Fuzzy Set Theory

Fuzzy Set Theory
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Dr.NaveenBansal,India,Teacher
Published Date:25-10-2017
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CHAPTER 1 Fuzzy Set Theory The classical set theory is built on the fundamental concept of “set” of which an individual is either a member or not a member. A sharp, crisp, and unambiguous distinction exists between a member and a nonmember for any well-defined “set” of entities in this theory, and there is a very precise and clear boundary to indicate if an entity belongs to the set. In other words, when one asks the question “Is this entity a member of that set?” The answer is either “yes” or “no.” This is true for both the deterministic and the stochastic cases. In probability and statistics, one may ask a question like “What is the probability of this entity being a member of that set?” In this case, although an answer could be like “The probability for this entity to be a member of that set is 90%,” the final outcome (i.e., conclusion) is still either “it is” or “it is not” a member of the set. The chance for one to make a correct prediction as “it is a member of the set” is 90%, which does not mean that it has 90% membership in the set and in the meantime it possesses 10% non-membership. Namely, in the classical set theory, it is not allowed that an element is in a set and not in the set at the same time. Thus, many real-world application problems cannot be described and handled by the classical set theory, including all those involving elements with only partial membership of a set. On the contrary, fuzzy set theory accepts partial memberships, and, therefore, in a sense generalizes the classical set theory to some extent. In order to introduce the concept of fuzzy sets, we first review the elementary set theory of classical mathematics. It will be seen that the fuzzy set theory is a very natural extension of the classical set theory, and is also a rigorous mathematical notion. I. CLASSICAL SET THEORY A. Fundamental Concepts Let S be a nonempty set, called the universe set below, consisting of all the possible elements of concern in a particular context. Each of these elements is called a member, or an element, of S. A union of several (finite or infinite) members of S is called a subset of S. To indicate that a member s of S belongs to a subset S of S, we write s∈ S. If s is not a member of S, we write s∉ S. To indicate that S is a subset of S, we write S⊂ S.2 Fuzzy Set Theory •• 1 Usually, this notation implies that S is a strictly proper subset of S in the sense that there is at least one member x∈ S but x∉ S. If it can be either S⊂ S or S = S, we write S⊆ S. An empty subset is denoted by ∅. A subset of certain members that have properties P , ... , P will be denoted by a capital letter, say A, as 1 n A = a a has properties P , ..., P . 1 n An important and frequently used universe set is the n-dimensional n n Euclidean space R . A subset A⊆ R that is said to be convex if ⎡ x ⎤ ⎡ y ⎤ 1 1 ⎢ ⎥ ⎢ ⎥ x = M ∈ A and y = M ∈ A ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ x y ⎣ n⎦ ⎣ n⎦ implies λx + (1 −λ)y∈ A for any λ∈ 0,1. Let A and B be two subsets. If every member of A is also a member of B, i.e., if a∈ A implies a∈ B, then A is said to be a subset of B. We write A⊂ B. If both A ⊂ B and B ⊂ A are true, then they are equal, for which we write A = B. If it can be either A ⊂ B or A = B, then we write A ⊆ B. Therefore, A ⊂ B is equivalent to both A⊆ B and A≠ B. The difference of two subsets A and B is defined by A− B = c c∈ A and c∉ B . In particular, if A = S is the universe set, then S − B is called the complement of B, and is denoted by B , i.e., B = S− B. Obviously, B = B, S = ∅, and ∅ = S. Let r∈ R be a real number and A be a subset of R. Then the multiplication of r and A is defined to be rA = ra a∈ A . The union of two subsets A and B is defined by A∪ B = B∪ A = c c∈ A or c∈ B . Thus, we always have A∪ S = S, A∪∅ = A, and A∪ A = S. The intersection of two subsets A and B is defined by A∩ B = B∩ A = c c∈ A and c∈ B . Obviously, A∩ S = A, A∩∅ = ∅, and A∩ A = ∅. Two subsets A and B are said to be disjoint if A∩ B = ∅. Basic properties of the classical set theory are summarized in Table 1.1, where A⊆ S and B⊆ S.1•• Fuzzy Set Theory 3 Table 1.1 Properties of Classical Set Operations Involutive law A = A Commutative law A∪ B = B∪ A A∩ B = B∩ A Associative law ( A∪ B ) ∪ C = A∪ ( B∪ C ) ( A∩ B ) ∩ C = A∩ ( B∩ C ) Distributive law A∩ ( B∪ C ) = ( A∩ B ) ∪ ( A∩ C ) A∪ ( B∩ C ) = ( A∪ B ) ∩ ( A∪ C ) A∪ A = A A∩ A = A A∪ ( A∩ B ) = A A∩ ( A∪ B ) = A A∪ ( A ∩ B ) = A∪ B A∩ ( A ∪ B ) = A∩ B A∪ S = S A∩∅ = ∅ A∪∅ = A A∩ S = A A∩ A = ∅ A∪ A = S DeMorgan’s law A∩B = A ∪ B A∪B = A ∩ B In order to simplify the notation throughout the rest of the book, if the universe set S has been specified or is not of concern, we simply call any of its subsets a set. Thus, we can consider two sets A and B in S, and if A ⊂ B then A is called a subset of B. For any set A, the characteristic function of A is defined by 1 if x∈A, ⎧ X (x) = A ⎨ 0 if x∉A. ⎩ It is easy to verify that for any two sets A and B in the universe set S and for any element x∈ S, we have X (x) = max X (x), X (x) , A∪B A B X (x) = min X (x), X (x) , A∩B A B X (x) = 1 − X (x). A A B .Elementary Measure Theory of Sets In this subsection, we briefly review the basic notion of measure in the classical set theory which, although may not be needed throughout this book, will be useful in further studies of some advanced fuzzy mathematics.4 Fuzzy Set Theory •• 1 Let S be the universe set and A a nonempty family of subsets of S. Let, moreover, µ: A→ 0,∞ be a nonnegative real-valued function defined on (subsets of) A, which may assume the value ∞. A set B in A, denoted as an element of A by B∈ A, is called a null set with respect to µ if µ(B) = 0, where µ(B) = µ(b) b∈ B . µ is said to be additive if n n µ( A ) = µ(A ) U U i i i=1 i=1 n for any finite collection A ,...,A of sets in A satisfying both A ∈ A and 1 n U i i=1 A ∩ A = ∅, i ≠ j, i,j=1,...,n. µ is said to be countably additive if n = ∞ in the i j above. Moreover, µ is said to be subtractive if A∈A, B∈A, A⊆ B, B− A∈A, and µ(B)∞ together imply µ(B− A) = µ(B)−µ(A). It can be verified, however, that if µ is additive then it is also subtractive. Now, µ is called a measure on A if it is countably additive and there is a nonempty set C∈A such that µ(C)∞. For example, if we define a function µ by µ(A) = 0 for all A ∈A, then µ is a measure on A, which is called the trivial measure. As the second example, suppose that A contains at least one finite set and define µ by µ(A) = the number of elements belonging to A. Then µ is a measure on A, which is called the natural measure. A measure µ on A has the following two simple properties: (i) µ(∅) = 0, and (ii) µ is finitely additive. Letµ be a measure on A. Then a set A∈ A is said to have a finite measure if µ(A)∞, and have a σ-finite measure if there is a sequence A of sets in i A such that ∞ … A⊆ A and µ(A )∞ for all i = 1,2, . U i i i=1 µ is finite (resp., σ-finite) on A if every set in A has a finite (resp., σ-finite) measure. A measure µ on A is said to be complete if B∈ A , A⊆ B , and µ(B) = 0 together imply µ(A) = 0. µ is said to be monotone if A∈ A , B∈ A , and A⊆ B together imply µ(A)≤µ(B). µ is said to be subadditive if µ(A)≤µ(A ) + µ(A ) 1 2 for any A, A , A ∈ A with A = A ∪ A . µ is said to be finitely subadditive if 1 2 1 21•• Fuzzy Set Theory 5 n µ(A)≤ µ(A ) ∑ i i=1 n for any finite collection A,A ,...,A of subsets in A satisfying A = A , 1 n U i i=1 andµ is said to be countably subadditive if n = ∞ in the above. It can be shown that if µ is countably subadditive and µ(∅) = 0, then it is also finitely subadditive. Let A∈ A. A measure µ on A is said to be continuous from below at A if A ⊂ A, A ⊆ A ⊆ ..., and lim A = A 1 2 i i i→∞ together imply lim µ(A ) = µ(A), i i→∞ andµ is said to be continuous from above at A if A ⊂ A, A ⊇ A ⊇ ..., µ(A )∞, and lim A = A 1 2 1 i i i→∞ together imply lim µ(A ) = µ(A). i i→∞ µ is continuous from below (resp., above) on A if and only if it is continuous from below (resp., above) at every set A ∈ A, and µ is said to be continuous if it is continuous both from below and from above (at A, or on A). Let A and A be families of subsets of A such that A ⊆ A , and let µ and 1 2 1 2 1 µ be measures on A and A , respectively. µ is said to be an extension of µ 2 1 2 2 1 ifµ (A) = µ (A) for every A∈ A . 1 2 1 For example, let A = (−∞,∞), A = a,b) −∞ a b∞ , A = 1 2 family of all finite, disjoint unions of bounded intervals of the form c,d), and a measure µ be defined on A by 1 1 µ (a,b)) = b− a. 1 Then µ is countably additive and so is a finite measure on A . This µ can be 1 1 1 extended to a finite measure µ on A by defining 2 2 µ (a,b)) = µ (a,b)) for all a,b)∈ A . 2 1 1 More generally, if f is a finite, nondecreasing, and left-continuous real-valued function of a real variable, then : µ (a,b)) = f(b)− f(a) for all a,b)∈ A , f 1 defines a finite measure on A , and it can be extended to be a finite measure µ 1 2 on A . 2 II. FUZZY SET THEORY In Section I.A, we have defined the characteristic function X of a set A by A 1 if x∈A, ⎧ X (x) = A ⎨ 0 if x∉A, ⎩ which is an indicator of members and nonmembers of the crisp set A. In the case that an element has only partial membership of the set, we need to6 Fuzzy Set Theory •• 1 generalize this characteristic function to describe the membership grade of this element in the set: larger values denote higher degrees of the membership. To give more motivation for this concept of partial membership, let us consider the following examples. Example 1.1. Let S be the set of all human beings, used as the universe set, and let S = s∈ S s is old . f Then S is a “fuzzy subset” of S because the property “old” is not well defined f and cannot be precisely measured: given a person who is 40 year old, it is not clear if this person belongs to the set S. Thus, to make the subset S well- f f defined, we have to quantify the concept “old,” so as to characterize the subset S in a precise and rigorous way. f For the time being, let us say, we would like to describe the concept “old” by the curve shown in Figure 1.1(a) using common sense, where the only people who are considered to be “absolutely old” are those 120 years old or older, and the only people who are considered to be “absolutely young” are those newborns. Meanwhile, all the other people are old as well as young, depending on their actual ages. For example, a person 40 years old is considered to be “old” with “degree 0.5” and at the same time also “young” with “degree 0.5” according to the measuring curve that we used. We cannot exclude this person from the set S described above, nor include him f completely. Thus, the curve that we introduce in Figure 1.1(a) establishes a mathematical measure for the “oldness” of a human being, and hence can be used to define the partial membership of any person relative to the subset S f described above. The curve shown in Figure 1.1(a), which is indeed a generalization of the classical characteristic function X (it can be used to S f conclude a person who either “is” or “is not” a member of the subset S ), is f called a membership function associated with the subset S . f Of course, one may also use the piecewise linear membership function shown in Figure 1.1(b) to describe the same concept of oldness for the same subset S , depending on whichever is more meaningful and more convenient f in one’s concern, where both are reasonable and acceptable in common sense. The reader may suggest many more good candidates for such a membership function for the subset S described above. There is yet no fixed, unique, and f universal rule or criterion for selecting a membership function for a particular “fuzzy subset” in general: a correct and good membership function is determined by the user based on his scientific knowledge, working experience, and actual need for the particular application in question. This selection is more or less subjective, but the situation is just like in the classical probability theory and statistics where if one says “we assume that the noise is Gaussian and white,” what he uses to start with all the rigorous mathematics is a subjective hypothesis that may not be very true, simply because the noise in question may not be exactly Gaussian and may not be perfectly white. Using the same approach, we can say, “we assume that the membership function that 1•• Fuzzy Set Theory 7 1.0 1.0 0.8 0.8 0.5 0.5 0 40 80 120 age 0 40 80 120 age Figure 1.1(a) An example of Figure 1.1(b) Another example membership functions. of membership functions. describes the oldness is the one given in Figure 1.1(a),” to start with all the rigorous mathematics in the rest of the investigation. The fuzzy set theory is taking the same logical approach as what people have been doing with the classical set theory: in the classical set theory, as soon as the two-valued characteristic function has been defined and adopted, rigorous mathematics follows; in the fuzzy set case, as soon as a multi-valued characteristic function (the membership function) has been chosen and fixed, a rigorous mathematical theory can be fully developed. Now, we return to the subset S introduced above. Suppose that the f membership function associated with it, say the one shown in Figure 1.1(a), has been chosen and fixed. Then, this subset S along with the membership f function used, which we will denote by µ (s) with s ∈ S, is called a fuzzy S f f subset of the universe set S. A fuzzy subset thus consists of two components: a subset and a membership function associated with it. This is different from the classical set theory, where all sets and subsets share the same (and the unique) membership function: the two-valued characteristic function mentioned above. Throughout this book, if no confusion would arise, we will simply call a fuzzy subset a fuzzy set, keeping in mind that it has to be a subset of some universe set and has to have a pre-described membership function associated with it. To familiarize this new concept, let us now discuss one more example. Example 1.2. Let S be the (universe) set of all real numbers, and let S = s∈ S s is positive and large . f This subset, S , is not well-defined in the classical set theory because, although f the statement “s is positive” is precise, the statement “s is large” is vague. However, if we introduce a membership function that is reasonable and meaningful for a particular application for the characterization or measure of the property “large,” say the one shown in Figure 1.2 quantified by the function ⎧ 0 if s≤0, µ (s) = S ⎨ f −s 1− e if s0, ⎩ then the fuzzy subset S, associated with this membership function µ (s), is f S f well defined. Similarly, a membership function for the subset8 Fuzzy Set Theory •• 1 1 . µ ( ) S f µ ( ) s S f 0 s R Figure 1.2 A membership function for a positive and large real number. S = s∈ S s is small f may be chosen to be the one shown in Figure 1.3, where the cutting edge E is determined by the user according to his concern in the application. Other commonly used membership functions for fuzzy sets that are convenient in various applications are shown in Figure 1.4, where we have normalized their maximum value to be 1, as usual, since 1 = 100% describes a full membership and is convenient to use. Obviously, a membership function is a nonnegative-valued function, which differs from the probability density functions in that the area under the curve of a membership function need not be equal to unity (in fact, it can be any value between 0 and ∞, including 0 and ∞). Another distinction between the fuzzy set theory and the classical one (actually, the entire theory of classical mathematics) is that a member of a fuzzy set may assume two or more (even conflicting) membership values. For example, if we use the two membership functions shown in Figure 1.5 to measure “positive and large” and “negative and small,” respectively, then a member s = 0.1 has the first membership value 0.095 and the second 0.08: they do not sum up to 1.0 nor cancel out to be 0. Moreover, the two concepts are conflicting: s is positive and in the meantime negative, a situation just like someone is old and also is young, which classical mathematics cannot accept. Such a vague and conflicting description of a fuzzy set is acceptable by the fuzzy mathematics, however, which turns out to be very useful in many real-world applications. More importantly, the use of conflicting membership functions like this will not cause any logical or mathematical problems in the consequence, provided that a correct approach is taken in the sequel. 1 . µ ( ) S f R -E 0 E Figure 1.3 A membership function for a real number of small magnitude.1•• Fuzzy Set Theory 9 1 1 1 1 1 1 Figure 1.4 Various shapes of commonly used membership functions. 1.0 R -1 1 2 Figure 1.5 The real number s = 0.1 is both “positive large” and “negative small” at the same time. III. INTERVAL ARITHMETIC In the last section, we introduced the concept of fuzzy (sub)sets, which consists of two parts: a (sub)set defined in the classical sense and a membership function defined on the (sub)set that is also defined in the ordinary sense. In this section, we first study some fundamental properties and operation rules pertaining to a special yet important kind of sets − intervals − and then in the next section, we will study properties and operations of membership functions defined on intervals. A. Fundamental Concepts Our concern here is the situation where the value of a member s of a set is uncertain. We assume, however, that the information on the uncertain value of s provides an acceptable range: s≤ s≤ s , where s , s ⊂ R is called the interval of confidence about the values of s. As a special case, when s = s , we have the certainty of confidence s , s = s , s = s. We mainly study closed intervals in this book; so an interval will always mean a closed and bounded interval throughout, unless otherwise indicated. In the two-dimensional case, an interval of confidence has a10 Fuzzy Set Theory •• 1 _ s 2 s 2 _ s s 1 1 Figure 1.6 An interval of confidence in the two-dimensional case. rectangular shape as shown in Figure 1.6, and is sometimes called the region of confidence. In the next subsection, we will introduce operational rules among intervals of confidence, which are important and useful in their own right in regards to engineering applications that are relative to intervals such as robust modeling, robust stability, and robust control. To prepare for that, we first give the following definitions. Definition 1.1. (a) Equality: Two intervals s , s and s , s are said to be equal: 1 2 1 2 s , s = s , s 1 2 1 2 if and only if s = s and s = s . 1 2 1 2 (b) Intersection: The intersection of two intervals s , s and s , s is 1 2 1 2 defined to be s , s ∩ s , s = maxs ,s , min s , s 1 2 1 2 1 2 1 2 and s , s ∩ s , s = ∅ 1 2 1 2 if and only if s s or s s . 1 2 2 1 (c) Union: The union of two intervals s , s and s , s is defined to 1 2 1 2 be s , s ∪ s , s = mins ,s , max s , s , 1 2 1 2 1 2 1 2 provided that s , s ∩ s , s ≠∅. Otherwise, it is undefined 1 2 1 2 (since the result is not an interval). (d) Inequality: Interval s , s is said to be less than (resp., greater 1 1 than) interval s , s , denoted by 2 2 s , s s , s (resp., s , s s , s ) 1 2 1 2 1 2 1 21•• Fuzzy Set Theory 11 if and only if s s (resp., s s ). Otherwise, they cannot be 2 1 1 2 compared. Note that the relations ≤ and ≥ are not defined for intervals. (e) Inclusion: The interval s , s is said to be included in s , s , 1 2 1 2 denoted by s , s ⊆ s , s 1 2 1 2 if and only if s ≤ s and s ≤ s . This is equivalent to saying that 2 1 1 2 the interval s , s is a subset or subinterval of s , s . 1 2 1 2 (f) Width: The width of an interval s , s is defined to be w s , s = s − s. Hence, a singleton s = s,s has a width zero: w s = w s,s = 0, for all s∈ R. (g) Absolute Value: The absolute value of an interval s , s is defined to be s , s = max s , s . Thus, the absolute value of a singleton s = s,s is its usual absolute value: s,s = s for all s∈ R. (h) Midpoint (mean): The midpoint (or mean) of an interval s , s is defined to be 1 m s , s = ( s + s ). 2 (i) Symmetry: Interval s , s is said to be symmetric if and only if s = − s or m s , s = 0. Example 1.3. For three given intervals, S = −1,0, S = −1,2, and S = 1 2 3 2,10, we have S ∩ S = −1,0∩ −1,2 = −1,0, 1 2 S ∩ S = −1,0∩ 2,10 = ∅, 1 3 S ∩ S = −1,2∩ 2,10 = 2,2 = 2, 2 3 S ∪ S = −1,0∪ −1,2 = −1,2, 1 2 S ∪ S = −1,0∪ 2,10 = undefined, 1 3 S ∪ S = −1,2∪ 2,10 = −1,10, 2 3 S = −1,0 2,10 = S , 1 3 S = −1,0⊂ −1,2 = S , 1 2 wS = w −1,0 = 0 − (−1) = 1, 1 wS = w −1,2 = 2 − (−1) = 3, 2 wS = w 2,10 = 10− 2 = 8, 3 S = −1,0 = max −1, 0 = 1, 1 S = −1,2 = max −1, 2 = 2, 2 S = 2,10 = max 2, 10 = 10, 3 1 1 mS = m −1,0 = (−1 + 0) = − , 1 2 212 Fuzzy Set Theory •• 1 1 1 mS = m −1,2 = (−1 + 2) = , 2 2 2 1 mS = m 2,10 = (2 + 10) = 6. 3 2 B. Interval Arithmetic Let s , s , s , s , and s , s be intervals. The basic arithmetic of 1 2 1 2 intervals is defined as follows. Definition 1.2. (1) Addition. s , s + s , s = s + s , s + s . 1 2 1 2 1 2 1 2 (2) Subtraction. s , s − s , s = s − s , s − s . 1 2 1 2 1 2 2 1 (3) Reciprocal. –1 If 0 ∉ s , s then s , s = 1/ s , 1/s ; –1 if 0 ∈ s , s then s , s is undefined. (4) Multiplication. . s , s s , s = p, p . 1 2 1 2 Here p = min s s , s s , ss , s s , 1 2 1 2 2 1 1 2 p = max s s , s s , ss , s s . 1 2 1 2 2 1 1 2 (5) Division. –1 . s , s / s , s = s , s s , s , 1 2 1 2 1 2 1 2 provided that 0 ∉ s , s . 2 2 Here, it is very important to note that interval arithmetic intends to obtain an interval as the result of an operation such that the resulting interval contains all possible solutions. Therefore, all these operational rules are defined in a conservative way in the sense that it intends to make the resulting interval as large as necessary to avoid loosing any true solution. For example, 1,2− 0,1= 0,2 means for any a∈1,2 and any b∈0,1 , it is guaranteed that a− b∈0,2. It is also very important to point out that the conservatism may cause some unusual results that could seem to be inconsistent with the ordinary numerical solutions. For instance, according to the subtraction rule (2), we have 1,2 – 1,2 = –1,1 ≠ 0,0 = 0. The result –1,1 here contains 0, but not only 0. The reason is that there can be other possible solutions: if we take 1.5 from the first interval and 1.0 from the second, then the result is 0.5 rather than 0; and 0.5 is indeed contained in –1,1. Thus, an interval subtract itself is equal to zero (a point) only if this interval is itself a point (a trivial interval). In general, we have the following: For any interval Z,1•• Fuzzy Set Theory 13 Z− Z = 0; or Z / Z = I (0∉ Z) only if wZ = 0, i.e., Z = z,z is a point, where I = 1,1. Associated with this is the following: For any intervals X, Y, and Z, X + Z = Y + Z ⇒ X = Y. Moreover, we have the following: For any interval Z, with 0 ∈ Z, 2 . . Z = Z Z = z, z z, z = p, p , where 2 2 p = min z , z z , z = z z , 2 2 2 2 p = max z , z z , z = max z , z . It should be noted that this is consistent with the definition of interval multiplication, Definition 1.2 (4). However, if one changed it to the following: 2 2 2 2 2 Z = ( z, z ) = z z∈ Z = 0, max z , z , then it would be more natural in the sense that a square is always nonnegative, but it is not consistent with the interval multiplication definition. Observe, moreover, that if we take a negative number from the first interval and a positive one from the second to multiply, a negative number does result. Therefore, we will use the first square rule shown above, although, oftentimes, it gives a more conservative result in an interval operation involving interval squares. For three intervals, X = x, x , Y = y, y , and Z = z, z , if we consider the interval operations of addition (+), subtraction (–), multiplication (⋅), and division (/) to be (set-variable and set-valued) functions, namely, . Z = f(X,Y) = X Y, ∈ +, −, , / , then it can be verified that all these four functions are continuous on compact sets such as intervals (see Section III-D below). Each function, f(X,Y), therefore, assumes a maximum and a minimum value, as well as all values in between, on any (closed and bounded) interval. Thus X Y is, again, a (closed and bounded) interval. The set of intervals is therefore closed under the four . operations +, −, , / defined above. It is also clear that the real numbers x, y, z, … . are isomorphic to intervals of the form x,x, y,y, z,z, ... For this reason, we will simplify the notation x,x Y of a point-interval operation to x Y. On the other hand, the . multiplication symbol “ ” will often be dropped, and the division symbol “/” may sometimes be replaced by “÷” for convenience. We collect together the most important and useful interval operation rules in the next subsection. C. Algebraic Properties of Interval Arithmetic Let X, Y, and Z be intervals. We first have the following simple but important rules.14 Fuzzy Set Theory •• 1 Theorem 1.1. The addition and multiplication operations of intervals are commutatitve and associative but not distributive. More precisely: (1) X + Y = Y + X; (2) Z + ( X + Y ) = ( Z + X ) + Y; (3) Z ( XY ) = ( ZX ) Y; (4) XY = YX; (5) Z + 0 = 0 + Z = Z and Z0 = 0Z = 0, where 0 = 0,0; (6) ZI = IZ = Z, where I = 1,1; (7) Z ( X + Y ) ≠ ZX + ZY, except when (a) Z = z,z is a point; or (b) X = Y = 0; or (c) xy≥ 0 for all x∈ X and y∈ Y. In general, we only have the subdistributive law: Z ( X + Y ) ⊆ ZX + ZY. . Proof. For (1) and (4), let ∈ +, . Then X Y = x y x∈ X, y∈ Y = y x y∈ Y, x∈ X = Y X. . For (2) and (3), let ∈ +, . Then Z ( X Y )= z a z∈ Z, a∈ X Y = z ( x y ) z∈ Z, x∈ X, y∈ Y = ( z x ) y z∈ Z, x∈ X, y∈ Y = b y b∈ Z X, y∈ Y =( Z X ) Y. . For (5) and (6), let ∈ +, . Then Z 0 = z 0 z∈ Z, 0 ∈ 0 = 0 z 0 ∈ 0, z∈ Z = 0 Z. . . Z I= z 1 z∈ Z, 1 ∈ I . = 1 z 1 ∈ I, z∈ Z = z z∈ Z = Z. For (7): . (a) z ( X + Y )= z a a∈ X + Y . = z ( x + y ) x∈ X, y∈ Y . . = z x + z y x∈ X, y∈ Y = zX + zY. (b) Z ( 0 + 0 )= Z 0 = 0 = Z 0 + Z 0 by (5)1•• Fuzzy Set Theory 15 (c) Without loss of generality, we only consider the case where x ≥ 0 and y ≥ 0 in X = x, x and Y = y, y . If z ≥ 0, then we have Z ( X + Y ) = z ( x + y ), z ( x + ) y and Z X + Z Y = z x, z x + z y, z y = z ( x + y ), z ( x + y ) , i.e., the equality holds. If z ≤ 0 then by considering −Z we have the same situation and result. Now, if z 0, then we z have Z ( X + Y ) = z ( x + y ), z ( x + y ) and Z X + Z Y = z x , z x + z y , z y = z ( x + y ), z ( x + y ) , which proves the final case. We note that in case (7), we do not even have the distributive law Z(x + y) = Zx + Zy for points x and y. We also note that more conditions under which this equality holds can be found in Theorem 1.3. A counterexample for the distributive law is the following. Example 1.4. Let Z = 1,2, X = I = 1,1, and Y = −I = −1,−1. Then we have . Z ( X + Y ) = 1,2 ( I − I ) = 1,2 0 = 0; . . Z X + Z Y = 1,2 1,1 + 1,2 −1,−1 = −1,1≠ 0. A more general rule for interval arithmetic operations is the following fundamental monotonic inclusion law. Theorem 1.2. Let X , X , Y , and Y be intervals such that 1 2 1 2 X ⊆ Y and X ⊆ Y . 1 1 2 2 . Then for the operations ∈ +, −, , / , we have X X ⊆ Y Y . 1 2 1 2 Proof. Since X ⊆ Y and X ⊆ Y , it follows that 1 1 2 2 X X = x x x ∈ X , x ∈ X 1 2 1 2 1 1 2 2 ⊆ y y y ∈ Y , y ∈ Y 1 2 1 1 2 2 = Y Y . 1 2 In particular, for any interval Z we have 0∈ Z− Z and 1∈ Z / Z (0∉ Z), as discussed before. A particularly useful special case of the theorem is the following. Corollary 1.1. Let X and Y be intervals with x∈ X and y∈ Y. Then x y∈ X Y . for any ∈ +, −, , / . 16 Fuzzy Set Theory •• 1 The following properties can be easily verified by following the method of proof of Theorem 1.1. Theorem 1.3. Let X, Y, and Z be symmetric intervals in the sense that the means mX = 0, mY = 0, and mZ = 0. Then 1 (1) Z = wZ; 2 (2) Z = Z –1,1; (3) X + Y = X – Y = ( X + Y ) –1,1; (4) XY = X Y –1,1; (5) Z ( X± Y ) = ZX± ZY = Z ( X + Y ) –1,1; 1 (6) Z = mZ + wZ –1,1; 2 (7) if X and Y are symmetric but Z is arbitrary, then (a) ZX = Z X; and (b) Z ( X + Y ) = ZX + ZY. We note that the equality X + Y = X− Y in (3) seems to be very strange, but it is true for symmetric intervals X and Y, which can be verified as follows: X− Y =x, x – y, y = minx–y,x– y , max x –y, x− y = min x+y, x−y , max x +y, x –y =x+y, x + y = X + Y, where one should recall that Y = −y, y because it is symmetric. We close this subsection by discussing the problem of solving the interval equation AX = B, where A and B are both given intervals with 0 ∉ A, and X is to be determined. Theorem 1.4. Let X be a solution of the interval equation AX = B,0∉ A. Then X⊆ B / A. Proof. For any x ∈ X, there exist a ∈ A and b ∈ B such that ax = b. Hence, x = b/a∈ B / A since 0 ∉ A. Note, however, that even if B/A is undefined, the interval equation AX = B may still have a solution. For example, the equation –1/3,1 X = –1,2 has 0 ∈ –1/3,1, so that –1,2/–1/3,1 is undefined. Yet X = –1,2 is its unique solution. This can be verified by direct multiplication –1/3,1⋅ –1,2, which gives –1,2. The uniqueness is verified by examining all possible solutions for X = x, x that satisfy −1/3,1⋅ x, x = –1,2 via multiplication operations. Note also that the above interval equation may not have a solution at all even if 0∉ A. One example is 1,2X = 2,3.1•• Fuzzy Set Theory 17 The matrix equation AX = B and its solvability will be further discussed in Section III.H. D. Measure Theory of Intervals In this subsection, we only introduce the notion of distance in the concern of measure theory for a family of intervals. Definition 1.3. Let X = x, x and Y = y, y be intervals. The distance between X and Y is defined by d(X,Y) = max x – y , x – y . . It can be verified that the set-variable function d( , ) satisfies the following properties: (1) d(X,Y)≥ 0, and d(X,Y) = 0 if and only if X = Y; (2) d(X,Y)≤ d(X,Z) + d(Z,Y) for any interval Z (the triangular inequality). The triangular inequality can be verified as follows: d(X,Z) + d(Z,Y) = max x− z , x − z + max z− y , z − y ≥ max x− z + z− y , x − z + z − y ≥ max x− y , x − y =d(X,Y). For real numbers x and y, this distance reduces to the standard one: d( x,x, y,y ) = x− y . . . It can also be verified that the interval distance function d( , ) defined here induces a metric, a Hausdorff metric, which is a generalization of the distance between two singleton points in a usual metric space. In fact, for any two nonempty compact sets X and Y of real numbers, including intervals, the Hausdorff distance is defined by h(X,Y) = max sup inf d(x,y), supinf d(x,y) . y∈Y x∈X x∈X y∈Y The introduction of a metric into the family of intervals, denoted I, makes it a metric space. Thus, the concepts of convergence and continuity can be defined and used in the standard way. ∞ Definition 1.4. Let X be a sequence of intervals in I. This sequence n n=1 is said to be convergent to an interval X ∈ I if the sequences of the upper and lower bounds of the individual members of the sequence of intervals converge to the corresponding bounds of X = x, x , namely, lim x = x and lim x = x , n n n→∞ n→∞ where X = x , x . In this case, we write n n n lim X = X. n n→∞ Then, it can be verified by a standard argument that every Cauchy sequence of intervals converges to an interval in I, that is, we have the following result.18 Fuzzy Set Theory •• 1 Theorem 1.5. The topological space I, when equipped with the metric defined by d(⋅,⋅), is a complete metric space (I,d). The next result characterizes the convergence behavior of an important class of interval sequences. ∞ Theorem 1.6. Let X be a sequence of intervals such that n n=1 X ⊇ X ⊇ X ⊇ ... 1 2 3 Then lim X = X, where n n→∞ ∞ X = X . I n n=1 Proof. Consider the sequence of bounds x ≤ x ≤ x ≤ ... ≤ x ≤ x ≤ x . 1 2 3 3 2 1 ∞ The sequence of the lower bounds of X is a monotonic nondecreasing n n=1 sequence of real numbers with an upper bound, say x ∞. Thus, it 1 converges to a real number, x. Similarly, the monotonic nonincreasing ∞ sequence of real numbers x converges to a real number, x , for which x n n=1 ≤ x . Hence, it follows that ∞ lim X = x, x = X = X . n I n n→∞ n= 1 Corollary 1.2. If X ⊇ X ⊇ X ⊇ ... ⊇ Y, 1 2 3 then lim X = X with X⊇ Y. n n→∞ The next important property is fundamental, as mentioned in the last subsection. . Theorem 1.7. The interval operations +, −, , / , introduced in Section III-B, are continuous functions of intervals. Proof. We only show the addition operation; the others are similar. ∞ ∞ Let X and Y be two sequences of intervals in I, with n n=1 n n=1 lim X = X and lim Y = Y. n n n→∞ n→∞ ∞ The sequence of interval sums X + Y then satisfies n n n=1 lim (X +Y)= lim x +y , x + y n n n n n n n→∞ n→∞ = lim (x +y ), lim ( x + y ) n n n n n→∞ n→∞ = x+y, x + y = X + Y. Corollary 1.3. Let f be an ordinary continuous function and X be an interval. Let f (X) = min f(x), max f(x) . I x∈X x∈X1•• Fuzzy Set Theory 19 Then f (X) is a continuous interval-variable and interval-valued function. I This corollary can be easily verified by using the continuity of the real function f, which guarantees the continuity of all important interval-variable n X and interval-valued functions like X , e , sin(X), X , etc. Theorem 1.8. Let X = x, x , Y = y, y , Z = z, z , and S = s, s be intervals in I. Then (1) d(X+Y,X+Z) = d(Y,Z); (2) d(X+Y,Z+S)≤ d(X,Z) + d(Y,S); (3) d(λX,λY) = λ d(X,Y), λ∈ R; (4) d(XY,XZ)≤ X d(Y,Z). . . Proof. For (1), it follows from the definition of d( , ) that d(X+Y,X+Z) = max (x + y)− (x + z) , ( x + y )− ( x + z ) = max y− z , y − z =d(Y,Z). For (2), using the triangular inequality, part (1) above, and the symmetry of . . d( , ), we have d(X+Y,Z+S) ≤ d(X+Y,Y+Z) + d(Z+S,Y+Z) ≤ d(X,Z) + d(Y,S). For (3), for any real number λ∈ R, we have d(λX,λY) = max λx−λy , λ x −λ y =λ max x− y , x − y =λ d(X,Y). For (4), for an interval A = a, a , we will use l(A) = a and u(A) = a for convenience in this proof. Then, what we need to show is max l(XY)− l(XZ) , u(XY)− u(XZ) ≤ X d(Y,Z). We only show that l(XY)− l(XZ) ≤ X d(Y,Z) and the inequality u(XY) − u(XZ) ≤ X d(Y,Z) can be verified in the same manner. Without loss of generality, assume that l(XY)≥ l(XZ); the case of l(XY) l(XZ) can be similarly analyzed. Then, since XZ = xz x∈ X, z∈ Z , there exists an x∈ X such that l(XZ) = l(xZ). On the other hand, we have xY⊆ XY, which implies that l(xY)≥ l(XY). Hence, we have l(xY)− l(xZ)≥ l(XY)− l(XZ)≥ 0, so that l(XY)− l(XZ) = l(XY)− l(XZ)20 Fuzzy Set Theory •• 1 ≤ l(xY)− l(xZ) = l(xY)− l(xZ) ≤ x d(Y,Z) ≤ X d(Y,Z). E. Properties of the Width of an Interval In this subsection, we summarize some interesting and useful properties of the width of an interval, which is defined in Definition 1.1 (f), as follows. For an interval S = s, s , the width of S is wS = s – s , which is equivalent to wS = max s – s . 1 2 s ,s ∈S 1 2 In addition to the properties listed in Problem P1.2, we have the following. Theorem 1.9. Let X and Y be intervals. Then (1) wXY≤ wX Y + X wY; (2) wXY≥ max X wY , Y wX ; n n–1 ... (3) wX ≤ n X wX, n=1,2, ; n n ... (4) w (X–x) ≤ 2 ( wX ) , x∈ X, n=1,2, . Proof. For (1), using the equivalent definition wS = max s – s , we 1 2 s ,s ∈S 1 2 have wXY= max x y – x y 1 1 2 2 x ,x ∈X ;y ,y ∈Y 1 2 1 2 = max x y – x y x y – x y 1 1 1 2 1 2 2 2 + x ,x ∈X ;y ,y ∈Y 1 2 1 2 ≤ max x ( y – y ) + y ( x – x ) 1 1 2 2 1 2 x ,x ∈X ;y ,y ∈Y 1 2 1 2 ≤ max x (y – y ) + max y (x – x ) 1 1 2 2 1 2 x ,x ∈X ;y ,y ∈Y x ,x ∈X ;y ,y ∈Y 1 2 1 2 1 2 1 2 =( max x ) ( max (y – y ) ) 1 1 2 x∈X y ,y ∈Y 1 1 2 + ( max y ) ( max (x – x ) ) 1 2 2 y ∈Y x ,x ∈X 2 1 2 =X wY + wX Y. For (2), we first have wXY= max x y – x y 1 1 2 2 x ,x ∈X ;y ,y ∈Y 1 2 1 2 ≥ max x y – x y 1 1 1 2 x ,x ∈X ;y ,y ∈Y 1 2 1 2 = max x y – y 1 1 2 x ,x ∈X ;y ,y ∈Y 1 2 1 2 =X wY. Similarly, we can show that wXY ≥ Y wX. Hence, the inequality (2) follows. For (3), we use mathematical induction. First, for n = 1, the inequality holds. If the inequality is true for n ≥ 1, then it follows from part (1) above that

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