By Janusz T. Starczewski
This publication generalizes fuzzy good judgment structures for various forms of uncertainty, together with - semantic ambiguity as a result of restricted belief or lack of know-how approximately certain club capabilities - loss of attributes or granularity bobbing up from discretization of actual information - obscure description of club features - vagueness perceived as fuzzification of conditional attributes. for this reason, the club uncertainty may be modeled via combining equipment of traditional and type-2 fuzzy common sense, tough set idea and risk conception. specifically, this publication presents a couple of formulae for imposing the operation prolonged on fuzzy-valued fuzzy units and provides a few simple buildings of generalized doubtful fuzzy good judgment structures, in addition to introduces numerous of easy methods to generate fuzzy club uncertainty. it truly is fascinating as a reference booklet for under-graduates in larger schooling, grasp and healthcare professional graduates within the classes of computing device technological know-how, computational intelligence, or fuzzy regulate and category, and is principally devoted to researchers and practitioners in undefined.
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Additional info for Advanced Concepts in Fuzzy Logic and Systems with Membership Uncertainty
T. Starczewski: Advanced Concepts in Fuzzy Logic and Systems, STUDFUZZ 284, pp. 33–76. com 34 2 Algebraic Operations on Fuzzy Valued Fuzzy Sets “seem very promising and are continuing” [Mendel 2007]. Operations on triangular [Starczewski and Rutkowski 2002; Starczewski 2009a] and Gaussian type-2 fuzzy sets [Starczewski 2005] have been just the ﬁrst steps we made in this direction. Therefore, it is necessary to supply analytical formulae for extensions of t-norms and t-conorms. This problem is studied and solved in this chapter.
Consider two fuzzy truth intervals with triangular membership F L mF +ΔF R −u and g (v) = , functions deﬁned as, f (u) = min u−mΔFF+Δ ΔF R L min v−mG +ΔGL mG +ΔGR −v , ΔGL ΔGR . 36) w (μ) = max (0, − (ΔF R +ΔGR ) μ + mF + mG + ΔF R +ΔGR − 1) . 37) Consequently, the extended Lukasiewicz t-norm based on the minimum is characterized by the two following cases presented in Fig. 5. 5 Fig. 38) otherwise μT˜L min (F,G) (w) = 1 w=0 mF +mG −1+ΔF R +ΔGR −w ΔF R +ΔGR elsewhere. 38) remains a triangular membership function.
E. T˜ min (F, G) . The use the upper μ pseudo-inverse ends the proof, since both the non-decreasing function w and the non-increasing function w are left-continuous. A detailed graphical explanation of the procedure described by this theorem can be found in Fig. 4. 2 to continuous triangular and trapezoidal fuzzy truth intervals. The extension of the Lukasiewicz tnorm partially leads to unexpected results. 2. Consider two fuzzy truth intervals with triangular membership F L mF +ΔF R −u and g (v) = , functions deﬁned as, f (u) = min u−mΔFF+Δ ΔF R L min v−mG +ΔGL mG +ΔGR −v , ΔGL ΔGR .