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Statistical Methods for Fuzzy Data by Reinhard Viertl

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Chapter 3

a. The δ-cuts of trapezoidal fuzzy numbers x* = t*(m,s,l,r) have the following form: Therefore, by applying the minimum t-norm, the δ-cuts of * are the Cartesian products Cδ(x1*) × Cδ(x2*).

By Cδ(x1*) = [1 + δ; 3 − δ] and Cδ(x2*) = [1 + δ; 5 − δ] we obtain

b. Denoting the δ-cuts of x1* and x2* by Cδ(x1*) = [aδ,1;bδ,1] and Cδ(x2*) = [aδ,2;bδ,2] for all δ ∈ (0; 1] the following cases have to be considered:

1. If all δ-cuts of both fuzzy intervals are subsets of +, the δ-cut of x1* x2* is given by

Unnumbered Display Equation

2. If the δ-cut of one fuzzy interval is a subset of +, for example Cδ(x1*) = [aδ,1;bδ,1] with aδ,1 ≥ 0, and the δ-cut of x2* is a subset of (− ∞; 0], Cδ(x2*) = [aδ,2;bδ,2] with bδ,2 ≤ 0, then the δ-cut of x1* x2* is given by

3. If both δ-cuts of x1* and x2* are subsets of (− ∞ ; 0] with ...

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