By Hans Bandemer, Wolfgang Näther

Fuzzy facts reminiscent of marks, rankings, verbal reviews, obscure observations, specialists' evaluations and gray tone images, are very common. In *Fuzzy info Analysis* the authors acquire their fresh effects supplying the reader with principles, techniques and techniques for processing such info whilst searching for sub-structures in wisdom bases for an review of sensible courting, e.g. on the way to specify diagnostic or keep watch over structures. The modelling provided makes use of rules from fuzzy set idea and the advised equipment resolve difficulties often tackled via information research if the information 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 software who've a few historical past in arithmetic and statistics.

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E. 114), I(a,b) is again a fuzzy number. If an L-R-type fuzzifying function, Le. 2. 7 Measuring the uncertainty Following Chapter 1, the uncertainty of data has many sources. If we stress impreciseness and vagueness then it is reasonable to model the data by fuzzy sets. 1) which give an answer to the question: How far is a given fuzzy set from well-defined crisp reference sets? If we stress variability and ignorance then it is reasonable to assume that the data live in an environment furnished with a fuzzy measure.

This is very simple if I is a one-to-one mapping, Le. if y E V is the image of a unique x E U. Then we simply take 1nB(Y) = 1nAU- 1 (y)) = 1nA(X). 56) Somewhat more care is necessary if I is not one-to-one, Le. if y E V may have more than one original. Then Zadeh proposes (see Fig. 57) 26 CHAPTER 2. 2) that y is an image by f. Note that a fuzzy singleton means a fuzzy set with one-point-support. g. 57) is widely accepted in fuzzy set theory. 57) more explicitely for an n-ary function f, Le. • X Un and the argument A is a Cartesian product A = A l x ...

98) 36 CHAPTER 2. 91) also can be written as Vx,yEU: If x#ythen mR(x,y)=Oor mR(y,x)=O. 11 R is called perfectly antisymmetric, iff Vx,yEU: If x # yand mR(x,y) > othen mR(y,x) = O. 101) Note that perfect antisymmetry implies antisymmetry. e. the strength of the relation between x and z must be greater than the strength of any indirect way x, y, z. 5. 102) Of specia.! g. 31). Let us introduce now some special fuzzy relations. nd require: i) "n object should be similar to itself. t. A and B be two objects.