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Statistical Methods for Fuzzy Data by Reinhard Viertl

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3.7 Problems

a. Calculate the δ-cuts of the fuzzy vector * which is obtained by applying the minimum t-norm to the two trapezoidal fuzzy numbers x1* = t*(2, 0, 1, 1) and x2* = t*(3, 1, 1, 1).

b. Prove the formula for the δ-cuts of x1* x2* in Remark 3.3 if x1* and x2* are fuzzy intervals.

c. Two fuzzy intervals x1* and x2* are given by their characterizing functions ξ1(·) and ξ2(·):

Calculate and draw the characterizing function of x1*x2*.

d. Calculate and draw the characterizing function of the product x1* x2* for x1* and x2* in (c).

e. Explain that the generalized integration of fuzzy valued functions reduces to the classical integration for classical real valued functions.

f. Calculate the fuzzy integral for a fuzzy constant function f*(x) ≡ y*.

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