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Statistical Methods for Fuzzy Data by Reinhard Viertl

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23.3 Generalized Hukuhara difference

The generalized difference operation for fuzzy numbers via the extension principle increases the fuzziness. For symmetric fuzzy numbers x* in LR-form with L(·) = R(·), l = r and m = 0 we obtain x* x* = x* ⊕ ( − x*) ≠ 0. In general for given x*y* and given x* the vector y* cannot be calculated by (x*y*) x*. Moreover the difference does not correspond to the difference in the functional space L2 (Sn − 1 × (0, 1]), i.e. the support function of y* x* is not equal to sy*sx*.

In order to find a difference which corresponds more to the usual difference of numbers the following argument is helpful: The difference xy of two real numbers x and y is the real number z for which y + z = x. This results in the so-called Hukuhara difference.

Definition 23.2:

For x*2(n) and y*2(n) the so-called Hukuhara difference y* Hx* is – if it exists – the solution ...

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