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Statistical Methods for Fuzzy Data by Reinhard Viertl

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2.4 Problems

a. Draw a graph of a characterizing function and construct five δ-cuts for this characterizing function, with δ = 0.2, 0.4, 0.5, 0.7 and 0.9.

b. Let the color intensity for a one-dimensional quantity be given by the following function:

Find the characterizing function of the fuzzy boundary between low color intensity and maximal color intensity using the method described in Section 2.1.

c. Calculate the δ-cuts of a fuzzy vector with vector-characterizing function

Unnumbered Display Equation

d. Explain that the combination of fuzzy numbers, based on the minimum t-norm, yields a fuzzy vector.

e. Prove the following: The combination of k fuzzy intervals, based on the minimum t-norm, yields a fuzzy k-dimensional interval.

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