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Statistical Methods for Fuzzy Data by Reinhard Viertl

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3.2 Addition of fuzzy numbers

Let x1* and x2* be two fuzzy numbers with corresponding characterizing functions ξ1(·) and ξ2(·). The generalized addition operation ⊕ for fuzzy numbers has to obey two demands: First it has to generalize the addition of real numbers, and secondly it has to generalize interval arithmetic.

For fuzzy intervals x1* and x2* the generalized addition can be defined using δ-cuts.

Let Cδ(x1*) = [aδ,1;bδ,1] and Cδ(x2*) = [aδ,2;bδ,2] ∀δ ∈ [0; 1] then the δ-cut of the fuzzy sum x1*x2* is given by

Unnumbered Display Equation

The characterizing function of x1*x2* is obtained by Lemma 2.1.

Remark 3.3:

The same result is obtained if the generalized sum is defined via the extension principle, applying the function +, defined on the Cartesian product × , where x1* and x2* are combined into a two-dimensional fuzzy interval by the minimum t-norm. This means

Unnumbered Display Equation

An example is given in Figure 3.2.

Figure 3.2 Addition of fuzzy intervals.

c03f002.eps

Remark 3.4:

The sum of two fuzzy intervals is again a fuzzy interval. ...

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