Lecture notes on Fuzzy set theory

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Fuzzy Set Theory-and Its Applications, Fourth Edition1 INTRODUCTION TO FUZZY SETS 1.1 Crispness, Vagueness, Fuzziness, Uncertainty Most of our traditional tools for formal modeling, reasoning, and computing are crisp, deterministic, and precise in character. By crisp we mean dichotomous, that is, yes-or-no-type rather than more-or-less type. In conventional dual logic, for instance, a statement can be true or false-and nothing in between. In set theory, an element can either belong to a set or not; and in optimization, a solution is either feasible or not. Precision assumes that the parameters of a model represent exactly either our perception of the phenomenon modeled or the features of the real system that has been modeled. Generally, precision also implies that the model is unequivocal, that is, that it contains no ambiguities. Certainty eventually indicates that we assume the structures and parameters of the model to be definitely known, and that there are no doubts about their values or their occurrence. If the model under consideration is a formal model Zimmermann 1980, p. 127, that is, if it does not pretend to model reality ade­ quately, then the model assumptions are in a sense arbitrary, that is, the model builder can freely decide which model characteristics he chooses. If, however, the model or theory asserts factuality Popper 1959; Zimmermann 1980, that is, if conclusions drawn from these models have a bearing on reality and are H.-J. Zimmermann, Fuzzy Set Theory - and Its Applications © Kluwer Academic Publishers 20012 FUZZY SET THEORY-AND ITS APPLICATIONS supposed to model reality adequately, then the modeling language has to be suited to model the characteristics of the situation under study appropriately. The utter importance of the modeling language is recognized by Apostel, when he says: The relationship between formal languages and domains in which they have models must in the empirical sciences necessarily be guided by two considerations that are by no means as important in the formal sciences: (a) The relationship between the language and the domain must be closer because they are in a sense produced through and for each other; (b) extensions of formalisms and models must necessarily be considered because everything introduced is introduced to make progress in the description of the objects studied. Therefore we should say that the formalization of the concept of approximate constructive necessary satisfaction is the main task of semantic study of models in the empirical sciences. Apostel 1961, p. 26 Because we request that a modeling language be unequivocal and nonredun­ dant on one hand and, at the same time, catch semantically in its terms all that is important and relevant for the model, we seem to have the following problem. Human thinking and feeling, in which ideas, pictures, images, and value systems are formed, first of all certainly has more concepts or comprehensions than our daily language has words. If one considers, in addition , that for a number of notions we use several words (synonyms) , then it becomes quite obvious that the power (in a set-theoretic sense) of our thinking and feeling is much higher than the power of a living language. If in tum we compare the power of a living lan­ guage with the logical language, then we will find that logic is even poorer. There­ fore it seems to be impossible to guarantee a one-to-one mapping of problems and systems in our imagination and in a model using a mathematical or logical language. One might object that logical symbols can arbitrarily be filled with semantic contents and that by doing so the logical language becomes much richer. It will be shown that it is very often extremely difficult to appropriately assign seman­ tic contents to logical symbols. The usefulness of the mathematical language for modeling purposes is undis­ puted. However, there are limits to the usefulness and the possibility of using classical mathematical language, based on the dichotomous character of set theory, to model particular systems and phenomena in the social sciences: "There is no idea or proposition in the field, which can not be put into mathematical lan­ guage, although the utility of doing so can very well be doubted" Brand 1961. Schwarz 1962 brings up another argument against the nonreflective use of math­ ematics when he states: "An argument, which is only convincing if it is precise loses all its force if the assumptions on which it is based are slightly changed,INTRODUCTION TO FUZZY SETS 3 while an argument, which is convincing but imprecise may well be stable under small perturbations of its underlying axioms." For factual models or modeling languages, two major complications arise: I. Real situations are very often not crisp and deterministic, and they cannot be described precisely. 2. The complete description of a real system often would require far more detailed data than a human being could ever recognize simultaneously, process, and understand. This situation has already been recognized by thinkers in the past. In 1923 the philosopher B. Russell 1923 referred to the first point when he wrote: All traditional logic habitually assumes that precise symbols are being employed. It is therefore not applicable to this terrestrial life but only to an imagined celestial existence. L. Zadeh referred to the second point when he wrote, "As the complexity of a system increases, our ability to make precise and yet significant statements about its behaviour diminishes until a threshold is reached beyond which preci­ sion and significance (or relevance) become almost mutually exclusive charac­ teristics." Zadeh 1973a Let us consider characteristic features of real-world systems again: Real situations are very often uncertain or vague in a number of ways. Due to lack of information, the future state of the system might not be known completely. This type of uncertainty (stochastic character) has long been handled appropriately by probability theory and statistics. This Kolmogoroff-type probability is essentially frequentistic and is based on set-theoretic considerations. Koopman's probability refers to the truth of statements and therefore is based on logic. In both types of probabilistic approaches, however, it is assumed that the events (elements of sets) or the statements, respectively, are well defined. We shall call this type of uncer­ tainty or vagueness stochastic uncertainty in contrast to the vagueness con­ cerning the description of the semantic meaning of the events, phenomena, or statements themselves, which we shall calljUzziness. Fuzziness can be found in many areas of daily life, such as in engineering see, for instance, Blockley 1980, medicine see Vila and Delgado 1983, meteorol­ ogy Cao and Chen 1983, manufacturing Mamdani 1981, and others. It is particularly frequent, however, in all areas in which human judgment, evaluation , and decisions are important. These are the areas of decision making, reasoning, learning, and so on. Some reasons for this fuzziness have already been mentioned . Others are that most of our daily communication uses "natural languages," and4 FUZZY SET THEORY-AND ITS APPLICATIONS a good part of our thinking is done in it. In these natural languages, the meaning of words is very often vague. The meaning of a word might even be well defined, but when using the word as a label for a set, the boundaries within which objects do or do not belong to the set become fuzzy or vague. Examples are words such as "birds" (how about penguins, bats, etc.?) or "red roses," but also terms such as "tall men," "beautiful women," and "creditworthy customers." In this context we can probably distinguish two kinds of fuzziness with respect to their origins: intrinsic fuzziness and informational fuzziness. The former is the fuzziness to which Russell's remark referred, and it is illustrated by "tall men." This term is fuzzy because the meaning of tall is fuzzy and dependent on the context (height of observer, culture, etc.). An example of the latter is the term "creditworthy customers" : A creditworthy customer can possibly be described completely and crisply if we use a large number of descriptors. These descriptors are more, however, than a human being could handle simultaneously. Therefore the term, which in psychology is called a "subjective category," becomes fuzzy. One could imagine that the subjective category "creditworthiness" is decomposed into two smaller subjective categories, each of which needs fewer descriptors to be completely described. This process of decomposition could be continued until the descriptions of the subjective categories generated are reasonably defined. On the other hand, the notion "creditworthiness" could be constructed by starting with the smallest subjective subcategories and aggregating them hierarchically. For creditworthiness the concept structure shown in figure 1-1, which has a symmetrical structure, was developed in consultation with 50 credit clerks of banks. Credit experts distinguish between the financial basis and the personality of an applicant. "Financial basis" comprises all realities, movables, assets, liquid funds, and others. The evaluation of the economic situation depends on the actual securities, that is, the difference between property and debts, and on the liquid­ ity, that is, the continuous difference between income and expenses. On the other hand, "personality" denotes the collection of traits by which a potent and serious person is distinguished. The achievement potential is based on mental and physical capacity as well as on the individual's motivation. The busi­ ness conduct includes economical standards. While the former means the setting of realistic goals, reasonable planning, and criteria of economic success, the latter is directed toward the applicant's disposition to obey business laws and mutual agreements. Hence a credit-worthy person lives in secure circumstances and guar­ antees a successful, profit-oriented cooperation (see figure 1-1). Before turning to fuzzy set theory it should, however, be stressed that uncer­ tainty is a multi-facetted phenomenon and that the modeling of it in application­ oriented models requires considerable investigations before we start the modelingINTRODUCTION TO FUZZY SETS 5 Figure 1-1. Concept hierarchy of creditworthiness. process. Also the available modeling tools do not only include probability theory and fuzzy set theory. We shall consider this fact in more detail in chapter 8. In chapter 16 we will return to this figure and elaborate on the type of aggregation. 1.2 Fuzzy Set Theory The first publications in fuzzy set theory by Zadeh 1965 and Goguen 1967, 1969 show the intention of the authors to generalize the classical notion of a set and a proposition statement to accommodate fuzziness in the sense described in section 1.1. Zadeh 1965, p. 339 writes, "The notion of a fuzzy set provides a convenient point of departure for the construction of a conceptual frame-work which paral­ lels in many respects the framework used in the case of ordinary sets, but is more general than the latter and, potentially, may prove to have a much wider scope of applicability, particularly in the fields of pattern classification and information processing. Essentially, such a framework provides a natural way of dealing with6 FUZZY SET THEORY-AND ITS APPLICATIONS problems in which the source of imprecision is the absence of sharply defined criteria of class membership rather than the presence of random variables." "Imprecision" here is meant in the sense of vagueness rather than the lack of knowledge about the value of a parameter (as in tolerance analysis) . Fuzzy set theory provides a strict mathematical framework (there is nothing fuzzy about fuzzy set theory) in which vague conceptual phenomena can be precisely and rigorously studied. It can also be considered as a modeling language well suited for situations in which fuzzy relations, criteria, and phenomena exist. Fuzziness has so far not been defined uniquely semantically, and probably never will be. It will mean different things, depending on the application area and the way it is measured. In the meantime, numerous authors have contributed to this theory. In 1984, as many as 4,000 publications have already existed and in 2000 there were already more than 30,000. The specialization of those publications conceivably increases, making it more and more difficult for newcomers to this area to find a good entry and to under­ stand and appreciate the philosophy, formalism, and applications potential of this theory. Roughly speaking, fuzzy set theory in the last two decades has developed along two lines: 1. As a formal theory that, when maturing, became more sophisticated and spec­ ified and was enlarged by original ideas and concepts as well as by "embrac­ ing" classical mathematical areas such as algebra, graph theory, topology, and so on by generalizing (fuzzifying) them. 2. As an application-oriented "fuzzy technology", i.e. as a tool for modeling, problem solving and data mining that has proven superior to existing methods in many cases and as an attractive "add-on" to classical approaches in other cases. In this context it may be useful to cite and comment the major goals of this tech­ nology briefly and to correct the still very common view that fuzzy set theory or fuzzy technology is exclusively or primarily useful to model uncertainty: a) Modeling of uncertainty This is certainly the best known and oldest goal. I am not sure, however, whether it can (still) be considered to be the most important goal of fuzzy set theory. Uncertainty has been a very important topic for several centuries. There are numerous methods and theories which claim to be the only proper tool to model uncertainties. In general, however, they do not even define sufficiently or only in a very specific and limited sense what is meant by "uncertainty". I believe that uncertainty, if considered as a subjective phenomenon, can and ought to beINTRODUCTION TO FUZZY SETS 7 modeled by very different theories, depending on the causes of uncertainty, the type and quantity of available information, the requirements of the observer etc. In this sense fuzzy set theory is certainly also one of the theories which can be used to model specific types of uncertainty under specific types of circumstances. It might then compete with other theories, but it might also be the most appro­ priate way to model this phenomenon for well-specified situations. It would certainly exceed the scope of this article to discuss this question in detail here Zimmermann 1997. b) Relaxation Classical models and methods are normally based on dual logic. They, therefore, distinguish between feasible and infeasible, belonging to a cluster or not, optimal or suboptimal etc. Often this view does not capture reality adequately. Fuzzy set theory has been used extensively to relax or generalize classical methods from a dichotomous to a gradual character. Examples of this are fuzzy mathematical pro­ gramming Zimmermann 1996, fuzzy clustering Bezdek and Pal 1992, fuzzy Petri Nets Lipp et al. 1989, fuzzy multi criteria analysis Zimmermann 1986. c) Compactification Due to the limited capacity of the human short term memory or of technical systems it is often not possible to either store all relevant data, or to present masses of data to a human observer in such a way, that he or she can perceive the information contained in these data. Fuzzy technology has been used to reduce the complexity of data to an acceptably degree usually either via linguistic vari­ ables or via fuzzy data analysis (fuzzy clustering etc.). d) Meaning Preserving Reasoning Expert system technology has already been used since two decades and has led in many cases to disappointment. One of the reasons for this might be, that expert systems in their inference engines, when they are based on dual logic, perform symbol processing (truth values true or false) rather than knowledge processing. In approximate reasoning meanings are attached to words and sentences via lin­ guistic variables. Inference engines then have to be able to process meaningful linguistic expressions, rather than symbols, and arrive at membership functions of fuzzy sets, which can then be retranslated into words and sentences via linguistic approximation.8 FUZZY SET THEORY-AND ITS APPLICATIONS e) Efficient Determination ofApproximate Solutions Already in the 70s Prof. Zadeh expressed his intention to have fuzzy set theory considered as a tool to determine approximate solutions of real problems in an efficient or affordable way. This goal has never really been achieved successfully. In the recent past, however, cases have become known which are very good examples for this goal. Bardossy 1996, for instance, showed in the context of water flow modeling that it can be much more efficient to use fuzzy rule based systems to solve the problems than systems of differential equations. Comparing the results achieved by these two alternative approaches showed that the accu­ racy of the results was almost the same for all practical purposes. This is partic­ ularly true if one considers the inaccuracies and uncertainties contained in the input data. It seems desirable that an introductory textbook be available to help students get started and find their way around. Obviously, such a textbook cannot cover the entire body of the theory in appropriate detail. The present book will there­ fore proceed as follows: Part I of this book, containing chapters 2 to 8, will develop the formal frame­ work of fuzzy mathematics. Due to space limitations and for didactical reasons, two restrictions will be observed: I. Topics that are of high mathematical interest but require a very solid math­ ematical background and those that are not of obvious relevance to applica­ tions will not be discussed. 2. Most of the discussion will proceed along the lines of the early concepts of fuzzy set theory. At appropriate times, however, the additional potential of fuzzy set theory that arises by using other axiomatic frameworks resulting in other operators will be indicated or described. The character of these chap­ ters will obviously have to be formal. Part II of the book, chapters 9 to 16, will then survey the most interesting applications of fuzzy set theory. At that stage the student should be in a position to recognize possible extensions and improvements of the applications presented.I FUZZY MATHEMATICS This first part of this book is devoted to the formal framework of the theory of fuzzy sets. Chapter 2 provides basic definitions of fuzzy sets and algebraic oper­ ations that will then serve for further considerations . Even though we shall use one version of terminology and one set of symbols consistently throughout the book, alternative ways of denoting fuzzy sets will be mentioned because they have become common. Chapter 3 extends the basic theory of fuzzy sets by intro­ ducing additional concepts and alternative operators. Chapter 4 is devoted to fuzzy measures, measures of fuzziness, and other important measures that are needed for applications presented either in Part II of this book or in the second volume on decision making in a fuzzy environment. Chapter 5 introduces the extension principle, which will be very useful for the following chapters and covers fuzzy arithmetic. Chapters 6 and 7 will then treat fuzzy relations, graphs, and functions. Chapter 8 focuses on uncertainty modeling and some special topics, such as the relationship between fuzzy set theory, probability theory, and other classical areas.2 FUZZY SETS-BASIC DEFINITIONS 2.1 Basic Definitions A classical (crisp) set is normally defined as a collection of elements or objects x E X that can be finite, countable, or overcountable. Each single element can either belong to or not belong to a set A, A s X. In the former case, the statement "x belongs to A" is true, whereas in the latter case this statement is false. Such a classical set can be described in different ways: one can either enu­ merate (list) the elements that belong to the set; describe the set analytically, for instance, by stating conditions for membership (A = xix:::; 5); or define the member elements by using the characteristic function, in which I indicates mem­ bership and 0 nonmembership. For a fuzzy set, the characteristic function allows various degrees of membership for the elements of a given set. Definition 2-1 If X is a collection of objects denoted generically by x, then e fuzz y set Ain X is a set of ordered pairs: H.-J. Zimmermann, Fuzzy Set Theory - and Its Applications © Kluwer Academic Publishers 200112 FUZZY SET THEORY-AND ITS APPLICATIONS A= (x, J.A(x))l x EX J.A(X) is called the membership function or grade of membership (also degree of compatibility or degree of truth) of x in Athat maps X to the membership space M (When M contains only the two points °and 1, A is nonfuzzy and J.A(X) is identical to the characteristic function of a nonfuzzy set). The range of the mem­ bership function is a subset of the nonnegative real numbers whose supremum is finite. Elements with a zero degree of membership are normally not listed. Example 2-1a A realtor wants to classify the house he offers to his clients. One indicator of comfort of these houses is the number of bedrooms in it. Let X = I, 2, 3,4, . . . , 10 be the set of available types of houses described by x = number of bedrooms in a house. Then the fuzzy set "comfortable type of house for a four-person family" may be described as A=(l, .2), (2, .5), (3, .8), (4, 1), (5, .7), (6, .3) In the literature one finds different ways of denoting fuzzy sets: 1. A fuzzy set is denoted by an ordered set of pairs, the first element of which denotes the element and the second the degree of membership (as in definition 2-1). Example 2-1b A= "real numbers considerably larger than 10" A= (x, J.A(x))l XEX where O, x::; 10 J. -(x) = A (l +(x -1O)-2r', x 10 Example 2-1c A= "real numbers close to 10" See figure 2-1. 2. A fuzzy set is represented solely by stating its membership function for instance, Negoita and Ralescu 1975.FUZZY SETS-BASIC DEFINITIONS 13 15 x 10 5 Figure 2-1 . Real numbers close to 10. n 3. A=IJ-A (XI)/ XI +IJ-A (X2)/ X2 . •. =L.IJ-A (X;)/ Xi ;=1 or LIJ-A()/X Example 2-1d A = "integers close to 10" A= 0.1/7 + 0.5/8+0.8/9+ 1/10+0.8/11 + 0.5/12 + O.1/13 Example 2-1e A= "real numbers close to 10" It has already been mentioned that the membership function is not limited to values between 0 and I. If SUpxIJ-A(X) = I, the fuzzy set Ais called normal. A non­ empty fuzzy set Acan always be normalized by dividing IJ-A(X) by SUpxIJ-A(X): As a matter of convenience, we will generally assume that fuzzy sets are normal­ ized. For the representation of fuzzy sets, we will use the notation I illustrated in examples 2-lb and 2-lc, respectively. A fuzzy set is obviously a generalization of a classical set and the member­ ship function a generalization of the characteristic function. Since we are gener­ ally referring to a universal (crisp) set X, some elements of a fuzzy set may have the degree of membership zero. Often it is appropriate to consider those elements of the universe that have a nonzero degree of membership in a fuzzy set.14 FUZZY SET THEORY-AND ITS APPLICATIONS Definition 2-2 The support of a fuzzy set A, S(,1), is the crisp set of all x E X such that IlA(X) O. Example 2-2 Let us consider example 2-1a again: The support of S(,1)= 1, 2, 3, 4, 5, 6. The elements (types of houses) 7, 8, 9, 1O are not part of the support of A A more general and even more useful notion is that of an a-level set. Definition 2-3 The (crisp) set of elements that belong to the fuzzy set Aat least to the degree a is called the a-level set: Au =xEXIIlA(x)2:a A = x E X IIlA(x) a is called "strong a-level set" or "strong a-cut." Example 2-3 We refer again to example 2-1a and list possible a -level sets: A. = I, 2, 3, 4, 5, 6 2 A.s=2, 3, 4, 5 A.&= 3, 4 AI =4 The strong a-level set for a =.8 is N& = 4. Convexity also plays a role in fuzzy set theory. By contrast to classical set theory, however, convexity conditions are defined with reference to the membership function rather than the support of the fuzzy set. Definition 2-4 A fuzzy set Ais convex if Ilti(I"x, +(1- A)X2)2:minllti (XI), Ilti(X2),X),X2 E X, AE 0, I Alternatively, a fuzzy set is convex if all a -level sets are convex.FUZZY SETS-BASIC DEFINITIONS 15 x Figure 2-2a. Convex fuzzy set. x Figure 2-2b. Nonconvex fuzzy set. Example 2-4 Figure 2- 2a depicts a convex fuzzy set, whereas figure 2-2b illustrates a non­ convex fuzzy set. One final feature of a fuzzy set, which we will use frequently in later chap­ ters, is its cardinality or "power" Zadeh 1981c).16 FUZZY SET THEORY-AND ITS APPLICATIONS Definition 2-5 For a finite fuzzy set A, the cardinality IAI is defined as IAI= L/lA(X) xeX IIAII = IIII is called the relative cardinality of A. Obviously, the relative cardinality of a fuzzy set depends on the cardinality of the universe . So you have to choose the same universe if you want to compare fuzzy sets by their relative cardinality. Example 2-5 For the fuzzy set "comfortable type of house for a four-person family" from example 2-1 a, the cardinality is IAI =.2 + .5 + .8 + 1 + .7 + .3 = 3.5 Its relative cardinality is - 35 IIAII =-' =0.35 10 The relative cardinality can be interpreted as the fraction of elements of X being in A, weighted by their degrees of membership in A. For infinite X, the cardinal­ ity is defined by IAI = fx/l..i(x) dx. Of course, IAI does not always exist. 2.2 Basic Set-Theoretic Operations for Fuzzy Sets The membership function is obviously the crucial component of a fuzzy set. It is therefore not surprising that operations with fuzzy sets are defined via their mem­ bership function s. We shall first present the concepts suggested by Zadeh in 1965 Zadeh 1965, p. 310. They constitute a consistent framework for the theory of fuzzy sets. They are, however, not the only possible way to extend classical set theory consistently. Zadeh and other authors have suggested alternative or addi­ tional definitions for set-theoretic operations, which will be discussed in chapter 3. Definition 2-6 The membership function /lc(x) of the intersection C=A n Bis pointwise defined by /lcCx)= min/lA(x), /lii(X), x E XFUZZY SETS-BASIC DEFINITIONS 17 Definition 2-7 The membership function 1l6(X) of the union fj = Au iJ is pointwise defined by Ilb(X) = maxu, (x), Ilb(X), x E X Definition 2-8 The membership function of the complement of a normalized fuzzy set A, /l¢A(X) is defined by Example 2-6 Let Abe the fuzzy set "comfortable type of house for a four-person family" from example 2-1a and iJ be the fuzzy set "large type of house" defined as iJ = (3, .2), (4, .4), (5, .6), (6, .8), (7, I), (8, l) The intersection C= An iJ is then C= (3, .2), (4, .4), (5, .6), (6, .3) The union fj =Au iJ is b = (l, .2), (2, .5), (3, .8), (4, I), (5, .7), (6, .8), (7, I), (8, l) The complement ¢iJ, which might be interpreted as "not large type of house," is ¢iJ = (l, I), (2, I), (3, .8), (4, .6), (5, .4), (6, .2), (9, I), (l0, l) Example 2-7 Let us assume that = "x is considerable larger than 10," and B "x is approximately 11," characterized by A= (x, IlA(x»lx E X where18 FUZZY SET THEORY-AND ITS APPLICATIONS o+ ==__L.- ,;::II. _ x 5 10 11 Figure 2-3. Union and intersection of fuzzy sets. and where Then __ ( )_ffiin(1+(x-IO) -2)-I,(I+(X-11)4) -I for x1O fAnB X - o for x; 10 (x is considerably larger than 10 and approximately 11) fAUi/(X) = max(l +(x -10) -2)-I, (I+(x -11)4) -1, x E X Figure 2-3 depicts the above. It has already been mentioned that min and max are not the only operators that could have been chosen to model the intersection or union, respectively, of fuzzy sets. The question arises, why those and not others? Bellman and Giertz addressed this question axiomatically in 1973 Bellman and Giertz 1973, p. 151. They argued from a logical point of view, interpreting the intersection as "logical and," the union as "logical or," and the fuzzy set A as the statement "The element x belongs to set A," which can be accepted as more or less true. It is very instruc­ tive to follow their line of argument, which is an excellent example for an axiomatic justification of specific mathematical models. We shall therefore sketch their reasoning: Consider two statements, Sand T, for which the truth values areFUZZY SETS-BASIC DEFINITIONS 19 Js and Jr. respectively, Js, JT E 0, I. The truth value of the "and" and "or" combination of these statements, J(S and T) and J(S or T), both from the inter­ val 0, I, are interpreted as the values of the membership functions of the inter­ section and union , respectively, of Sand T. We are now looking for two real-valued functions j and g such that JSandT = j(J s, JT) JSorT = g(Js, JT) Bellman and Giertz feel that the following restrictions are reasonably imposed onjand g: I. j and g are nondecreasing and continuous in Js and JT. ii. j and g are symmetric, that is, j(J S, JT) = j(JT, Js) g(Js, JT)= g(JT , Js) iii. f(Js, Js) and g(Js, Js) are strictly increasing in Js· IV. j(Js, JT) ::;min (Js, JT) and g(Js, JT) max (Js, JT). This implies that accept­ ing the truth of the statement "S and T" requires more, and accepting the truth of the statement "S or T" less than accepting S or T alone as true. v. j(l, 1) = 1 and g(O, 0) =O. VI. Logically equivalent statements must have equal truth values, and fuzzy sets with the same contents must have the same membership functions, that is, is equivalent to (St and Sz) or(St and S3) and therefore must be equally true. Bellman and Giertz now formalize the above assumptions as follows : Using the symbols /\ for "and" (= intersection) and v for "or" (= union), these assump­ tions amount to the following seven restrictions, to be imposed on the two com­ mutative (see (ii) and associative (see (vi» binary compositions /\ and v on the closed interval 0, 1, which are mutually distributive (see (vi» with respect to one another. 1. Js /\ JT = JT/\ Js Js V JT= JT V Js 2. (Js /\ JT) /\ Ju = Js /\ (JT/\ Ju) (Js v JT) v Ju = Js V (JT V Ju)20 FUZZY SET THEORY-AND ITS APPLICATIONS 3. s /\ (T V u) =(s /\ T) V (s /\ u) s V (T /\ u) = (s V T) /\ (s V u) 4. s /\ T and s v T are continuous and nondecreasing in each component 5. s /\ s and s v s are strictly increasing in s (see (iii» 6. s /\ T :::; min (s, T) s V T 2: max (s, T) (see (ivj) 7. 1 /\ 1 = 1 o v 0 = 0 (see (v» Bellman and Giertz then prove mathematically see Bellman and Giertz 1973, p. 154 that SAT =min(s,T ) and SvT =max(s,T ) For the complement, it would be reasonable to assume that if statement "S" is true, its complement "non S" is false, or if s = 1 then nonS =0 and vice versa. The function h (as complement in analogy ofjand g for intersection and union) should also be continuous and monotonically decreasing, and we would like the complement of the complement to be the original statement (in order to be in line with traditional logic and set theory). These requirements, however, are not enough to determine uniquely the mathematical form of the complement. Bellman and Giertz require in addition that s(1/2) = 1/2. Other assumptions are certainly possible and plausible. Exercises 1. Model the following expressions as fuzzy sets: a. Large integers b. Very small numbers c. Medium-sized men d. Numbers approximately between 10 and 20 e. High speeds for racing cars 2. Determine all a-level sets and all strong a-level sets for the following fuzzy sets: a. A = (3, 1), (4, .2), (5, .3), (6, .4), (7, .6), (8, .8), (10, 1), (12, .8), (14, .6) 1 b. B= (x, B(X) =(l + (x - 1O)2r ) for a = .3, .5, .8 c. C= (x, c(x»lx E R where c(x) = 0 for x:::; 10 2r1 c (x) = (1 + (x - IOr for x 10

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