By Maciej Wygralak

Counting belongs to the main undemanding and common psychological actions of humans. Its effects are a foundation for coming to a choice in loads of events and dimensions of our lifestyles. This e-book offers a singular method of the complex and complex case, referred to as clever counting, within which the gadgets of counting are imprecisely, fuzzily designated.

Formally, this collapses to counting in fuzzy units, interval-valued fuzzy units or I-fuzzy units (Atanassov's intuitionistic fuzzy sets).

The monograph is the 1st one exhibiting and emphasizing that the awarded equipment of clever counting are human-consistent: are reflections and formalizations of actual, human counting methods played less than imprecision and, in all likelihood, incompleteness of knowledge. different purposes of clever counting in a variety of parts of clever platforms and choice help can be mentioned, too.

The entire presentation is self-contained, systematic, and outfitted with many examples, figures and tables. laptop and data scientists, researchers, engineers and practitioners, utilized mathematicians, and postgraduate scholars attracted to details imprecision are the objective readers.

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Extra resources for Intelligent Counting Under Information Imprecision: Applications to Intelligent Systems and Decision Support

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1), a sŁ b = 1 v ( a + b), • (algebraic or product t-norm) (algebraic or product t-conorm) (Łukasiewicz t-norm) (Łukasiewicz t-conorm) a td b = (drastic t-norm) a sd b = (drastic t-conorm) Fig. 70) where a, b 0 [0, 1]. e. v is a unique idempotent t-norm, while w forms a unique idempotent t-conorm. 72) [œa, b, c 0 [0, 1]: a s ( b t c) = ( a s b) t ( a s c)] ] t = v. A t-norm is thus distributive with respect to a t-conorm only if that t-conorm is w and, on the other hand, a t-conorm is distributive with respect to a t-norm t only if t = v.

5 = νŁ, ν= c 6 ν( for c 6 0, ν= c 6 ν* for c 6 1. 48 2 Basic Notions of the Language of Fuzzy Sets Fig. 4. The reader is also referred to GOTTWALD (2001) and FODOR/YAGER (2000) for more details about and further references to negations. 2 Complements Based on Negations For a fuzzy set A: U 6[0, 1] and a negation ν, the complement Aν of A induced by ν is a fuzzy set defined as Aν( x) = ν( A( x)) for each x 0 U. The standard complement AN is thus that induced by the standard negation νŁ. 8 is given below (cf.

75) for t-norms t1 and t2. 4). If t and s are νŁ-dual, one says that they are associated, and one writes s = t* or, equivalently, t = s*. Thus a t* b = 1 ! (1 ! a) t (1 ! b) and a s* b = 1 ! (1 ! a) s (1 ! b). 76) v and w, ta and sa, tŁ and sŁ as well as td and sd are all examples of a t-norm and the associated t-conorm. Further instances are given below. 6. Selected parameterized families of t-norms and associated t-conorms: • a tS, p b = [0 w ( a p + b p ! 1)]1/p, (Schweizer t-norms) p p 1/p a sS, p b = 1!

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