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we might assume that a sentence, in the logical sense of the term, might have values other than falsehood or truth. A sentence, of which we do not know whether it is false or true, might have no value determined as truth or falsehood, but might have some third, undetermined, value. We might, for instance, consider that the sentence in a year from now I shall be in Warsaw is neither true or false and has a third, undetermined value, which can be symbolized as 1/2. We might go still further and ascribe to sentences infinitely many values contained between falsehood and truth... In the logic of infinitely many values, it is assumed that sentences can take on values represented by rational numbers x that satisfy the condition 0 ≤ × ≤ 1. As the need for many-valued logic is evident, similar is the rationale for the generalization of set theory culminating in the form of fuzzy sets. 3.2. Basic definitionAs profoundly emphasized in the literature right from the inception of the theory (Zadeh, 1965, 1973, 1981; Kandel, 1982, 1986; Klir and Folger, 1988), a fuzzy set A is defined as a collection of objects with membership values between 0 (complete exclusion) and 1 (complete membership). The membership values express the degrees with which each object is compatible with the properties or features that are distinctive to the collection. More formally, fuzzy sets are defined as follows. fuzzy set (Zadeh, 1965). A fuzzy set is characterized by a membership function mapping the elements of a domain, space or an universe of discourse X to the unit interval [0, 1], that is Thus, a fuzzy set A in X may be represented as a set of ordered pairs of a generic element x ∈ X and its grade of membership: A = {A(x)| x ∈ X}. Clearly, a fuzzy set is a generalization of the concept of a set whose membership function only takes two values {0, 1}, as discussed previously. The value of A(x) describes a degree of membership of x in A. For instance, consider the concept of high temperature in, say, environmental context with temperatures distributed in the interval [0,50] defined in °C. Clearly 0° C is not understood as a high temperature value and we may assign a zero degree to express its grade of compatibility with the high temperature concept. In other words, the membership degree of 0° C to the class of high temperatures is zero. Likewise, 30° C and over are certainly high temperatures and we may assign a full degree of compatibility with the concept. Fuzzy set can be regarded as an elastic constraint imposed on the elements of a universe or a domain. An interesting analogy which helps to grasp this idea can be developed by experimenting with an elastic rubber band. To encircle any new point, the band should be stretched. The extent of this stretching depends upon the location of the point. The membership value could be sought as inversely proportional to the force needed to stretch the rubber band and embrace this point. The higher the force, the lower the membership degree. The points not requiring the stretching at all naturally fit into the category. This example illustrates the essence of fuzzy sets as dealing primarily with a concept of elasticity, or absence of sharply defined boundary (Pedrycz, 1995). Fuzzy sets can be defined either in finite or infinite universes. Based on that, the reader should be aware of different notations being used throughout the literature. If an universe X is discrete and finite, with cardinality n, then a fuzzy set is given in a form of a n-dimensional vector whose entries denote grades of membership of the corresponding elements of X. Sometimes used is a sum notation. This allows us to enumerate only elements of X with nonzero grades of membership to the fuzzy set. For instance, if X = {x1, x2, , xn}, then the fuzzy set A = {(a1 /xi) | xi ∈ X }, where ai = A(xi), i = 1, , n, may be denoted by (Zadeh, 1965; Kandel, 1986; Negoita, 1981) In this notation the sum should not be confused with the standard algebraic summation; the only purpose of the symbol in the above expression is to denote the set of the ordered pairs. Also, note that when A = {a/x}, that is , there is only one point x in an universe for which the membership degree is nonzero, we have a fuzzy singleton. In this sense, we may also interpret the summation symbol as union of singletons. Equivalently, one can summarize A as a vector meaning that A=[a1 a2 an]. When the universe X is continuous, we use, to represent a fuzzy set, the following expression where a = A(x) and the integral symbol should be interpreted in the same manner as the sum given above. We conclude that points of the real line (numerical information) form a very specific form of a fuzzy set described through the delta function where x, p ∈ R. 3.3. Types of membership functionsIn principle any (continuous) function of the form A : X → [0,1] qualifies to describe a membership function associated with a fuzzy set. Let us note, however, that each fuzzy set comes with some meaning (semantics) attached to it and this narrows down a class of plausible membership functions. For instance, multimodal membership functions are somewhat eliminated as not being interpretable in an obvious way. Quite often the membership functions come equipped with parameters (parameterized membership functions) that allow us to adjust their form. The list of commonly used membership functions is included below. Note that the parameters of some of these functions are helpful in calibrating linguistic terms. Triangular membership functions Here m is a modal value while the lower and upper bounds for nonzero values of A(x) are denoted by a and b, respectively. The equivalent notation of the triangular membership function reads as Γ- membership function or where k > 0. S- membership function The point Trapezoidal membership function The equivalent notation reads as Gaussian membership function where k > 0. Exponential-like function k > 1 or with k > 0.
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