Classical sets and Fuzzy sets and Fuzzy relations

classical set theory vs fuzzy set theory and difference between classical sets and fuzzy sets and how to create fuzzy sets and how to draw fuzzy sets
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Published Date:25-10-2017
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CHAPTER 2 CLASSICAL SETS AND FUZZY SETS Philosophicalobjectionsmay be raisedby thelogical implicationsof buildingamathematical structureonthepremiseoffuzziness,sinceitseems(atleastsuperficially)necessarytorequire that an object be or not be an element of a given set. From an aesthetic viewpoint, this may be the most satisfactory state of affairs, but to the extent that mathematical structures are used to model physical actualities, it is often an unrealistic requirement.... Fuzzy sets have an intuitively plausible philosophical basis. Once this is accepted, analytical and practical considerations concerning fuzzy sets are in most respects quite orthodox. JamesBezdek Professor, Computer Science, 1981 As alluded to in Chapter 1, the universe of discourse is the universe of all available information on a given problem. Once this universe is defined we are able to define certain events on this information space. We will describe sets as mathematical abstractions of these events and of the universe itself. Figure 2.1a shows an abstraction of a universe of discourse, say X, and a crisp (classical) set A somewhere in this universe. A classical set is defined by crisp boundaries, i.e., there is no uncertainty in the prescription or location of the boundaries of the set, as shown in Fig. 2.1a where the boundary of crisp set A is an unambiguous line. A fuzzy set, on the other hand, is prescribed by vague or ambiguous properties;henceitsboundariesareambiguouslyspecified,asshownbythefuzzyboundary for set A in Fig. 2.1b. ∼ In Chapter 1 we introduced the notion of set membership, from a one-dimensional viewpoint.Figure 2.1againhelpstoexplainthis idea,butfromatwo-dimensional perspec- tive. Point a in Fig. 2.1a is clearly a member of crisp set A; point b is unambiguously not a member of set A. Figure 2.1b shows the vague, ambiguous boundary of a fuzzy set A on the same universe X: the shaded boundary represents the boundary region of A.In ∼ ∼ the central (unshaded) region of the fuzzy set, point a is clearly a full member of the set. Fuzzy Logic with Engineering Applications, Second Edition T. J. Ross  2004 John Wiley & Sons, Ltd ISBNs: 0-470-86074-X (HB); 0-470-86075-8 (PB) www.MatlabSite.comCLASSICAL SETS 25 X (Universe of discourse) X (Universe of discourse) c A A a a b b (a)(b) FIGURE2.1 Diagrams for (a) crisp set boundary and (b) fuzzy set boundary. Outside the boundary region of the fuzzy set, point b is clearly not a member of the fuzzy set. However, the membership of point c, which is on the boundary region, is ambiguous. If completemembership in aset (such as point a in Fig. 2.1b) is representedby the number 1, and no-membership in a set (such as point b in Fig. 2.1b) is represented by 0, then point c in Fig. 2.1b must have some intermediate value of membership (partial membership in fuzzy set A) on the interval 0,1. Presumably the membership of point c in A approaches ∼ ∼ a value of 1 as it moves closer to the central (unshaded) region in Fig. 2.1b of A,and ∼ the membership of point c in A approaches a value of 0 as it moves closer to leaving the ∼ boundary region of A. ∼ In this chapter, the precepts and operations of fuzzy sets are compared with those of classical sets. Several good books are available for reviewing this basic material see for example, Dubois and Prade, 1980; Klir and Folger, 1988; Zimmermann, 1991; Klir and Yuan, 1995. Fuzzy sets embrace virtually all (with one exception, as will be seen) of the definitions, precepts, and axioms that define classical sets. As indicated in Chapter 1, crisp sets are a special form of fuzzy sets; they are sets without ambiguity in their membership (i.e., they are sets with unambiguous boundaries). It will be shown that fuzzy set theory is a mathematically rigorous and comprehensive set theory useful in characterizing concepts (sets) with natural ambiguity. It is instructive to introduce fuzzy sets by first reviewing the elements of classical (crisp) set theory. CLASSICALSETS Define a universe of discourse, X, as a collection of objects all having the same character- istics. The individual elements in the universe X will be denoted as x. The features of the elements in X can be discrete, countable integers or continuous valued quantities on the real line. Examples of elements of various universes might be as follows: The clock speeds of computer CPUs The operating currents of an electronic motor The operating temperature of a heat pump (in degrees Celsius) The Richter magnitudes of an earthquake The integers 1 to 10 www.MatlabSite.com26 CLASSICAL SETS AND FUZZY SETS Mostreal-worldengineeringprocessescontainelementsthatarerealandnon-negative. The first four items just named are examples of such elements. However, for purposes of modeling, most engineering problems are simplified to consider only integer values of the elements in a universe of discourse. So, for example, computer clock speeds might be measured in integer values of megahertz and heat pump temperatures might be measured in integer values of degrees Celsius. Further, most engineering processes are simplified to consider only finite-sized universes. Although Richter magnitudes may not have a theoretical limit, we have not historically measured earthquake magnitudes much above 9; this value might be the upper bound in a structural engineering design problem. As another example, suppose you are interested in the stress under one leg of the chair in which you are sitting. You might argue that it is possible to get an infinite stress on one leg of the chair by sitting in the chair in such a manner that only one leg is supporting you and by letting the area of the tip of that leg approach zero. Although this is theoretically possible, in reality the chair leg will either buckle elastically as the tip area becomes very small or yield plastically and fail because materials that have infinite strength have not yet been developed. Hence, choosing a universe that is discrete and finite or one that is continuous and infinite is a modeling choice; the choice does not alter the characterization of sets defined on the universe. If elements of a universe are continuous, then sets defined on the universe will be composed of continuous elements. For example, if the universe of discourse is defined as all Richter magnitudesuptoavalueof9,thenwecandefineaset of ‘‘destructive magnitudes,’’ which might be composed (1) of all magnitudes greater than or equal to a value of 6 in the crisp case or (2) of all magnitudes ‘‘approximately 6 and higher’’ in the fuzzy case. A useful attribute of sets and the universes on which they are defined is a metric knownasthecardinality,orthecardinalnumber.Thetotalnumberofelementsinauniverse Xiscalleditscardinalnumber,denoted n ,where x againisalabelforindividual elements x in the universe. Discrete universes that are composed of a countably finite collection of elements will have a finite cardinal number; continuous universes comprised of an infinite collection of elements will have an infinite cardinality. Collections of elements within a universe arecalled sets, and collections of elements within sets are called subsets. Sets and subsets are terms that are often used synonymously, since any set is also a subset of the universal set X. The collection of all possible sets in the universe is called the whole set. For crisp sets A and B consisting of collections of some elements in X, the following notation is defined: x ∈ X ⇒ x belongs to X x ∈ A ⇒ x belongs to A x ∈ A ⇒ x does not belong to A For sets A and B on X, we also have A ⊂ B ⇒ A is fully contained in B (if x ∈ A,then x ∈ B) A ⊆ B ⇒ A is contained in or is equivalent to B (A ↔ B) ⇒ A ⊆ B and B ⊆ A (A is equivalent to B) We define the null set, ∅, as the set containing no elements, and the whole set, X, as the setof all elements in the universe.Thenull set is analogous to animpossible event,and www.MatlabSite.comCLASSICAL SETS 27 the whole set is analogous to a certain event. All possible sets of X constitute a special set called the power set, P(X). For a specific universe X, the power set P(X) is enumerated in the following example. Example2.1. We have a universe comprised of three elements, X=a, b, c, so the cardinal number is n = 3. The power set is x P(X)=∅, a, b, c, a,b, a,c, b, c, a,b,c The cardinality of the power set, denoted n , is found as P(X) n 3 X n = 2 = 2 = 8 P(X) Notethatifthecardinalityoftheuniverseisinfinite,thenthecardinalityofthepowersetis also infinity, i.e., n =∞⇒ n =∞. X P(X) OperationsonClassicalSets LetAandBbetwosetsontheuniverseX.Theunionbetweenthetwosets,denotedA ∪B, represents all those elements in the universe that reside in (or belong to) the set A, the set B, or both sets A and B. (This operation is also called the logical or; another form of the union is the exclusive or operation. The exclusive or will be described in Chapter 5.) The intersection of the two sets, denoted A ∩B, represents all those elements in the universe X that simultaneously reside in (or belong to) both sets A and B. The complement of a set A, denoted A, is defined as the collection of all elements in the universe that do not reside in the set A. The difference of a set A with respect to B, denoted A B,isdefined as the collection of all elements in the universe that reside in A and that do not reside in B simultaneously. These operations are shown below in set-theoretic terms. Union A ∪B=x x ∈ A or x ∈ B (2.1) Intersection A ∩B=x x ∈ A and x ∈ B (2.2) Complement A=x x ∈ A,x ∈ X (2.3) Difference A B=x x ∈ A and x ∈ B (2.4) These four operations are shown in terms of Venn diagrams in Figs. 2.2–2.5. A B FIGURE2.2 Union of sets A and B (logical or). www.MatlabSite.com28 CLASSICAL SETS AND FUZZY SETS A B FIGURE2.3 Intersection of sets A and B. A FIGURE2.4 Complement of set A. A B FIGURE2.5 Difference operation A B. PropertiesofClassical(Crisp)Sets Certain properties of sets are important because of their influence on the mathematical manipulationofsets.Themostappropriatepropertiesfordefiningclassicalsetsandshowing their similarity to fuzzy sets are as follows: Commutativity A ∪B = B ∪A A ∩B = B ∩A (2.5) www.MatlabSite.comCLASSICAL SETS 29 Associativity A ∪ (B ∪C) = (A ∪B) ∪C A ∩ (B ∩C) = (A ∩B) ∩C (2.6) Distributivity A ∪ (B ∩C) = (A ∪B) ∩ (A ∪C) A ∩ (B ∪C) = (A ∩B) ∪ (A ∩C) (2.7) Idempotency A ∪A = A A ∩A = A (2.8) Identity A∪∅ = A A ∩X = A A∩∅ =∅ (2.9) A ∪X = X Transitivity If A ⊆BandB ⊆ C, then A ⊆ C (2.10) Involution A = A (2.11) The double-cross-hatched area in Fig. 2.6 is a Venn diagram example of the asso- ciativity property for intersection, and the double-cross-hatched areas in Figs. 2.7 and 2.8 AB A B C C (a) (b) FIGURE2.6 Venn diagrams for (a) (A ∩B) ∩Cand(b)A ∩ (B ∩C). A B A B C C (a) (b) FIGURE2.7 Venn diagrams for (a) (A ∪B) ∩Cand(b) (A ∩C) ∪ (B ∩C). www.MatlabSite.com30 CLASSICAL SETS AND FUZZY SETS = A C = B C AB A B C C (a) (b) FIGURE2.8 Venn diagrams for (a) (A ∩B) ∪Cand(b) (A ∪C) ∩ (B ∪C). are Venn diagram examples of the distributivity property for various combinations of the intersection and union properties. Two special properties of set operations are known as the excluded middle axioms and De Morgan’s principles. These properties are enumerated here for two sets A and B. The excluded middle axioms are very important because these are the only set operations described here that are not valid for both classical sets and fuzzy sets. There are two excluded middle axioms (given in Eqs. (2.12)). The first, called the axiom of the excluded middle, deals with the union of a set A and its complement; the second, called the axiom of contradiction, represents the intersection of a set A and its complement. Axiom of the excluded middle A ∪A = X (2.12a) Axiom of the contradiction A ∩A=∅ (2.12b) De Morgan’s principles are important because of their usefulness in proving tautolo- gies and contradictions in logic, as well as in a host of other set operations and proofs. De Morgan’s principles are displayed in the shaded areas of the Venn diagrams in Figs. 2.9 and 2.10 and described mathematically in Eq. (2.13). A ∩B = A ∪B (2.13a) A ∪B = A ∩B (2.13b) In general, De Morgan’s principles can be stated for n sets, as provided here for events, E : i E ∪E ∪···∪E = E ∩E ∩···∩E (2.14a) 1 2 n 1 2 n E ∩E ∩···∩E = E ∪E ∪···∪E (2.14b) 1 2 n 1 2 n From the general equations, Eqs. (2.14), for De Morgan’s principles we get a duality relation: the complement of a union or an intersection is equal to the intersection or union, respectively, of the respective complements. This result is very powerful in dealing with www.MatlabSite.comCLASSICAL SETS 31 A B FIGURE2.9 De Morgan’s principle (A ∩B). A B FIGURE2.10 De Morgan’s principle (A ∪B). Load Arch members FIGURE2.11 A two-member arch. set structures since we often have information about the complement of a set (or event), or the complement of combinations of sets (or events), rather than information about the sets themselves. Example2.2. A shallow arch consists of two slender members as shown in Fig. 2.11. If either member fails, then the arch will collapse. If E = survival of member 1 and 1 E = survival of member 2, then survival of the arch = E ∩E , and, conversely, collapse 2 1 2 of the arch = E ∩E .Logically,collapseof the arch willoccur if eitherof the members fails, 1 2 i.e., when E ∪E . Therefore, 1 2 E ∩E = E ∪E 1 2 1 2 which is an illustrationof De Morgan’s principle. www.MatlabSite.com32 CLASSICAL SETS AND FUZZY SETS Source line A 1 Pump Junction 3 C 2 Source line B FIGURE2.12 Hydraulic hose system. AsEq. (2.14)suggests,DeMorgan’sprinciplesareveryusefulforcompoundevents, as illustrated in the following example. Example2.3. For purposes of safety, the fluid supply for a hydraulic pump C in an airplane comes from two redundant source lines, A and B. The fluid is transported by high-pressure hoses consisting of branches 1, 2, and 3, as shown in Fig. 2.12. Operating specifications for the pump indicate that either source line alone is capable of supplying the necessary fluid pressure to the pump. Denote E = failure of branch 1, E = failure of branch 2, and 1 2 E = failure of branch 3. Then insufficient pressure to operate the pump would be caused by 3 (E ∩E ) ∪E , and sufficient pressure would be the complement of this event. Using De 1 2 3 Morgan’s principles,we can calculate the conditionof sufficient pressure to be (E ∩E ) ∪E = (E ∪E ) ∩E 1 2 3 1 2 3 inwhich (E ∪E )meanstheavailabilityofpressureatthejunction,andE meanstheabsence 1 2 3 of failure in branch 3. MappingofClassicalSetstoFunctions Mapping is an important concept in relating set-theoretic forms to function-theoretic representations of information. In its most general form it can be used to map elements or subsetsononeuniverseofdiscoursetoelementsorsetsinanotheruniverse.SupposeXand Y are two different universes of discourse (information). If an element x is contained in X and corresponds to an element y contained in Y, it is generally termed a mapping from X to Y, or f :X → Y. As a mapping, the characteristic (indicator) function χ is defined by A  1,x ∈ A χ (x) = (2.15) A 0,x ∈/ A where χ expresses ‘‘membership’’ in set A for the element x in the universe. This A membership idea is a mapping from an element x in universe X to one of the two elements in universe Y, i.e., to the elements 0 or 1, as shown in Fig. 2.13. For any set A defined on the universe X, thereexists a function-theoretic set, called a valueset,denotedV(A),underthemappingofthecharacteristicfunction, χ.Byconvention, the null set ∅ is assigned the membership value 0 and the whole set X is assigned the membership value 1. www.MatlabSite.comCLASSICAL SETS 33 χ 1 A x 0 FIGURE2.13 Membership function is a mapping for crisp set A. Example2.4. Continuing with the example (Example 2.1) of a universe with three elements, X=a,b,c,wedesiretomap theelementsofthepowersetof X,i.e.,P(X), toa universe,Y, consistingof only two elements (the characteristic function), Y=0,1 As before, the elements of the power set are enumerated. P(X) = ∅, a,b,c,a,b,b,c, a,c,a,b,c Thus, the elements in the value set V(A) as determined from the mapping are VP(X)= 0,0,0,1,0,0,0,1,0,0,0,1, 1,1,0,0,1,1,1,0,1,1,1,1 For example, the third subsetin the power setP(X) is the element b. For this subsetthere is no a,soavalueof0goesinthefirstpositionofthedatatriplet;thereisa b,soavalueof1goesin thesecondpositionofthedatatriplet;andthereisno c,soavalueof0goesinthethirdposition ofthedatatriplet.Hence,thethirdsubsetofthevaluesetisthedatatriplet, 0,1,0,asalready seen. The value set has a graphical analog that is described in Chapter 1 in the section ‘‘Sets as Points in Hypercubes.’’ Now,definetwosets,AandB,ontheuniverseX.Theunionofthesetwosetsinterms of function-theoretic terms is given as follows (the symbol ∨ is the maximum operator and ∧ is the minimum operator): Union A ∪B −→ χ (x) = χ (x) ∨ χ (x) = max(χ (x), χ (x)) (2.16) A∪B A B A B The intersection of these two sets in function-theoretic terms is given by Intersection A ∩B −→ χ (x) = χ (x) ∧ χ (x) = min(χ (x), χ (x)) (2.17) A∩B A B A B The complement of a single set on universe X, say A, is given by Complement A −→ χ (x) = 1 − χ (x) (2.18) A A For two sets on the same universe, say A and B, if one set (A) is contained in another set (B), then Containment A ⊆ B −→ χ (x) ≤ χ (x) (2.19) A B www.MatlabSite.com34 CLASSICAL SETS AND FUZZY SETS Function-theoretic operators for union and intersection (other than maximum and minimum, respectively) are discussed in the literature Gupta and Qi, 1991. FUZZYSETS Inclassical,or crisp,sets the transition foranelement in the universebetweenmembership and nonmembership in a given set is abrupt and well-defined (said to be ‘‘crisp’’). For an elementin auniversethat contains fuzzysets, this transition canbe gradual.This transition among various degrees of membership can be thought of as conforming to the fact that the boundaries of the fuzzy sets are vague and ambiguous. Hence, membership of an element from the universe in this set is measured by a function that attempts to describe vagueness and ambiguity. Afuzzyset,then,isasetcontainingelementsthathavevaryingdegreesofmembership intheset.Thisideaisincontrastwithclassical,orcrisp,setsbecausemembersofacrispset would not be members unless their membership was full, or complete, in that set (i.e.,their membership is assigned a value of 1). Elements in a fuzzy set, because their membership need not be complete, can also be members of other fuzzy sets on the same universe. Elements of a fuzzy set are mapped to a universe of membership values using a function-theoretic form. As mentioned in Chapter 1 (Eq. (1.2)), fuzzy sets are denoted in thistextbyasetsymbolwithatildeunderstrike;so,forexample,AwouldbethefuzzysetA. ∼ ThisfunctionmapselementsofafuzzysetAtoarealnumberedvalueontheinterval0to1. ∼ If an element in the universe, say x, is a member of fuzzy set A, then this mapping is given ∼ by Eq. (1.2), or µ (x) ∈ 0,1. This mapping is shown in Fig. 2.14 for a typical fuzzy set. A ∼ A notation convention for fuzzy sets when the universe of discourse, X, is discrete and finite, is as follows for a fuzzy set A: ∼     µ (x ) µ (x )  µ (x ) A 1 A 2 A i ∼ ∼ ∼ A = + +··· = (2.20) ∼ x x x 1 2 i i When the universe, X, is continuous and infinite, the fuzzy set A is denoted by ∼    µ (x) A ∼ A = (2.21) ∼ x In both notations, the horizontal bar is not a quotient but rather a delimiter. The numerator in each term is the membership value in set A associated with the element of the universe ∼ µ A 1 0 x FIGURE2.14 Membership function for fuzzy set A. ∼ www.MatlabSite.comFUZZY SETS 35 indicatedinthedenominator.Inthefirstnotation,thesummationsymbolisnotforalgebraic summationbutratherdenotesthecollectionoraggregationofeachelement;hencethe‘‘+’’ signs in the first notation are not the algebraic ‘‘add’’ but are an aggregation or collection operator.Inthesecondnotationtheintegralsignisnotanalgebraicintegralbutacontinuous function-theoretic aggregation operator for continuous variables. Both notations are due to Zadeh 1965. FuzzySetOperations Define three fuzzy sets A,B,andC on the universe X. For a given element x of the ∼ ∼ ∼ universe, the following function-theoretic operations for the set-theoretic operations of union, intersection, and complement are defined for A,B,andC on X: ∼ ∼ ∼ Union µ (x) = µ (x) ∨ µ (x) (2.22) A∪B A B ∼ ∼ ∼ ∼ Intersection µ (x) = µ (x) ∧ µ (x) (2.23) A∩B A B ∼ ∼ ∼ ∼ Complement µ (x) = 1 − µ (x) (2.24) A A ∼ ∼ Venn diagrams for these operations, extended to consider fuzzy sets, are shown in Figs. 2.15–2.17. The operations given in Eqs. (2.22)–(2.24) are known as the standard fuzzy operations. There are many other fuzzy operations, and a discussion of these is given later in this chapter. AnyfuzzysetAdefinedonauniverseXisasubsetofthatuniverse.Alsobydefinition, ∼ just as with classical sets, the membership value of any element x in the null set ∅ is 0, µ A B 1 0 x FIGURE2.15 Union of fuzzy sets A and B. ∼ ∼ µ 1 A B 0 x FIGURE2.16 Intersection of fuzzy sets A and B. ∼ ∼ www.MatlabSite.com36 CLASSICAL SETS AND FUZZY SETS µ A 1 A 0 x FIGURE2.17 Complement of fuzzy set A. ∼ and the membership value of any element x in the whole set X is 1. Note that the null set and the whole set are not fuzzy sets in this context (no tilde understrike). The appropriate notation for these ideas is as follows: A ⊆ X ⇒ µ (x) ≤ µ (x) (2.25a) A X ∼ ∼ For all x ∈ X,µ (x) = 0 (2.25b) ∅ For all x ∈ X,µ (x) = 1 (2.25c) X The collection of all fuzzy sets and fuzzy subsets on X is denoted as the fuzzy power set P(X). It should be obvious, based on the fact that all fuzzy sets can overlap, that the ∼ cardinality, n , of the fuzzy power set is infinite; that is, n =∞. P(X) P(X) DeMorgan’sprinciples forclassicalsetsalso hold forfuzzysets,asdenotedby these expressions: A ∩B = A ∪B (2.26a) ∼ ∼ ∼ ∼ A ∪B = A ∩B (2.26b) ∼ ∼ ∼ ∼ As enumerated before, all other operations on classical sets also hold for fuzzy sets, except for the excluded middle axioms. These two axioms do not hold for fuzzy sets since theydonotformpartofthebasicaxiomaticstructureoffuzzysets(seeAppendixA);since fuzzy sets can overlap, a set and its complement can also overlap. The excluded middle axioms, extended for fuzzy sets, are expressed by A ∪A = X (2.27a) ∼ ∼ A ∩A =∅ (2.27b) ∼ ∼ Extended Venn diagrams comparing the excluded middle axioms for classical (crisp) sets and fuzzy sets are shown in Figs. 2.18 and 2.19, respectively. PropertiesofFuzzySets Fuzzy sets follow the same properties as crisp sets. Because of this fact and because the membership values of a crisp set are a subset of the interval 0,1, classical sets can be www.MatlabSite.comFUZZY SETS 37 χ χ 1 1 A A 0 x 0 x (a) (b) χ 1 0 x (c) FIGURE2.18 Excluded middle axioms for crisp sets. (a) Crisp set A and its complement; (b)crispA ∪A = X (axiom of excluded middle); (c)crispA ∩A=∅ (axiom of contradiction). µ µ A 1 1 A 0 x 0 x (a) (b) µ 1 0 x (c) FIGURE2.19 Excluded middle axioms for fuzzy sets. (a) Fuzzy set A and its complement; (b) fuzzy A ∪A = X ∼ ∼ ∼ (axiom of excluded middle); (c) fuzzy A ∩A =∅ (axiom of contradiction). thought ofasaspecialcaseoffuzzysets.Frequentlyusedpropertiesoffuzzysetsarelisted below. Commutativity A ∪B = B ∪A ∼ ∼ ∼ ∼ A ∩B = B ∩A (2.28) ∼ ∼ ∼ ∼    Associativity A ∪ B ∪C = A ∪B ∪C ∼ ∼ ∼ ∼ ∼ ∼    A ∩ B ∩C = A ∩B ∩C (2.29) ∼ ∼ ∼ ∼ ∼ ∼ www.MatlabSite.com38 CLASSICAL SETS AND FUZZY SETS      Distributivity A ∪ B ∩C = A ∪B ∩ A ∪C ∼ ∼ ∼ ∼ ∼ ∼ ∼      A ∩ B ∪C = A ∩B ∪ A ∩C (2.30) ∼ ∼ ∼ ∼ ∼ ∼ ∼ Idempotency A ∪A = A and A ∩A = A (2.31) ∼ ∼ ∼ ∼ ∼ ∼ Identity A∪∅ = A and A ∩ X = A ∼ ∼ ∼ ∼ A∩∅ =∅ and A ∪ X = X (2.32) ∼ ∼ Transitivity If A ⊆ B and B ⊆ C,thenA ⊆ C (2.33) ∼ ∼ ∼ ∼ ∼ ∼ Involution A = A (2.34) ∼ ∼ Example2.5. Consider a simple hollowshaft of approximately 1 m radius and wall thickness 1/(2π) m. The shaft is built by stacking a ductile section, D, of the appropriate cross section over a brittle section, B, as shown in Fig. 2.20. A downward force P and a torque T are simultaneouslyappliedtotheshaft.Becauseofthedimensionschosen,thenominalshearstress on any element in the shaft is T (pascals) and the nominal vertical component of stress in the shaft is P (pascals). We also assume that the failure properties of both B and D are not known with any certainty. WedefinethefuzzysetAtobetheregionin(P,T)spaceforwhichmaterialDis‘‘safe’’ ∼ 2 2 1/2 using as a metric the failure function µ = f(P +4T ). Similarly, we define the set B A ∼ to be the region in (P,T) space for which material B is ‘‘safe,’’ using as a metric the failure function µ = g(P − βT ),where β is an assumed material parameter. The functions f and B g will, of course, be membership functions on the interval 0, 1. Their exact specification is not important at this point. What is useful, however, prior to specifying f and g,istodiscuss the basic set operations in the context of this problem. This discussion is summarized below: 1. A ∪B is the set of loadings for which one expects that either material B or material D will ∼ ∼ be ‘‘safe.’’ 2. A ∩B is the set of loadings for which one expects that both material B and material D are ∼ ∼ ‘‘safe.’’ 3. AandBarethesetsofloadingsforwhichmaterialDandmaterialBareunsafe,respectively. ∼ ∼ P T Radius R = 1m D 1 Wall thickness = m 2π B (a)(b) FIGURE2.20 (a) Axial view and (b) cross-sectional view of example hollow shaft. www.MatlabSite.comFUZZY SETS 39 4. A Bisthesetofloadingsforwhichtheductilematerialissafebutthebrittlematerialisin ∼ ∼ jeopardy. 5. B Aisthesetofloadingsforwhichthebrittlematerialissafebuttheductilematerialisin ∼ ∼ jeopardy. 6. DeMorgan’sprincipleA ∩B = A ∪Bassertsthattheloadingsthatarenotsafewithrespect ∼ ∼ ∼ ∼ to both materials are the union of those that are unsafe with respect to the brittle material with those that are unsafe with respect to the ductile material. 7. DeMorgan’sprincipleA ∪B = A ∩Bassertsthattheloadsthataresafeforneithermaterial ∼ ∼ ∼ ∼ D nor material B are the intersection of those that are unsafe for material D with those that are unsafe for material B. To illustrate these ideas numerically, let’s say we have two discrete fuzzy sets, namely,   1 0.5 0.3 0.2 0.5 0.7 0.2 0.4 A = + + + and B = + + + ∼ ∼ 2 3 4 5 2 3 4 5 We can now calculate several of the operations just discussed (membership for element 1 in both A and B is implicitly0): ∼ ∼  1 0 0.5 0.7 0.8 Complement A = + + + + ∼ 1 2 3 4 5  1 0.5 0.3 0.8 0.6 B = + + + + ∼ 1 2 3 4 5  1 0.7 0.3 0.4 Union A ∪B = + + + ∼ ∼ 2 3 4 5  0.5 0.5 0.2 0.2 Intersection A ∩B = + + + ∼ ∼ 2 3 4 5  0.5 0.3 0.3 0.2 Difference A B = A ∩B = + + + ∼ ∼ ∼ ∼ 2 3 4 5  0 0.5 0.2 0.4 B A = B ∩A = + + + ∼ ∼ ∼ ∼ 2 3 4 5  1 0 0.3 0.7 0.6  De Morgan s A ∪B = A ∩B = + + + + ∼ ∼ ∼ ∼ 1 2 3 4 5 principles  1 0.5 0.5 0.8 0.8 A ∩B = A ∪B = + + + + ∼ ∼ ∼ ∼ 1 2 3 4 5 Example2.6. Continuing from the chemical engineering case described in Problem 1.13 of Chapter 1, suppose the selection of an appropriate analyzer to monitor the ‘‘sales gas’’ sour gas concentration is important. This selection process can be complicated by the fact that one type of analyzer, say A, does not provide an average suitable pressure range but it does give a borderlinevalueof instrumentdead time;incontrastanotheranalyzer,say B,may givea good value of process dead time but a poor pressure range. Suppose for this problem we consider three analyzers: A, B and C. Let  0.7 0.3 0.9 P = + + ∼ A B C www.MatlabSite.com40 CLASSICAL SETS AND FUZZY SETS represent the fuzzy set showing the pressure range suitability of analyzers A, B, and C (a membership of 0 is not suitable, a value of 1 is excellent). Also let  0.5 0.9 0.4 OT = + + ∼ A B C represent the fuzzy set showing the instrument dead time suitability of analyzers A, B, and C (again, 0 is not suitable and 1 is excellent). PandOTwillshowtheanalyzersthatarenotsuitableforpressurerangeandinstrument ∼ ∼ dead time, respectively:   0.3 0.7 0.1 0.5 0.1 0.6 P = + + and OT = + + , ∼ ∼ A B C A B C  0.3 0.1 0.1 therefore P ∩OT = + + ∼ A B C P ∪OT will show which analyzer is most suitable in either category: ∼ ∼  0.7 0.9 0.9 P ∪OT = + + ∼ ∼ A B C P ∩OT will show which analyzer is suitable in both categories: ∼ ∼  0.5 0.3 0.4 P ∩OT = + + ∼ ∼ A B C Example2.7. One of the crucial manufacturing operations associated with building the external fuel tank for the Space Shuttle involves the spray-on foam insulation (SOFI) process, which combines two critical component chemicals in a spray gun under high pressure and a precisetemperatureandflowrate.Controloftheseparameterstonearsetpointvaluesiscrucial for satisfying a number of important specification requirements. Specification requirements consist of aerodynamic, mechanical, chemical, and thermodynamic properties. Fuzzy characterization techniques could be employed to enhance initial screening experiments; for example, to determine the critical values of both flow and temperature. The true levels can only be approximated in the real world. If we target a low flow rate for 48 ◦ lb/min, it may be 38 to 58 lb/min. Also, if we target a high temperature for 135 F, it may be ◦ 133 to 137 F. Howtheimprecisionoftheexperimentalsetupinfluencesthevariabilitiesofkeyprocess end results could be modeled using fuzzy set methods, e.g., high flow with high temperature, low flow with low temperature, etc. Examples are shown in Fig. 2.21, for low flow rate and high temperature. Suppose we have a fuzzy set for flow, normalized on a universe of integers 1, 2, 3, 4, 5 anda fuzzysetfor temperature,normalizedona universeofintegers1,2,3,4,asfollows:   0 0.5 1 0.5 0 0 1 0 F = + + + + and D = + + ∼ ∼ 1 2 3 4 5 2 3 4 Further suppose that we are interested in how flow and temperature are related in a pairwise sense; we could take the intersection of these two sets. A three-dimensional image should be constructed when we take the union or intersection of sets from two different universes. For example, the intersection of F and D is given in Fig. 2.22. The idea of combining membership ∼ ∼ functions from two different universes in an orthogonal form, as indicated in Fig. 2.22, is associated with what is termed noninteractive fuzzy sets, and this will be described below. www.MatlabSite.comFUZZY SETS 41 µ µ Low flow rate High temperature 1 1 38 48 58 133 135 137 Flow rate (lb/min) Temperature (°F) (a) (b) FIGURE2.21 Foam insulationmembership function for (a) low flow rate and (b) high temperature. µ( f, d) f 1 5 4 3 2 1 0 1 2 3 4 d FIGURE2.22 Three-dimensional image of the intersection of two fuzzy sets, i.e., F ∩D. ∼ ∼ NoninteractiveFuzzySets Later in the text, in Chapter 8 on simulation, we will make reference to noninteractive fuzzy sets. Noninteractive sets in fuzzy set theory can be thought of as being analogous to independent events in probability theory. They always arise in the context of relations or in n-dimensional mappings Zadeh, 1975; Bandemer and Nather, ¨ 1992. A noninteractive fuzzy set can be defined as follows. Suppose we define a fuzzy set A on the Cartesian ∼ space X = X ×X.ThesetA is separable into two noninteractive fuzzy sets, called its 1 2 ∼ orthogonal projections, if and only if A = Pr (A) ×Pr (A)(2.35a) X X 1 2 ∼ ∼ ∼ where µ (x ) = max µ (x ,x ), ∀x ∈ X (2.35b) Pr (A) 1 A 1 2 1 1 X 1 ∼ ∼ x ∈X 2 2 µ (x ) = max µ (x ,x ), ∀x ∈ X (2.35c) Pr (A) 2 A 1 2 2 2 X 2 ∼ ∼ x ∈X 1 1 are the membership functions for the projections of A on universes X and X,respec- 1 2 ∼ tively. Hence, if Eq. (2.35a) holds for a fuzzy set, the membership functions µ (x ) Pr (A) 1 X 1 ∼ www.MatlabSite.com42 CLASSICAL SETS AND FUZZY SETS x x µ µ 2 2 A × A 1 2 1 1 x x 1 1 (a)(b) FIGURE2.23 Fuzzy sets: (a) interactive and (b) noninteractive. and µ (x ) describe noninteractive fuzzy sets, i.e., the projections are noninteractive Pr (A) 2 X 2 ∼ fuzzy sets. SeparabilityornoninteractivityoffuzzysetAdescribesakindofindependenceofthe ∼ components(x and x ):Acanbeuniquelyreconstructedbyitsprojections;thecomponents 1 2 ∼ of the fuzzy set A can vary without consideration of the other components. As an example, ∼ the two-dimensional planar fuzzy set shown in Fig. 2.22 comprises noninteractive fuzzy sets (F and D), because it was constructed by the Cartesian product (intersection in this ∼ ∼ case) of the two fuzzy sets F and D, whereas a two-dimensional fuzzy set comprising ∼ ∼ curved surfaces will be nonseparable, i.e., its components will be interactive. Interactive components are characterized by the fact that variation of one component depends on the values of the other components. See Fig. 2.23. AlternativeFuzzySetOperations The operations on fuzzy sets listed as Eqs. (2.22–2.24) are called the standard fuzzy operations. These operations are the same as those for classical sets, when the range of membership values is restricted to the unit interval. However, these standard fuzzy operations are not the only operations that can be applied to fuzzy sets. For each of the three standard operations, there exists a broad class of functions whose members can be considered fuzzy generalizations of the standard operations. Functions that qualify as fuzzy intersections and fuzzy unions are usually referred to in the literature as t-norms and t-conorms (or s-norms), respectively e.g., Klir and Yuan, 1995; Klement et al., 2000. These t-norms and t-conorms are so named because they were originally introduced as triangular norms and triangular conorms, respectively, by Menger 1942 in his study of statistical metric spaces. The standard fuzzy operations have special significance when compared to all of the other t-norms and t-conorms. The standard fuzzy intersection, min operator, when applied to a fuzzy set produces the largest membership value of all the t-norms, and the standard fuzzy union, max operator, when applied to a fuzzy set produces the smallest membership value of all the t-conorms. These features of the standard fuzzy intersection and union are www.MatlabSite.comREFERENCES 43 significant because they both prevent the compounding of errors in the operands Klir and Yuan, 1995. Most of the alternative norms lack this significance. Aggregation operations on fuzzy sets are operations by which several fuzzy sets are combined in a desirable way to produce a single fuzzy set. For example, suppose a computer’s performance in three test trials is described as excellent, very good, and nominal, and each of these linguistic labels is represented by a fuzzy set on the universe 0, 100. Then, a useful aggregation operation would produce a meaningful expression, in terms of a single fuzzy set, of the overall performanceof the computer. The standard fuzzy intersections and unions qualify as aggregation operations on fuzzy sets and, although they are defined for only two arguments, the fact that they have a property of associativity provides a mechanism for extending their definitions to three or more arguments. Other common aggregation operations, such as averaging operations and ordered weighted averaging operations, can be found in the literature see Klir and Yuan, 1995. The averaging operations have their own range that happens to fill the gap between the largest intersection (the min operator) and the smallest union (the max operator). These averaging operations on fuzzy sets have no counterparts in classical set theory and, because of this, extensions of fuzzy sets into fuzzy logic allows for the latter to be much more expressive in natural categories revealed by empirical data or required by intuition Belohlavek et al., 2002. SUMMARY Inthis chapterwehavedevelopedthebasicdefinitions for,propertiesof,andoperationson crisp sets and fuzzy sets. It has been shown that the only basic axioms not common to both crisp and fuzzy sets are the two excluded middle axioms; however, these axioms are not part of the axiomatic structure of fuzzy set theory (see Appendix A). All other operations detailed here are common to both crisp and fuzzy sets; however, other operations such as aggregation and averaging operators that are allowed in fuzzy sets have no counterparts in classical set theory. For many situations in reasoning, the excluded middle axioms present constraints on reasoning (see Chapters 5 and 15). Aside from the difference of set membership being an infinite-valued idea as opposed to a binary-valued quantity, fuzzy sets are handled and treated in the same mathematical form as are crisp sets. The principle of noninteractivity between sets was introduced and is analogous to the assumption of independence in probability modeling. Noninteractive fuzzy sets will become a necessary idea in fuzzy systems simulation when inputs from a variety of universes are aggregated in a collective sense to propagate an output; Chapters 5 and 8 will discuss this propagation process in more detail. Finally, it was pointed out that there are many other operations, called norms, that can be used to extend fuzzy intersections, unions, and complements, but such extensions are beyond the scope of this text. REFERENCES Bandemer, H. and Nather, ¨ W. (1992). Fuzzy data analysis, Kluwer Academic, Dordrecht. Belohlavek, R., Klir, G., Lewis, H., and Way, E. (2002). ‘‘On the capability of fuzzy set theory to represent concepts’’, Int. J. Gen. Syst., vol. 31, pp. 569–585.

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