The primary purpose of this book is to provide the reader with a comprehensive coverage of theoretical foundations of fuzzy set theory and fuzzy logic, as well as . Fuzzy Sets and Fuzzy Logic Theory and Applications - George j. Klir, Bo Yuan - Free ebook download as PDF File .pdf), Text File .txt) or read book online for. Reflecting the tremendous advances that have taken place in the study of fuzzy set theory and fuzzy logic from to the present, this book not only details the .

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FUZZY SETS AND FUZZY LOGIC Theory and ApplicationsGEORGE J. KLIR AND BO YUANFor baoic and boastDre faarmatlart. -n. fuzzy sets and fuzzy logic. Pages·· MB·5, Downloads. FUZZY SETS AND FUZZY LOGIC. Theory and Applications. GEORGE J. KLIR AND BO. Fuzzy Sets and Fuzzy Logic: Theory and Applications with ISBN is a book written by George J. Klir, Bo Yuan. We have this.

Theoretical aspects of fuzzy set theory and fuzzy logic are covered in Part I of the t Reflecting the tremendous advances that have taken place in the study of fuzzy set theory and fuzzy logic from to the present, this book not only details the theoretical advances in these areas, but considers a broad variety of applications of fuzzy sets and fuzzy logic as well. Theoretical aspects of fuzzy set theory and fuzzy logic are covered in Part I of the text, including: basic types of fuzzy sets; connections between fuzzy sets and crisp sets; the various aggregation operations of fuzzy sets; fuzzy numbers and arithmetic operations on fuzzy numbers; fuzzy relations and the study of fuzzy relation equations. Part II is devoted to applications of fuzzy set theory and fuzzy logic, including: various methods for constructing membership functions of fuzzy sets; the use of fuzzy logic for approximate reasoning in expert systems; fuzzy systems and controllers; fuzzy databases; fuzzy decision making; and engineering applications. For everyone interested in an introduction to fuzzy set theory and fuzzy logic.

Captured in the book are not only theoretical advances in these areas, but a broad variety of applications of fuzzy sets and fuzzy logic as well. The primary purpose of the book is to facilitate education in the increasingly important areas of fuzzy set theory and fuzzy logic. It is written as a text for a course at the graduate or upper-division undergraduate level. Although there is enough material in the text for a two-semester course, relevant material may be selected, according to the needs of each individual program, for a one-semester course.

The text is also suitable for self- study and for short, intensive courses of continuing education. No previous knowledge of fuzzy settheoryor fuzzy logic is required for an understanding of the material in this text. Although we assume that the reader is familiar with the basic notions of classical nonfuzzy set theory, classical two-valued logic,.

Basic ideas of neural networks, genetic algorithms, and rough sets, which are occasionally needed in the text, are provided in Appendices A-C. This makes the book virtually self- contained. Theoretical aspects of fuzzy set theory and fuzzy logic are covered in the first nine chapters, which are designated Part I of the text.

Elementary concepts, including basic types of fuzzy sets, are introduced in Chapter 1, which also contains a discussion of the meaning and significance of the emergence of fuzzy set theory.

Connections between fuzzy sets and crisp sets are examined in Chapter 2. It shows how fuzzy sets can be represented by families of crisp sets and how classical mathematical functions can be fuzzified. Chapter 3 deals with the various aggregation operations on fuzzy sets. It covers general fuzzy complements, fuzzy intersections f-norms , fuzzy unions r-conorms , and averaging operations.

Fuzzy numbers and arithmetic operations on fuzzy numbers are covered in Chapter 4, where also the concepts of linguistic variables and fuzzy equations are introduced and examined. Basic concepts of fuzzy relations are introduced in Chapter 5 and employed in Chapter 6 for the study of fuzzy relation equations, an important tool for many applications of fuzzy set theory.

I Prerequisite dependencies among chapters of this book. Chapter 7 deals with possibility theory and its intimate connection with fuzzy set theory. The position of possibility theory within the broader framework of fuzzy measure theory is also examined. Chapter 8 overviews basic aspects of fuzzy logic, including its connection to classical multivalued logics, the various types of fuzzy propositions, and basic types of fuzzy inference rules. Chapter 9, the last chapter in Part I, is devoted to the examination of the connection between uncertainty and information, as represented by fuzzy sets, possibility theory, or evidence theory.

The chapter shows how relevant uncertainty and uncertainty-based information can be measured and how these uncertainty measures can be utilized. Part n, which is devoted to applications of fuzzy set theory and fuzzy logic, consists of the remaining eight chapters. Chapter 10 examines various methods for constructing membership functions of fuzzy sets, including the increasingly popular use of neural networks.

Chapter 11 is devoted to the use of fuzzy logic for approximate reasoning in expert systems. It includes a thorough examination of the concept of a fuzzy implication. Fuzzy systems are covered in Chapter 12, including fuzzy controllers, fuzzy automata, and fuzzy neural networks. Fuzzy techniques in the related areas of clustering, pattern recognition, and image processing are Preface xv overviewed in Chapter Fuzzy databases, a well-developed application area of fuzzy set theory, and the related area of fuzzy retrieval systems are covered in Chapter Basic ideas of the various types of fuzzy decision making are summarized in Chapter Engineering applications other than fuzzy control are, touched upon in Chapter 16, and applications in various other areas medicine, economics, etc.

The prerequisite dependencies among the individual chapters and some appendices are expressed by the diagram in Fig. Following the diagram, the reader has ample flexibility in studying the material. For example, Chapters 3, 5 and 6 may be studied prior to Chapters 2 and 4; Chapter 10 and Appendix A may be studies prior to Chapter 2 and Chapters 4 through 9; etc.

In order to avoid interruptions in the main text, virtually all bibliographical, historical, and other remarks are incorporated in the notes that follow each individual chapter.

These notes are uniquely numbered and are only occasionally referred to in the text. The notes are particularly important in Part II, where they contain ample references, allowing the interested reader to pursue further study in the application area of concern. When the book is used at the upper-division undergraduate level, coverage of some or all proofs of the various mathematical theorems may be omitted, depending on the background of the students.

At the graduate level, on the other hand, we encourage coverage of most of these proofs in order to effect a deeper understanding of the material. In all cases, the relevance of the material to the specific area of student interest can be emphasized with additional application- oriented readings guided by relevant notes in Part II of the text. Each chapter is followed by a set of exercises, which are intended to enhance an understanding of the material presented in the chapter.

The solutions to a selected subset of these exercises are provided in the instructor's manual, which also contains further suggestions for use of the text under various circumstances. The book contains an extensive bibliography, which covers virtually all relevant books and significant papers published prior to It also contains a Bibliographical Index, which consists of reference lists for selected application areas and theoretical topics. This index should be particularly useful for graduate, project-oriented courses, as well as for both practitioners and researchers.

Each book in the bibliography is emphasized by printing its year of publication in bold. George J. In science, this change has been manifested by a gradual transition from the traditional view, which insists that uncertainty is undesirable in science and should be avoided by all possible means, to an alternative view, which is tolerant of uncertainty and insists that science cannot avoid it.

According to the traditional view, science should strive for certainty in all its manifestations precision, specificity, sharpness, consistency, etc.

According to the alternative or modem view, uncertainty is considered essential to science; it is not only an unavoidable plague, but it has, in fact, a great utility.

The first stage of the transition from the traditional view to the modem view of uncertainty began in the late 19th century, when physics became concerned with processes at the molecular level.

Although precise laws of Newtonian mechanics were relevant to the study of these processes, their actual application to the enormous number of entities involved would have resulted in computational demands that were far beyond existing computational capabilities and, as we realize now, exceed even fundamental computational limits. That is, these precise laws are denied applicability in this domain not only in practice based on existing computer technology but in principle. The need for a fundamentally different approach to the study of physical processes at the molecular level motivated the development of relevant statistical methods, which turned out to be applicable not only to the study of molecular processes statistical mechanics , but to a host of other areas such as the actuarial profession, design of large telephone exchanges, and the like.

In statistical methods, specific manifestations of microscopic entities molecules, individual telephone sites, etc. While analytic methods based upon the calculus are applicable only to problems involving a very small number of variables that are related to one.

These two types of methods are thus highly complementary. When one type excels, the other totally fails. Despite their complementarity, these types of methods cover, unfortunately, only problems that are clustered around the two extremes of complexity and randomness scales. In his well-known paper, Warren Weaver [] refers to them as problems of organized simplicity and disorganized complexity.

He argues that these types of problems represent only a tiny fraction of all systems problems. Most problems are. Weaver calls them problems of organized complexity; they are typical in life, cognitive, social, and environmental sciences, as well as in applied fields such as modern technology or medicine. The emergence of computer technology in World War II and its rapidly growing power in the second half of this century made it possible to deal with increasingly complex problems, some of which began to resemble the notion of organized complexity.

Initially, it was the common belief of many scientists that the level of complexity we can handle is basically a matter of the level of computational power at our disposal. In this book, we present an introduction to the major developments of the theory as well as to some of the most successful applications of the theory.

The aim of this text is to introduce the main components of fuzzy set theory and to overview some of its applications.

To distinguish between fuzzy sets and classical nonfuzzy sets, we refer to the latter as crisp sets. This name is now generally accepted in the literature. In our presentation, we assume that the reader is familiar with fundamentals of the theory of crisp sets.

We include this section into the text solely to refresh the basic concepts of crisp sets and to introduce notation and terminology useful in our discussion of fuzzy sets. The following general symbols are employed, as needed, throughout the text:. In addition, we use "iff" as a shorthand expression of "if and only if," and the standard symbols 3 and V are used for the existential quantifier and the universal quantifier, respectively.

Sets are denoted in this text by upper-case letters and their members by lower-case letters. The letter X denotes the universe of discourse, or universal set. This set contains all the possible elements of concern in each particular context or application from which sets can be formed. The set that contains no members is called the empty set and is denoted by 0.

To indicate that an individual object x is a member or element of a set A, we write xeA. There are three basic methods by which sets can be defined within a given universal set X:. A set is defined by naming all its members the list method. A set is defined by a property satisfied by its members the rule method. It is required that the property P be such that for any given l e i , the proposition P x is either true of false.

A set is defined by a function, usually called a characteristic function, that declares which elements of X are members of the set and which are not. A set whose elements are themselves sets is often referred to as a family of sets. It can be defined in the form.

Because the index i is used to reference the sets Ait the family of sets is also called an indexed set. In this text, families of sets are usually denoted by script capital letters.

For example,.

If every member of set A is also a member of set B i. Every set is a subset of itself, and every set is a subset of the universal set. A, then A and B contain the same members. In this case, A is called a proper subset of B, which is denoted by AcB. When A c B, we also say that A is included in 5. The family of all subsets of a given set A is called the power set of A, and it is usually denoted by V A. The family of all subsets of?

Similarly, higher order power sets T3 A , y 4 A ,. The relative complement of a set A with respect to set B is the set containing all the members of B that are not also members of A. This can be written B — A. If the set B is the universal set, the complement is absolute and is usually denoted by A.

The absolute complement is always involutive; that is, taking the complement of a complement yields the original set, or. The absolute complement of the empty set equals the universal set, and the absolute complement of the universal set equals the empty set.

That is,. The union of sets A and B is the set containing all the elements that belong either to set A alone, to set B alone, or to both set A and set B. This is denoted by A U B. The union operation can be generalized for any number of sets. The intersection of sets A and B is the set containing all the elements belonging to both set A and set B.

It is denoted by A n B. P X of a universal set X. Note that all the equations in this table that involve the set union and intersection are arranged in pairs.

The second equation in each pair can be obtained from the first by replacing 0 , U, and n with X, n, and U, respectively, and vice versa. We are thus concerned with pairs of dual equations. They exemplify a general principle of duality: TABLE 1. Elements of the power set 7 X of a universal set X or any subset of X can be ordered by the set inclusion c.

This ordering, which is only partial, forms a lattice in which the join least upper bound, supremum and meet greatest lower bound, infimum of any pair of sets A, B e CP X is given by A U B and A n B, respectively. This lattice is distributive due to the distributive properties of U and n listed in Table 1.

X ; it is usually called a Boolean lattice or a Boolean algebra. Any two sets that have no common members are called disjoint.

A family of pairwise disjoint nonempty subsets of a set A is called a partition on A if the union of these subsets yields the original, set A. We denote a partition on A by the symbol jr A. Members of a partition TT A , which are subsets of A, are usually referred to as blocks of the partition. Each member of A belongs to one and only one block of n A.

Then, A is called a nested family, and the sets Ai and An are called the innermost set and the outermost set, respectively.

This definition can easily be extended to infinite families. The Cartesian product of two sets—say, A and B in this order —is the set of all ordered pairs such that the first element in each pair is a member of A, and the second element is a member of B.

It is written as either Ai x A 2 x. Subsets of Cartesian products are called relations. They are the subject of Chapter 5. A set whose members can be labelled by the positive integers is called a countable set.

If such labelling is not possible, the set is called uncountable. Every uncountable set is infinite; countable sets are classified into finite and countably infinite also called denumerable. An important and frequently used universal set is the set of all 'points in the n- dimensional Euclidean vector space M.

Sets defined in terms of R" are often required to possess a property referred to as convexity. In other words, a set A in R" is convex iff, for every pair of points r and s in A, all points located on the straight-line segment connecting r and s are also in A. Examples of convex and nonconvex sets in K2 are given in Fig. In R, any set defined by a single interval of real numbers is convex; any set defined by more than one interval that does not contain some points between the intervals is not convex.

Let R denote a set of real numbers R c R. Figure 1. Basic Types. For any set of real numbers R that is bounded below, a real number s is called the infimum of R iff:. As denned in the previous section, the characteristic function of a crisp set assigns a vaiue of either 1 or 0 to each individual in the universal set, thereby discriminating between members and nonmembers of the crisp set under consideration.

This function can be generalized such that the values assigned to the elements of the universal set fall within a specified range and indicate the membership grade of these elements in the set in question. Larger values denote higher degrees of set membership. Such a function is called a membership function, and the set defined by it a fuzzy set. The most commonly used range of values of membership functions is the unit interval [0,1].

In this case, each membership function maps elements of a given universal set X, which is always a crisp set, into real numbers in [0,1].

Two distinct notations are most commonly employed in the literature to denote membership functions. A; that is, HA: X — [0,1], In the other one, the function is denoted by A and has, of course, the same form: According to the first notation, the symbol label, identifier, name of the fuzzy set A is distinguished from the symbol of its membership function GAA.

According to the second notation, this distinction is not made, but no ambiguity results from this double use of the same symbol.

Each fuzzy set is completely and uniquely defined by one particular membership function; consequently, symbols of membership functions may also be used as labels of the associated fuzzy sets. In this text, we use the second notation. That is, each fuzzy set and the associated membership function are denoted by the same capital letter.

Since crisp sets and the associated characteristic functions may be viewed, respectively, as special fuzzy sets and membership functions, the same notation is used for crisp sets as well. As discussed in Sec. For example, applying the concept of high temperature in one context to weather and in another context to a nuclear reactor would necessarily be represented by very different fuzzy sets. That would also be the case, although to a lesser degree, if the concept were applied to weather in different seasons, at least in some climates.

Even for similar contexts, fuzzy sets representing the same concept may vary consider- ably. In this case, however, they also have to be similar in some key features.

As an exam- ple, let us consider four fuzzy sets whose membership functions are shown in Fig. Each of these fuzzy sets expresses, in a particular form, the general conception of a class of real num- bers that are close to 2.

In spite of their differences, the four fuzzy sets are simi- lar in the sense that the following properties are possessed by each Ai i e N These properties are necessary in order to properly represent the given conception. A Any additional fuzzy sets attempting to represent the same conception would have to possess them as well.

Basic Types The four membership functions in Fig. This similarity does not reflect the conception itself, but rather the context in which it is used. The functions are manifested by very different shapes of their graphs. Whether a particular shape is suitable or not can be determined only in the context of a particular application. It turns out, however, that many applications are not overly sensitive to variations in the shape. In such cases, it is convenient to use a simple shape, such as the triangular shape of Aj.

Each function in Fig. Functions in Fig. Fuzzy sets in Fig. Let us consider now, as a simple example, three fuzzy sets defined within a finite universal set that consists of seven levels of education:. Membership functions of the three fuzzy sets, which attempt to capture the concepts of little-educated, highly educated, and very highly educated people are defined in Fig.

Thus, for example, a person who has a bachelor's degree but no higher degree is viewed, according to these definitions, as highly educated to the degree of 0. Several fuzzy sets representing linguistic concepts such as low, medium, high, and so on are often employed to define states of a variable. In Fig. States of the fuzzy variable are fuzzy sets representing five linguistic concepts: Graphs of these functions have trapezoidal shapes, which, together with triangular shapes such as Ai in Fig.

States of the corresponding traditional variable are crisp sets defined by the right-open intervals of real numbers shown in Fig. The significance of fuzzy variables is that they facilitate gradual transitions between states and, consequently, possess a natural capability to express and deal with observation and measurement uncertainties.

Traditional variables, which we may refer to as crisp variables, do not have this capability. Although the definition of states by crisp sets is mathematically correct, it is unrealistic in the face of unavoidable measurement errors.

A measurement that falls into a close neighborhood of each precisely defined border between states of a crisp variable is taken as evidential support for only one of the states, in spite of the inevitable uncertainty involved in this decision. The uncertainty reaches its maximum at each border, where any measurement should be regarded as equal evidence for the two states on either side of the border. Very low Low Medium High Very high. Since fuzzy variables capture measurement uncertainties as part of experimental data, they are more attuned to reality than crisp variables.

It is an interesting paradox that data based on fuzzy variables provide us, in fact, with more accurate evidence about real phenomena than data based upon crisp variables. This important point can hardly be expressed better than by the following statement made by Albert Einstein in So far as laws of mathematics refer to reality, they are not certain.

And so far as they are certain, they do not refer to reality. Although mathematics based on fuzzy sets has far greater expressive power than classical mathematics based on crisp sets, its usefulness depends critically on our capability to construct appropriate membership functions for various given concepts irr various contexts. This capability, which was rather weak at the early stages of fuzzy set theory, is now well developed for many application areas.

However, the problem of constructing meaningful membership functions is a difficult one, and a lot of additional research work will have to be done on it to achieve full satisfaction. We discuss the problem and overview currently available construction methods in Chapter Thus far, we introduced only one type of fuzzy set. Given a relevant universal set X, any arbitrary fuzzy set of this type say, set A is denned by a function of the form.

Fuzzy sets of this type are by far the most common in the literature as well as in the various successful applications of fuzzy set theory.

However, several more general types of fuzzy sets have also been proposed in the literature. Let fuzzy sets of the type thus far discussed be called ordinary fuzzy sets to distinguish them from fuzzy sets of the various generalized types.

The primary reason for generalizing ordinary fuzzy sets is that their membership functions are often overly precise. They require that each element of the universal set be assigned a particular real number. However, for some concepts and contexts in which they are applied, we may be able to identify appropriate membership functions only approximately.

For example, we may only be able to identify meaningful lower and upper bounds of membership grades for each element of the universal set. In such cases, we may basically take one of two possible approaches. We may either suppress the identification uncertainty by choosing reasonable values between the lower and upper bounds e. A membership function based on the latter approach does not assign to each element of the universal set one real number, but a closed interval of real numbers between the identified lower and upper bounds.

Fuzzy sets denned by membership functions of this type are called interval-valued fuzzy sets. These sets are defined formally by functions of the. An example of a membership function of this type is given in Fig. For each x, A x is represented by the segment between the two curves, which express the identified lower and upper bounds.

Membership functions of interval-valued fuzzy sets are not as specific as their counter-. Their advantage is that they allow us to express our uncertainty in identify- ing a particular membership function. This uncertainty is involved when interval-valued fuzzy sets are processed, making results of the processing less specific but more credi- ble.

The primary disadvantage of interval-valued fuzzy sets is that this processing, when compared with ordinary fuzzy sets, is computationally more demanding. Since most current applications of fuzzy set theory do not seem to be overly sensitive to minor changes in relevant membership functions, this disadvantage of interval-valued fuzzy sets usually outweighs their advantages.

Interval-valued fuzzy sets can further be generalized by allowing their intervals to be fuzzy. Each interval now becomes an ordinary fuzzy set defined within the universal set [0,1]. Since membership grades assigned to elements of the universal set by these generalized fuzzy sets are ordinary fuzzy sets, these sets are referred to as fuzzy sett of type 2.

Their membership functions have the form. The concept of a type 2 fuzzy set is illustrated in Fig. It is assumed here that membership functions of all fuzzy intervals involved are of trapezoidal shapes and, consequently, each of them is fully defined by four numbers. For each x, these numbers are produced by four functions, represented in Fig. Fuzzy sets of type 2 possess a great expressive power and, hence, are conceptually quite appealing.

However, computational demands for dealing with them are even greater than those for dealing with interval-valued fuzzy sets. This seems to be the primary reason why they have almost never been utilized in any applications. Assume now that the membership grades assigned by a type 2 fuzzy set e.

A Grand. Paradigm Shift Chap. Then, we obtain a fuzzy set of type 3, and it is easy to see that fuzzy sets of still higher types could be obtained recursively in the same way.

However, there is little rationale for further pursuing this line of generalization, at least for the time being, since computational complexity increases significantly for each higher type.

When we relax the requirement that membership grades must be represented by numbers in the unit interval [0,1] and allow them to be represented by symbols of an arbitrary set L that is at least partially ordered, we obtain fuzzy sets of another generalized type. They are called L- fuzzy sets, and their membership functions have the form A: By allowing only a partial ordering among membership grades, L -fuzzy sets are very general.

In fact, they capture all the other types introduced thus far as special cases. A different generalization of ordinary fuzzy sets involves fuzzy sets defined within a universal set whose elements are ordinary fuzzy sets.

These fuzzy sets are known as level 2 fuzzy sets. Level 2 fuzzy sets allow us to deal with situations in which elements of the universal set cannot be specified precisely, but only approximately, for example, by fuzzy sets expressing propositions of the form "x is close to r," where x is a variable whose values are real numbers, and r is a particular real number.

In order to determine the membership grade of some value of x in an ordinary fuzzy set A, we need to specify the value say, r precisely. Assuming that the proposition "x is close to r" is represented by an ordinary fuzzy set B, the membership grade of a value of x that is known to be close to r in the level 2 fuzzy set A is given by A B. Level 2 fuzzy sets can be generalized into level 3 fuzzy sets by using a universal set whose elements are level 2 fuzzy sets.

Higher-level fuzzy sets can be obtained recursively in the same way. We can also conceive of fuzzy sets that are of type 2 and also of level 2. Their membership functions have the form A: A formal treatment of fuzzy sets of type 2 and level 2 as well as higher types and levels is closely connected with methods of fuzzification, which are discussed in Sec. Except for this brief overview of the various types of generalized fuzzy sets, we do not further examine their properties and the procedures by which they are manipulated.

Their detailed coverage is beyond the scope of this text. Since these generalized types of fuzzy sets have not as yet played a significant role in applications of fuzzy set theory, this omission is currently of no major consequence. The generalized fuzzy sets are introduced in this section for two reasons.

Basic Concepts. Second, we feel that the practical significance of at least some of the generalized types will gradually increase, and it is thus advisable that the reader be familiar with the basic ideas and terminology pertaining to them. We may occasionally refer to some of the generalized types of fuzzy sets later in the text. By and large, however, the rest of the text is devoted to the study of ordinary fuzzy sets.

Unless otherwise stated, the term "fuzzy set" refers in this text to ordinary fuzzy sets. For the sake of completeness, let us mention that ordinary fuzzy sets may also be viewed as fuzzy sets of type 1 and level 1. In this section, we introduce some basic concepts and terminology of fuzzy sets. To illustrate the concepts, we consider three fuzzy sets that represent the concepts of a young,, middle-aged, and old person. These functions are defined on the interval [0, 80] as follows: A possible discrete approximation, D2, of function A2, is also shown in Fig.

Such approximations are important because they are typical in computer representations of fuzzy sets. One of the most important concepts of fuzzy sets is the concept of an a-cut and its variant, a strong a-cut. A f Middle age: A 2 Old: Shown discrete approximation Di of Ai is defined numerically in Table The set of all levels a e [0,1] that represent distinct a-cuts of a given fuzzy set A is called a level set of A.

Basic Concepts An important property of both a-cuts and strong a-cuts, which follows immediately from their definitions, is that the total ordering of values of a in [0,1] is inversely preserved by set inclusion of the corresponding a-cuts as well as strong a-cuts.

Notice, for example, that the intervals representing the a-cuts and the strong a-cuts of the fuzzy sets Alt A2, and A3 in Fig. Since level sets of Ai, A2, and A3 are all [0,1], clearly, the families of all a-cuts and all strong a-cuts are in this case infinite for each of the sets. The families of all a-cuts and all strong a-cuts of D 2 are shown for some convenient values of a in Fig.

The support of a fuzzy set A within a universal set X is the crisp set that contains all the elements of X that have nonzero membership grades in A. The height, h A , of a fuzzy set A is the largest membership grade obtained by any element in that set.

This property is viewed as a generalization of the classical concept of convexity of crisp sets. Since all the a- cuts are convex, the resulting fuzzy set is also viewed as convex. To avoid confusion, note that the definition of convexity for fuzzy sets does not mean that the membership function of a convex fuzzy set is a convex function. In fact, membership functions of convex fuzzy sets are functions that are, according to standard definitions, concave and not convex.

We now prove a useful theorem that provides us with an alternative formulation of convexity of fuzzy sets. For the sake of simplicity, we restrict the theorem to fuzzy sets on K, which are of primary interest in this text. Theorem 1. We need to prove that for any a g 0,1], "A is convex. Now for any xlt x2 e "A i. Hence, A is convex.

Convexity of fuzzy sets, as defined above, is an example of a cutworthy property and, as can be proven, also a strong cutworthy property. The three basic operations on crisp sets—the complement, intersection and union—can be generalized to fuzzy sets in more than one way. However, one particular generalization, which results in operations that are usually referred to as standard fuzzy set operations, has a special significance in fuzzy set theory.

In the following, we introduce only the standard operations. For the standard complement, clearly, membership grades of equilibrium points are 0. For example, the equilibrium points of A2 in Fig. Due to the associativity of min and max, these definitions can be extended to any finite number of fuzzy sets. Applying these standard operations to the fuzzy sets in Fig. The equation makes good sense: Another example based on the same sets is shown in Fig. Normality and convexity may thus be lost when we operate on fuzzy sets by the standard operations of intersection and complement.

The lattice is distributed and complemented under the standard fuzzy complement. It satisfies all the properties of the Boolean lattice listed in Table 1. Such a lattice is often referred to as a De Morgan lattice or a De Morgan algebra.

Hence, the law of contradiction is satisfied only for crisp sets. We have to consider two cases: Other laws of the De Morgan lattice can be verified similarly. This equivalence makes a connection between the two definitions of the lattice.

For example, the scalar, cardinality of the fuzzy set D2 defined in Table 1. For any pair of fuzzy subsets defined on a finite universal set X, the degree of subsethood, S A, B , of A in B is defined by the formula. To conclude this section, let us introduce a special notation that is often used hi the literature for defining fuzzy sets with a finite support. For the case in which a fuzzy set A is defined on a universal set that is finite or countable, we may write, respectively,.

Jx Again, the integral sign does not have, in this notation, the usual meaning; it solely indicates that all the pairs of x and A x in the interval X collectively form A. It is interesting and conceptually useful to interpret ordinary fuzzy subsets of a finite universal set X with n elements as points in the n-dimensional unit cube [0,1]". That is, the entire cube represents the fuzzy power set 3 X , and its vertices represent the crisp power set IP X.

This interpretation suggests that a suitable distance be defined between fuzzy sets. Observe that probability distributions are represented by sets whose cardinality is 1. Hence, the set of all probability distributions that can be defined on X is represented by an n — 1 -dimensional simplex of the n- dimensional unit cube.

Examples of this simplex are shown in Fig. Before embarking on deeper study of fuzzy sets, let us reflect a little more on the transition from the traditional view to the modern view of uncertainty, which "is briefly mentioned in Sec. It is increasingly recognized that this transition has characteristics typical of processes, usually referred to as paradigm shifts, which appear periodically throughout the history of science.

The concept of a scientific paradigm was introduced by Thomas Kuhn in his important and highly influential book The Structure of Scientific Revolutions Univ. Using this concept, Kuhn characterizes scientific development as a process in which periods of normal science, based upon a particular paradigm, are interwoven with periods of paradigm shifts, which are referred to by Kuhn as scientific revolutions.

In his book, Kuhn illustrates the notion of a paradigm shift by many well-documented examples from the history of science. Some of the most visible paradigm shifts are associated with the names of Copernicus astronomy , Newton mechanics , Lavoisier chemistry , Darwin biology , Maxwell electromagnetism , Einstein mechanics , and Godel mathematics.

Although paradigm shifts vary from one another in their scope, pace, and other features, they share a few general characteristics: Each paradigm shift is initiated by emerging problems that are difficult or impossible to: Each paradigm, when proposed, is initially rejected in various forms it is ignored, ridiculed, attacked, etc.

Those who usually support the new paradigm are either very young or very new to! Since the paradigm is initially not well-developed, the position of its proponents is weak. The paradigm eventually gains its status on pragmatic grounds by demonstrating that it is more successful than the existing paradigm in dealing with problems that are generally recognized as acute. As a rule, the greater the scope of a paradigm shift, the longer it takes for the new paradigm to be generally accepted. The paradigm shift initiated by the concept of a fuzzy set and the idea of mathematics based upon fuzzy sets, which is currently ongoing, has similar characteristics to other paradigm shifts recognized in the history of science.

It emerged from the need to bridge the gap between mathematical models and their empirical interpretations. This gap has become increasingly disturbing, especially in the areas of biological, cognitive, and social sciences, as well as in applied sciences, such as modern technology and medicine. The need to bridge the gap between a mathematical model and experience is well characterized in a penetrating study by the American philosopher Max Black []:.

It is a paradox, whose importance familiarity fails to diminish, that the most highly developed and useful scientific theories are ostensibly expressed in terms of objects never encountered in experience. The line traced by a draftsman, no matter how accurate, is seen beneath the microscope as a kind of corrugated trench, far removed from the ideal line of pure geometry.

And the "point-planet" of astronomy, the "perfect gas" of thermodynamics, or the "pure species" of genetics axe equally remote from exact realization. While the mathematician constructs a theory in terms of "perfect" objects, the experimental scientist observes objects of which the properties demanded by theory are and can, in the very nature of measurement, be only approximately true.

Mathematical deduction is not useful to the physicist if interpreted rigorously. It is necessary to know that its validity is unaltered when the premise and conclusion are only "approximately true.

To do so, however, replaces the original mathematical deduction by a more complicated mathematical theory in respect of whose interpretation the same problem arises, and whose exact nature is in any case unknown, This lack of exact correlation between a scientific theory and its empirical interpretation can be blamed either upon the world or upon the theory.

We can regard the shape of an orange or a tennis ball as imperfect copies of an ideal form of which perfect knowledge is to be had in pure geometry, or we can regard the geometry of spheres as a simplified and imperfect version of the spatial relations between the members of a certain class of physical objects. On either view there remains a gap between scientific theory and its application which ought to be, but is not, bridged.

To say that all language symbolism, or thought is vague is a favorite method for evading the problems involved and lack of analysis has the disadvantage of tempting even the most eminent thinkers into the appearance of absurdity.

We shall not assume that "laws" of logic or mathematics prescribe modes of existence to which intelligible discourse must necessarily conform. It will be argued, on the contrary, that deviations from the logical or mathematical standards of precision are all pervasive in symbolism; that to label them as subjective aberrations sets an impassable gulf between formal laws and experience and leaves the usefulness of the formal sciences an insoluble mystery.

The same need w a s expressed by Zadeh [], three years before he actually proposed the new paradigm of mathematics based upon the concept of a fuzzy set:. There are some who feel this gap reflects the fundamental inadequacy of the conventional mathematics—the mathematics of precisely-defined points, functions, sets, probability measures, etc.

Indeed, the need for such mathematics is becoming increasingly apparent even in the realm of inanimate systems, for in most practical cases the a priori data as well as the criteria by which the performance of a man-made system is judged are far from being precisely specified or having accurately known probability distributions.

When the new paradigm was proposed [Zadeh, b], the usual process of a paradigm shift began. The concept of a fuzzy set, which underlies this new paradigm, was initially ignored, ridiculed, or attacked by many, while it was supported only by a few, mostly young and not influential.

In spite of the initial lack of interest, skepticism, or even open hostility, the new paradigm persevered with virtually no support in the s, matured significantly and gained some support in the s, and began to demonstrate its superior pragmatic utility in the s. This is not surprising, since the scope of the paradigm shift is enormous.

The new paradigm does not affect any particular field of science, but the very foundations of science. In fact, it challenges the most sacred element of the foundations—the Aristotelian two-valued logic, which for millennia has been taken for granted and viewed as inviolable.

The acceptance of such a radical challenge is surely difficult for most scientists; it requires an open mind, enough time, and considerable effort to properly comprehend the meaning and significance of the paradigm shift mvolved. At this time, we can recognize at least four features that make the new paradigm superior to the classical paradigm:.

The new paradigm allows us to express irreducible observation and measurement uncertainties in their various manifestations and make these uncertainties intrinsic to empirical data. Such data, which are based on graded distinctions among states of relevant variables, are usually called fuzzy data. When fuzzy data are processed, their intrinsic uncertainties are processed as well, and the results obtained are more meaningful, in both epistemological and pragmatic terms, than, their counterparts obtained by processing the usual crisp data.

For the reasons briefly discussed in Sec. The general experience is that the more complex the problem involved, the greater the superiority of fuzzy methods. The new paradigm has considerably greater expressive power; consequently, it can effectively deal with a broader class of problems.

In particular, it has the capability to capture and deal with meanings of sentences expressed in natural language. This capability of the new paradigm allows us to deal in mathematical terms with problems that require the use of natural language. The new paradigm has a greater capability to capture human common-sense reasoning, decision making, and other aspects of human cognition. When employed in machine design, the resulting machines are human-friendlier. The reader may not be able at this point to comprehend the meaning and significance of the described features of the new paradigm.

This hopefully will be achieved after his or her study of this whole text is completed. For a general background on crisp sets and classical two-valued logic, we recommend the book Set Theory and Related Topics by S. Lipschutz Shaum, New York, The book covers all topics that are needed for this text and contains many solved examples. For a more advanced treatment of the topics, we recommend the book Set Theory and Logic by R.

Stoll W. Freeman, San Francisco, The concept of L-fuzzy sets was introduced by Goguen [].

The concept of fuzzy sets of level k, which is due to Zadeh [b], was investigated by Gottwald []. Convex fuzzy sets were studied in greater detail by Lowen [] and Liu [].

The geometric interpretation of ordinary fuzzy sets as points in the n -dimensional unit cube was introduced and pursued by Kosko [, , a], 1. An alternative set theory, which is referred to as the theory of semisets, was proposed and developed by Vopenka and Hajek [] to represent sets with imprecise boundaries. Unlike fuzzy sets, however, semisets may be defined in terms of vague properties and not necessarily - by explicit membership grade functions.

While semisets are more general than fuzzy sets, they are required to be approximated by fuzzy sets in practical situations. The relationship between semisets and fuzzy sets is well characterized by Novak []. The concept of semisets leads into a formulation of an alternative nonstandard set theory [Vopenka, ]. Explain the difference between randomness and fuzziness. Find some examples of prospective fuzzy variables in daily life.

Describe the concept of a fuzzy set in your own words. Find some examples of interval-valued fuzzy sets, L -fuzzy sets, level 2 fuzzy sets, and type 2 fuzzy sets. Explain why we need fuzzy set theory. Explain why the law of contradiction and the law of exclusive middle are violated in fuzzy set theory under the standard fuzzy sets operations. What is the significance of this?

Compute the scalar cardinalities for each of the following fuzzy sets: Let A, B be fuzzy sets defined on a universal set X. Prove that. Determine mathematical formulas and graphs of the membership grade functions of each of the following sets: Calculate the a-cuts and strong a-cuts of the three fuzzy sets in Exercise 1. Hania rated it it was amazing May 20, Mike rated it really liked it Mar 26, Steve rated it really liked it Sep 12, Suganya K rated it liked it Aug 15, Neoasimov rated it really liked it Feb 28, Alexandre Teixeira Mafra rated it really liked it Nov 26, Ideva rated it liked it Jan 30, Lovelur rated it really liked it Dec 09, Yates Buckley rated it really liked it Apr 22, Abhishek Singh rated it it was amazing Nov 05, Deeksha rated it liked it Dec 03, Rami rated it it was amazing Apr 10, Brad Thompson rated it it was amazing Oct 07, Oliver Laas rated it it was amazing Oct 24, Saleh rated it really liked it Oct 02, John rated it really liked it Sep 30, Kartik rated it really liked it Dec 11, Alejandra rated it liked it Sep 10, Bitopan rated it it was amazing Feb 08, There are no discussion topics on this book yet.

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