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By Hans Bandemer, Wolfgang Näther

Fuzzy information equivalent to marks, rankings, verbal reviews, obscure observations, specialists' critiques and gray tone photographs, are very common. In Fuzzy facts Analysis the authors acquire their fresh effects delivering the reader with rules, ways and strategies for processing such info whilst searching for sub-structures in wisdom bases for an assessment of practical dating, e.g. with a purpose to specify diagnostic or keep an eye on structures. The modelling awarded makes use of rules from fuzzy set concept and the urged equipment resolve difficulties frequently tackled by means of info research if the knowledge are genuine numbers.
Fuzzy facts Analysis is self-contained and is addressed to mathematicians orientated in the direction of purposes and to practitioners in any box of program who've a few history in arithmetic and data.

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This extreme case is called dissonant. e. • , B s } with B 1 C B 2 C ... CBs. Then it can be shown that PI (A) = Poss (A), Bel (A) = Nee (A) . 144) Hence we can say that a probability is a dissonant plausibility, and a possibility is a consonant plausibility. The general case, however, is that a body of evidence has more or less overlapping pieces of evidence. The more the degree of overlapping, the more the" distance" from probability. According to the probability assignment p the subsets of U can be considered as random sets.

A second type of fuzziness measure is the so-called energy measure e (see DE LUCA/TERMINI 1972) which evaluates deviations from the empty set, Le. A o = 0 and e(0) = O. e. A ~ B), then Due to the inclusion-monotonicity of e, in general, also crisp sets A :I 0 have positive energy e(A). 11)). 123». A third possibility for specifying impreciseness or vagueness consists in measuring deviations from the "one-point-sharpness", Le. measuring deviation from the type of (ordinary) one-point sets. This leads to so-called measures of nonspecifity.

4. 66) dHs(A, B) = sup dH(A Ot , BOt) , OtE[O,I] which are indeed metrics if some further (technical) assumptions are satisfied (see PURIjRALEsCU 1986). 5. 6. 58) extends given "point" functions f by extending domain and image space (both from "points" to fuzzy sets). 63) extends a given set function only by extending the domain (from sets to fuzzy sets). 2 Extended real operations An important field of application for the extension principle is given by algebraic operations such as addition, multiplication, minimum, maximum.

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