engineering and natural sciences; medicine; management; and behavioral, cognitive and social sciences. This book will be useful for practitioners of fuzzy technology, scientists and students who are looking for applications of their models and methods, for topics of their theses, and even for venture capitalists who look for attractive possibilities for investments.

The book systematically develops the nonlinear potential theory connected with the weighted Sobolev spaces, where the weight usually belongs to Muckenhoupt's class of Ap weights. These spaces occur as solutions spaces for degenerate elliptic partial differential equations. The Sobolev space theory covers results concerning approximation, extension, and interpolation, Sobolev and Poincaré inequalities, Maz'ya type embedding theorems, and isoperimetric inequalities. In the chapter devoted to potential theory, several weighted capacities are investigated. Moreover, "Kellogg lemmas" are established for various concepts of thinness. Applications of potential theory to weighted Sobolev spaces include quasi continuity of Sobolev functions, Poincaré inequalities, and spectral synthesis theorems.

Information is precious. It reduces our uncertainty in making decisions. Knowledge about the outcome of an uncertain event gives the possessor an advantage. It changes the course of lives, nations, and history itself. Information is the food of Maxwell's demon. His power comes from know ing which particles are hot and which particles are cold. His existence was paradoxical to classical physics and only the realization that information too was a source of power led to his taming. Information has recently become a commodity, traded and sold like or ange juice or hog bellies. Colleges give degrees in information science and information management. Technology of the computer age has provided access to information in overwhelming quantity. Information has become something worth studying in its own right. The purpose of this volume is to introduce key developments and results in the area of generalized information theory, a theory that deals with uncertainty-based information within mathematical frameworks that are broader than classical set theory and probability theory. The volume is organized as follows."

In the beginning of 1983, I came across A. Kaufmann's book "Introduction to the theory of fuzzy sets" (Academic Press, New York, 1975). This was my first acquaintance with the fuzzy set theory. Then I tried to introduce a new component (which determines the degree of non-membership) in the definition of these sets and to study the properties of the new objects so defined. I defined ordinary operations as "n," "U," "+" and ."" over the new sets, but I had began to look more seriously at them since April 1983, when I defined operators analogous to the modal operators of "necessity" and "possibility." The late George Gargov (7 April 1947 - 9 November 1996) is the "god father" of the sets I introduced - in fact, he has invented the name "intu itionistic fuzzy," motivated by the fact that the law of the excluded middle does not hold for them. Presently, intuitionistic fuzzy sets are an object of intensive research by scholars and scientists from over ten countries. This book is the first attempt for a more comprehensive and complete report on the intuitionistic fuzzy set theory and its more relevant applications in a variety of diverse fields. In this sense, it has also a referential character."

The explanation of the formal duality of Kerdock and Preparata codes is one of the outstanding results in the field of applied algebra in the last few years. This result is related to the discovery of large sets of quad riphase sequences over Z4 whose correlation properties are better than those of the best binary sequences. Moreover, the correlation properties of sequences are closely related to difference properties of certain sets in (cyclic) groups. It is the purpose of this book to illustrate the connection between these three topics. Most articles grew out of lectures given at the NATO Ad vanced Study Institute on "Difference sets, sequences and their correlation properties." This workshop took place in Bad Windsheim (Germany) in August 1998. The editors thank the NATO Scientific Affairs Division for the generous support of this workshop. Without this support, the present collection of articles would not have been realized."

The explanation of the formal duality of Kerdock and Preparata codes is one of the outstanding results in the field of applied algebra in the last few years. This result is related to the discovery of large sets of quad riphase sequences over Z4 whose correlation properties are better than those of the best binary sequences. Moreover, the correlation properties of sequences are closely related to difference properties of certain sets in (cyclic) groups. It is the purpose of this book to illustrate the connection between these three topics. Most articles grew out of lectures given at the NATO Ad vanced Study Institute on "Difference sets, sequences and their correlation properties." This workshop took place in Bad Windsheim (Germany) in August 1998. The editors thank the NATO Scientific Affairs Division for the generous support of this workshop. Without this support, the present collection of articles would not have been realized."

Mathematical Principles of Fuzzy Logic provides a systematic study of the formal theory of fuzzy logic. The book is based on logical formalism demonstrating that fuzzy logic is a well-developed logical theory. It includes the theory of functional systems in fuzzy logic, providing an explanation of what can be represented, and how, by formulas of fuzzy logic calculi. It also presents a more general interpretation of fuzzy logic within the environment of other proper categories of fuzzy sets stemming either from the topos theory, or even generalizing the latter. This book presents fuzzy logic as the mathematical theory of vagueness as well as the theory of commonsense human reasoning, based on the use of natural language, the distinguishing feature of which is the vagueness of its semantics.

Fuzzy Sets, Logics and Reasoning about Knowledge reports recent results concerning the genuinely logical aspects of fuzzy sets in relation to algebraic considerations, knowledge representation and commonsense reasoning. It takes a state-of-the-art look at multiple-valued and fuzzy set-based logics, in an artificial intelligence perspective. The papers, all of which are written by leading contributors in their respective fields, are grouped into four sections. The first section presents a panorama of many-valued logics in connection with fuzzy sets. The second explores algebraic foundations, with an emphasis on MV algebras. The third is devoted to approximate reasoning methods and similarity-based reasoning. The fourth explores connections between fuzzy knowledge representation, especially possibilistic logic and prioritized knowledge bases. Readership: Scholars and graduate students in logic, algebra, knowledge representation, and formal aspects of artificial intelligence.

The transition from school mathematics to university mathematics is seldom straightforward. Students are faced with a disconnect between the algorithmic and informal attitude to mathematics at school, versus a new emphasis on proof, based on logic, and a more abstract development of general concepts, based on set theory. The authors have many years' experience of the potential difficulties involved, through teaching first-year undergraduates and researching the ways in which students and mathematicians think. The book explains the motivation behind abstract foundational material based on students' experiences of school mathematics, and explicitly suggests ways students can make sense of formal ideas. This second edition takes a significant step forward by not only making the transition from intuitive to formal methods, but also by reversing the process- using structure theorems to prove that formal systems have visual and symbolic interpretations that enhance mathematical thinking. This is exemplified by a new chapter on the theory of groups. While the first edition extended counting to infinite cardinal numbers, the second also extends the real numbers rigorously to larger ordered fields. This links intuitive ideas in calculus to the formal epsilon-delta methods of analysis. The approach here is not the conventional one of 'nonstandard analysis', but a simpler, graphically based treatment which makes the notion of an infinitesimal natural and straightforward. This allows a further vision of the wider world of mathematical thinking in which formal definitions and proof lead to amazing new ways of defining, proving, visualising and symbolising mathematics beyond previous expectations.

In this presentation of the Galois correspondence, modern theories of groups and fields are used to study problems, some of which date back to the ancient Greeks. The techniques used to solve these problems, rather than the solutions themselves, are of primary importance. The ancient Greeks were concerned with constructibility problems. For example, they tried to determine if it was possible, using straightedge and compass alone, to perform any of the following tasks? (1) Double an arbitrary cube; in particular, construct a cube with volume twice that of the unit cube. (2) Trisect an arbitrary angle. (3) Square an arbitrary circle; in particular, construct a square with area 1r. (4) Construct a regular polygon with n sides for n > 2. If we define a real number c to be constructible if, and only if, the point (c, 0) can be constructed starting with the points (0,0) and (1,0), then we may show that the set of constructible numbers is a subfield of the field R of real numbers containing the field Q of rational numbers. Such a subfield is called an intermediate field of Rover Q. We may thus gain insight into the constructibility problems by studying intermediate fields of Rover Q. In chapter 4 we will show that (1) through (3) are not possible and we will determine necessary and sufficient conditions that the integer n must satisfy in order that a regular polygon with n sides be constructible.

The analysis and control of complex systems have been the main motivation for the emergence of fuzzy set theory since its inception. It is also a major research field where many applications, especially industrial ones, have made fuzzy logic famous. This unique handbook is devoted to an extensive, organized, and up-to-date presentation of fuzzy systems engineering methods. The book includes detailed material and extensive bibliographies, written by leading experts in the field, on topics such as: Use of fuzzy logic in various control systems. Fuzzy rule-based modeling and its universal approximation properties. Learning and tuning techniques for fuzzy models, using neural networks and genetic algorithms. Fuzzy control methods, including issues such as stability analysis and design techniques, as well as the relationship with traditional linear control. Fuzzy sets relation to the study of chaotic systems, and the fuzzy extension of set-valued approaches to systems modeling through the use of differential inclusions. Fuzzy Systems: Modeling and Control is part of The Handbooks of Fuzzy Sets Series. The series provides a complete picture of contemporary fuzzy set theory and its applications. This volume is a key reference for systems engineers and scientists seeking a guide to the vast amount of literature in fuzzy logic modeling and control.

Fuzzy Sets in Decision Analysis, Operations Research and Statistics includes chapters on fuzzy preference modeling, multiple criteria analysis, ranking and sorting methods, group decision-making and fuzzy game theory. It also presents optimization techniques such as fuzzy linear and non-linear programming, applications to graph problems and fuzzy combinatorial methods such as fuzzy dynamic programming. In addition, the book also accounts for advances in fuzzy data analysis, fuzzy statistics, and applications to reliability analysis. These topics are covered within four parts: Decision Making, Mathematical Programming, Statistics and Data Analysis, and Reliability, Maintenance and Replacement. The scope and content of the book has resulted from multiple interactions between the editor of the volume, the series editors, the series advisory board, and experts in each chapter area. Each chapter was written by a well-known researcher on the topic and reviewed by other experts in the area. These expert reviewers sometimes became co-authors because of the extent of their contribution to the chapter. As a result, twenty-five authors from twelve countries and four continents were involved in the creation of the 13 chapters, which enhances the international character of the project and gives an idea of how carefully the Handbook has been developed.

Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory is a major attempt to provide much-needed coherence for the mathematics of fuzzy sets. Much of this book is new material required to standardize this mathematics, making this volume a reference tool with broad appeal as well as a platform for future research. Fourteen chapters are organized into three parts: mathematical logic and foundations (Chapters 1-2), general topology (Chapters 3-10), and measure and probability theory (Chapters 11-14). Chapter 1 deals with non-classical logics and their syntactic and semantic foundations. Chapter 2 details the lattice-theoretic foundations of image and preimage powerset operators. Chapters 3 and 4 lay down the axiomatic and categorical foundations of general topology using lattice-valued mappings as a fundamental tool. Chapter 3 focuses on the fixed-basis case, including a convergence theory demonstrating the utility of the underlying axioms. Chapter 4 focuses on the more general variable-basis case, providing a categorical unification of locales, fixed-basis topological spaces, and variable-basis compactifications. Chapter 5 relates lattice-valued topologies to probabilistic topological spaces and fuzzy neighborhood spaces. Chapter 6 investigates the important role of separation axioms in lattice-valued topology from the perspective of space embedding and mapping extension problems, while Chapter 7 examines separation axioms from the perspective of Stone-Cech-compactification and Stone-representation theorems. Chapters 8 and 9 introduce the most important concepts and properties of uniformities, including the covering and entourage approaches and the basic theory of precompact orcomplete [0,1]-valued uniform spaces. Chapter 10 sets out the algebraic, topological, and uniform structures of the fundamentally important fuzzy real line and fuzzy unit interval. Chapter 11 lays the foundations of generalized measure theory and representation by Markov kernels. Chapter 12 develops the important theory of conditioning operators with applications to measure-free conditioning. Chapter 13 presents elements of pseudo-analysis with applications to the Hamilton&endash;Jacobi equation and optimization problems. Chapter 14 surveys briefly the fundamentals of fuzzy random variables which are [0,1]-valued interpretations of random sets.

Between the two world wars, Stanislaw Lesniewski (1886-1939), created the famous and important system of foundations of mathematics that comprises three deductive theories: Protothetic, Ontology, and Mereology. His research started in 1914 with studies on the general theory of sets (later named `Mereology'). Ontology followed between 1919 and 1921, and was the next step towards an integrated system. In order to combine these two systematically he constructed Protothetic - the system of `first principles'. Together they amount to what Z. Jordan called `... most thorough, original, and philosophically significant attempt to provide a logically secure foundation for the whole of mathematics'. The volume collects many of the most significant commentaries on, and contributions to, Protothetic. A Protothetic Bibliography is included.

This book is devoted to some results from the classical Point Set Theory and their applications to certain problems in mathematical analysis of the real line. Notice that various topics from this theory are presented in several books and surveys. From among the most important works devoted to Point Set Theory, let us first of all mention the excellent book by Oxtoby [83] in which a deep analogy between measure and category is discussed in detail. Further, an interesting general approach to problems concerning measure and category is developed in the well-known monograph by Morgan [79] where a fundamental concept of a category base is introduced and investigated. We also wish to mention that the monograph by Cichon, W";glorz and the author [19] has recently been published. In that book, certain classes of subsets of the real line are studied and various cardinal valued functions (characteristics) closely connected with those classes are investigated. Obviously, the IT-ideal of all Lebesgue measure zero subsets of the real line and the IT-ideal of all first category subsets of the same line are extensively studied in [19], and several relatively new results concerning this topic are presented. Finally, it is reasonable to notice here that some special sets of points, the so-called singular spaces, are considered in the classi

Introduction Model theorists have often joked in recent years that the part of mathemat- ical logic known as "pure model theory" (or stability theory), as opposed to the older and more traditional "model theory applied to algebra" , turns out to have more and more to do with other subjects ofmathematics and to yield gen- uine applications to combinatorial geometry, differential algebra and algebraic geometry. We illustrate this by presenting the very striking application to diophantine geometry due to Ehud Hrushovski: using model theory, he has given the first proof valid in all characteristics of the "Mordell-Lang conjecture for function fields" (The Mordell-Lang conjecture for function fields, Journal AMS 9 (1996), 667-690). More recently he has also given a new (model theoretic) proof of the Manin-Mumford conjecture for semi-abelian varieties over a number field. His proofyields the first effective bound for the cardinality ofthe finite sets involved (The Manin-Mumford conjecture, preprint). There have been previous instances of applications of model theory to alge- bra or number theory, but these appl~cations had in common the feature that their proofs used a lot of algebra (or number theory) but only very basic tools and results from the model theory side: compactness, first-order definability, elementary equivalence...

This book presents a systematic treatment of deductive aspects and structures of fuzzy logic understood as many valued logic sui generis. It aims to show that fuzzy logic as a logic of imprecise (vague) propositions does have well-developed formal foundations and that most things usually named 'fuzzy inference' can be naturally understood as logical deduction. It is for mathematicians, logicians, computer scientists, specialists in artificial intelligence and knowledge engineering, and developers of fuzzy logic.

It is the business of science not to create laws, but to discover them. We do not originate the constitution of our own minds, greatly as it may be in our power to modify their character. And as the laws of the human intellect do not depend upon our will, so the forms of science, of (1. 1) which they constitute the basis, are in all essential regards independent of individual choice. George Boole 10, p. llJ 1. 1 Comparison with Traditional Logic The logic of this book is a probability logic built on top of a yes-no or 2-valued logic. It is divided into two parts, part I: BP Logic, and part II: M Logic. 'BP' stands for 'Bayes Postulate'. This postulate says that in the absence of knowl edge concerning a probability distribution over a universe or space one should assume 1 a uniform distribution. 2 The M logic of part II does not make use of Bayes postulate or of any other postulates or axioms. It relies exclusively on purely deductive reasoning following from the definition of probabilities. The M logic goes an important step further than the BP logic in that it can distinguish between certain types of information supply sentences which have the same representation in the BP logic as well as in traditional first order logic, although they clearly have different meanings (see example 6. 1. 2; also comments to the Paris-Rome problem of eqs. (1. 8), (1. 9) below)."

During the past 25 years, set theory has developed in several interesting directions. The most outstanding results cover the application of sophisticated techniques to problems in analysis, topology, infinitary combinatorics and other areas of mathematics. This book contains a selection of contributions, some of which are expository in nature, embracing various aspects of the latest developments. Amongst topics treated are forcing axioms and their applications, combinatorial principles used to construct models, and a variety of other set theoretical tools including inner models, partitions and trees. Audience: This book will be of interest to graduate students and researchers in foundational problems of mathematics.

Fuzzy hardware developments have been a major force driving the applications of fuzzy set theory and fuzzy logic in both science and engineering. This volume provides the reader with a comprehensive up-to-date look at recent works describing new innovative developments of fuzzy hardware. An important research trend is the design of improved fuzzy hardware. There is an increasing interest in both analog and digital implementations of fuzzy controllers in particular and fuzzy systems in general. Specialized analog and digital VLSI implementations of fuzzy systems, in the form of dedicated architectures, aim at the highest implementation efficiency. This particular efficiency is asserted in terms of processing speed and silicon utilization. Processing speed in particular has caught the attention of developers of fuzzy hardware and researchers in the field. The volume includes detailed material on a variety of fuzzy hardware related topics such as: Historical review of fuzzy hardware research Fuzzy hardware based on encoded trapezoids Pulse stream techniques for fuzzy hardware Hardware realization of fuzzy neural networks Design of analog neuro-fuzzy systems in CMOS digital technologies Fuzzy controller synthesis method Automatic design of digital and analog neuro-fuzzy controllers Electronic implementation of complex controllers Silicon compilation of fuzzy hardware systems Digital fuzzy hardware processing Parallel processor architecture for real-time fuzzy applications Fuzzy cellular systems Fuzzy Hardware: Architectures and Applications is a technical reference book for researchers, engineers and scientists interested in fuzzy systems in general and in building fuzzy systems in particular.