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A first kind of fuzzy function arises, if ordinary functions just "carry" the fuzziness of their arguments without generating extra fuzziness themselves. 57). 45)). )L (see Fig. 13) A second kind of fuzzy function can be described as uncertain functions of nonfuzzy arguments. Uncertainty should be understood here as introduced in Chapter 1, but restricted, first of all, to the two sources impreciseness and vagueness. e. we exclude in this introduction the consideration of random processes. 6. 13: Fuzzy extension of a nonfuzzy straight line map a crisp point x E U into a fuzzy set B = f(x) E F(V), where F(V) denotes the set of all fuzzy sets on the universe V, Le.

E. 50) where B is a positive definite (k x k)-matrix. Note, that all a-cuts of the "bean" are k-dimensional ellipsoids (see Fig. 5). A further example for a fuzzy vector is the Cartesian product of fuzzy numbers. g. the fuzzy set A = Ak, Ai i=l, ... •. 51) is a fuzzy vector with the central point M = (M}, ... , Mk)T, and all a-cuts of A are k-dimensional rectangles (see Fig. 6). Sometimes, from a high-dimensional fuzzy set A, we are only interested in a lower-dimensional "shadow". 3. 6: Cartesian product of fuzzy numbers 24 CHAPTER 2.

F. t. p. More illustrative: F(p) contains all "pieces" of knowledge on the uncertainty behaviour on U. Note that any B E F(p) carries only that part of probability (or evidence) which is not carried by subsets of B. Formally, this is ensured by the fact that p is a measure on B( U) and not on U, Le. for different A, B E F(p) we have {A} n {B} = 0 while we may have A n B i= 0. Thus, the focal subsets need not be disjoint, nor form a covering of U. Note further that the case of total ignorance is represented by F(p) = {U}, Le.

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