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Statistical Methods for Fuzzy Data by Reinhard Viertl

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Chapter 2

a.

b.

c.

d. By Proposition 2.1 the δ-cuts of the combined fuzzy vector are the Cartesian products of the δ-cuts of the fuzzy numbers ξi(·), i = 1(1)n. Since the δ-cuts of the fuzzy numbers are finite unions of closed intervals, also the Cartesian products are finite unions of so-called rectangles which are closed sets.

e. The δ-cuts of fuzzy intervals are compact intervals of real numbers. Application of the minimum t-norm yields as δ-cuts of the fuzzy combined vector the Cartesian products of the mentioned compact intervals. Therefore the δ-cuts of the combined fuzzy vector are k-dimensional compact intervals.

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