Imagine you're sorting your clothes. Some are definitely shirts, others are clearly pants, but what about items like t-shirts with long sleeves? Are they shirts or light jackets? This is where fuzzy sets come in handy. Unlike traditional sets where an item is either in or out (like being either a shirt or a pant), fuzzy sets allow for degrees of belonging.In a fuzzy set, each item has a membership score between 0 and 1. A score close to 1 means the item strongly belongs to the set, while a score near 0 means it hardly belongs. For example, a regular t-shirt might have a high score in the \'shirt\' category, but a long-sleeve t-shirt might have a moderate score in both \'shirt\' and \'light jacket\' categories.Fuzzy sets are useful because they reflect how we naturally categorize things in the real world – not always black and white, but often in shades of gray. They extend traditional set concepts like union (combining sets), intersection (what's common between sets), and others, to accommodate this gradation. This is especially helpful in complex decision-making scenarios where strict \'yes or no\' categories don't quite capture the nuances of real-life situations.
Consider now a converse problem in which A is a given fuzzy set in X, and T, as before, is a mapping from X to Y. The question is: What. is the membership function for the fuzzy set B in Y which is induced by this mapping? If T is not one-one, then an ambiguity arises when two or more distinct points in X, say xl and z2, with different grades of membership in A, are mapped into the same poirtt y in Y. In this case, the question is: What grade of membership in B should be assigned to y? To resolve this ambiguity, we agree to assign the larger of the two grades of membership to y. More generally, the membership function for B will be defined by .f~(y) = Max~r-~(~)fA(x), y C Y where T-~(y) is the set of points in X which are mapped into y by T.
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V. CONVEXITY As will be seen in the sequel, the notion of convexity can readily be extended to fuzzy sets in such a way as to preserve many of the properties which it has in the context of ordinary sets. This notion appears to be particularly useful in applications involving pattern classification, optimization and related problems. (22) (23) FUZZY SETS ~ fA(x) onvex fuzzy set vnon-convex fuzzy set ., . \p...IfAXx,+('-X)xa x I x 2 x FIG. 4. Convex and noneonvex fuzzy sets in E ~ hi what follows, we assume for concreteness t h a t X is a real Euclidean space E ~. DEFINITIONS Convexity. A fuzzy set A is convex if and only if the sets I~, defined by r . = {x lyA(x) => ~} are convex for all a in the interval (0, 1. An alternative and more direct definition of convexity is the fo!towingS: A is convex if and only if
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~Xxl + (1 X)x2 > Min ~'.~(xl), fA(x2) for all xl and x2 in X and all X in 0, l. Note t h a t this definition does not imply t h a t f ~ ( x ) must be a convex function of x. This is illustrated in Fig. 4 for n = 1. To show the equivalence between the above definitions note that if A is convex in the sense of the first definition and c~ -f ~ ( x l ) < f~(x~), (1 X)x2 ~ iP~ b y the convexity of F~. Hence then x2 ~ r , and Xxl + .hXx~ + (1 -h)x2 > a = JS(x~) = Min fA(x~),j~(x~). Conversely, if A is convex in the sense of the second definition and = f~ (x~), then 17~ m a y be regarded as the set of all points x~ for which form (1 X ) x 2 , 0 < X _-< 1, is also in r~ and hence r~ is a convex f~(x~) > Xxl + set. Q.E.D. f~(x~). I n virtue of (25), every point of the A basic p r o p e r t y of convex fuzzy sets is expressed b y the THEOreM. I f A and B are convex, so is the# intersection. This way of expressing convexity was suggested to the writer by his colleague,
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E. Berlekamp. 347 (24) (25) 3 4 8 ZADEtt Proof: Let C = A fl B. Then fcXxl t(1 X)x2 = Min fAiXx~ ~(1 X)x2,f,hxl ~ (1 k)x~. Now, since A and B are convex f~Xxl + (1 -X)x2 > lVIin L(x~),L(x2) f , Dtxl-~(1 X)x~ ~ ~ i n ~B(Xl),f,(x2) and hence fc~.Xl -~(1 -X)X2 > :~Vlin Min fA(z~), fA(x~), Min f.(x~), f.(z~) or equivalently fcxx~ + (I x)x~ > Min Min fA(x~), f,(x~), Min IrA(x2), f,(x2)
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