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Statistical Methods for Fuzzy Data by Reinhard Viertl

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Chapter 9

a. If all δ-cuts of ψ (·) are finite unions of compact intervals then the convex hull of ψ (·) is ψ (·) itself. If some δ-cuts of ψ (·) are not compact or not finite unions of compact intervals then the function ξ(·) is defined by

Unnumbered Display Equation

A concrete example is the following:

The -cut of ψ (·) is (a; b] which is not closed. Therefore the convex hull ξ(·) is the following

b. From β < δ it follows Cβ(x*) ⊇ Cδ(x*) by definition of δ-cuts. Therefore we obtain

Unnumbered Display Equation

In order to prove the equality of both sets assume xCβ(x*) for all β > δ. Then ξx*(x) ≥ β for all β < δ. Assuming ξx*(x) < δ yields a contradiction. Therefore and the equality follows.

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