By Janusz T. Starczewski
This e-book generalizes fuzzy common sense structures for various sorts of uncertainty, together with - semantic ambiguity as a result of restricted notion or lack of awareness approximately specified club services - loss of attributes or granularity coming up from discretization of genuine information - vague description of club services - vagueness perceived as fuzzification of conditional attributes. for that reason, the club uncertainty should be modeled by way of combining tools of traditional and type-2 fuzzy good judgment, tough set thought and danger idea. specifically, this publication offers a couple of formulae for imposing the operation prolonged on fuzzy-valued fuzzy units and offers a few easy constructions of generalized doubtful fuzzy common sense platforms, in addition to introduces a number of of tips on how to generate fuzzy club uncertainty. it's fascinating as a reference publication for under-graduates in better schooling, grasp and healthcare professional graduates within the classes of desktop technological know-how, computational intelligence, or fuzzy keep an eye on and category, and is mainly devoted to researchers and practitioners in undefined.
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Extra info for Advanced Concepts in Fuzzy Logic and Systems with Membership Uncertainty
On the contrary, the concept of fuzzy sets introduces an intrinsic vagueness of an object, as for example we cannot describe a yellow color in therms of its wavelength. Some colors may be even ambivalent. Therefore roughness is the problem connected with granularity of X, while fuzziness is the problem of ill-deﬁnition of the object itself. The following properties are satisﬁed by rough sets. 1. 2. 3. 4. 5. R (X) ⊆ X ⊆ R (X), R (∅) = R (∅) = ∅ and R (X) = R (X) = X, R (X ∩ Y ) = R (X) ∩ R (Y ) and R (X ∪ Y ) = R (X) ∪ R (Y ), X ⊆ Y implies that R (X) ⊆ R (Y ) and R (X) ⊆ R (Y ), R (X − X) = X−R (X) and R (X − X) = X−R (X).
1) where φ−1 (w) is the inverse image of w. The extension principle in this ﬁrst formulation rely on the use of minimum in aggregation of memberships. However any t-norm, T∗ , in the place of the minimum has diﬀerent properties, as for example an interactivity distinguishes the product t-norm from the minimum [Dubois and Prade 1980]. In this book we mostly look into the extension principle in its more general form [Nguyen 1978]. According to the extension principle, the extended operation is expressed as the least upper bound of the Cartesian product with the non-extended operation as a constraint.
34). 1) to the minimum, the maximum and the continuous, strictly monotone and involutive complement neg(u) ≡ 1 − u. 13) ˜ is characterized where A˜ is characterized by its membership grades Fx , and B by Gx , ∀Fx , Gx ∈ F ([0, 1]), x ∈ R. 2) to model the alternative operations on fuzzy-valued fuzzy sets. 3. Let F and G be fuzzy truth values, with their membership functions f and g, respectively, at x ∈ R, where for simplicity x is omitted, and let T and T∗ be arbitrary t-norms. 15) S(u,v)=w and an extended complementation N is characterized by μN (F ) (w) = sup f (u) .
Advanced Concepts in Fuzzy Logic and Systems with Membership Uncertainty by Janusz T. Starczewski