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If w ∈ (nF , nG ], instead of f we obtain g. If w ∈ (nG , 1], T∗ is maximal when u = v = w since both f and g are non-increasing. 17), the rest of the proof follows. 8 1 Fig. 3 Extended minimum t-norm based on the Lukasiewicz t-norm. 1. 2 where /x/ stands for max (0, min (1, x)). The objective is to ﬁnd an analytical formula for the extended minimum t-norm based on the Lukasiewicz t-norm TL (a, b) = /a + b − 1/. We calculate mF = 3, nF = 4, and mG = nG = 5. We do not have to change the order of arguments, since nF nG .

Let two Gaussian fuzzy truth numbers be given by their membership functions f (u) = exp − 21 g (v) = exp − 12 v−mG σG u−mF σF 2 and 2 . 51) (w) = max ⎜ (F,G) 2 ⎠ ⎝ P TD F mG exp − 12 w−m mF σG = exp − 1 2 w−mF mG max(mG σF ,mF σG ) 2 . 53) 2 if w ∈ [0, min (mF , mG )] if w ∈ (mF , mG ] if w ∈ (mG , mF ] otherwise. As it can be seen ‘in Fig. 7, the extended product based on the weakest tnorm preserves the Gaussian shape on [0, min(mF , mG )] and on[min(mF , mG ), max (mF , mG )], separately. Therefore, some approximation of this result can be applied to adaptive network fuzzy inference systems with small computational costs.

Possibility Theory. : Resolution principles in possibilistic logic. : Rough fuzzy sets and fuzzy rough sets. : Fuzzy sets in approximate reasoning, part 1: inference with possibility distributions. : Interval-valued fuzzy sets, possibility theory and imprecise probability. In: Proceedings of International Conference in Fuzzy Logic and Technology, pp. : Decision-theoretic foundations of qualitative possibility theory. : An information-vased discussion of vagueness. , Lefebvre, C. ) Handbook of Categorization in Cognitive Science, ch.