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Fuzzy Systems

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Consider the problem of designing a set of all tall people, and assigning all the people you know to this set. Consider classical set theory where an element is either a member of the set or not. Suppose all tall people are described as those with height greater than 1.75m. Then, clearly a person of height 1.78m will be an element of the set tall, and someone with height 1.5m will not belong to the set of tall people. But, the same will apply to someone of height 1.73m, which implies that someone who falls only 2cm short is not considered as being tall. Also, using two-valued set theory, there is no distinction among members of the set of tall people. For example, someone of height 1.78m and one of height 2.1m belongs equally to the set! Thus, no semantics are included in the description of membership. The alternative, fuzzy set theory, has no problem with this situation. In this case all the people you know will be members of the set tall, but to different degrees. For example, a person of height 2.1m may be a member of the set to degree 0.95, while someone of length 1.7m may belong to the set with degree 0.4. Fuzzy logic is an extension of Boolean logic to handle the concept of partial truth, which enables the modeling of the uncertainties of natural language. The vagueness in natural language is further emphasized by linguistic terms used to describe objects or situations. For example, the phrase when it is very cloudy, it will most probably rain, has the linguistic terms very and most probably - which are understood by the human brain. Fuzzy logic arid fuzzy sets give the tools to also write software which enables computing systems to understand such vague terms, and to reason with these terms. This chapter formally introduces fuzzy sets and fuzzy logic. Section 18.1 defines fuzzy sets, while membership functions are discussed in Section 18.2. Operators that can be applied to fuzzy sets are covered in Section 18.3. Characteristics of fuzzy sets are summarized in Section 18.4. The concepts of linguistic variables and hedges are discussed in Section 18.5. The chapter is concluded with a discussion of

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CHAPTER 18. FUZZY SYSTEMS

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the differences between fuzziness and probability in Section 18.6.

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Fuzzy Sets

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Different to classical sets, elements of a fuzzy set have membership degrees to that set. The degree of membership to a fuzzy set indicates the certainty (or uncertainty) we have that the element belongs to that set. Formally defined, suppose X is the domain, or universe of discourse, and x 6 X is a specific element of the domain X. Then, the fuzzy set A is characterized by a membership mapping function

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Therefore, for all x G X, A(x) indicates the certainty to which element x belongs to fuzzy set A. For two-valued sets, HA(Z) is either 0 or 1. Fuzzy sets can be defined for discrete (finite) or continuous (infinite) domains. The notation used to denote fuzzy sets differ based on the type of domain over which that set is defined. In the case of a discrete domain X, the fuzzy set can either be expressed in the form of an n-dimensional vector or using the sum notation. If X = {x1, x2, xn} then, using vector notation, A = {(nA(xi)/Xi)|xi Using sum notation,

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X,i = l , - - - , n }

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A = HA(x1I)/x1 + pA(x2)/x2 +

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1- p,A(xn}/xn =

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where the sum should not be confused with algebraic summation. The use of sum notation above simply serves as an indication that A is a set of ordered pairs. A continuous fuzzy set, A, is denoted as

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A= I n(x}/x Jx

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Again, the integral notation should not be algebraically interpreted.

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Membership Functions

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The membership function is the essence of fuzzy sets. A membership function, also referred to as the characteristic function of the fuzzy set, defines the fuzzy set. The function is used to associate a degree of membership of each of the elements of the domain to the corresponding fuzzy set. Two-valued sets are also characterized by

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