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# Consider the fuzzy functionfor triangular fuzzy numbers

16 CHAPTER 2. FUZZY SETS

Equation (2.23) defines the membership function of Z for any triangular (shaped) fuzzy number X in [a, b].

z1(α) = min{ h(x) | x ∈ X[α] }, max{ h(x) | x ∈ X[α] }, (2.24)
= (2.25)
z2(α)
Z(z) = sup

� �

X(x), Y (y)

| h(x, y) = z
(2.26)
x,y
z1(α) = min{ h(x, y) | x ∈ X[α], y ∈ Y [α] }, max{ h(x, y) | x ∈ X[α], y ∈ Y [α] }, (2.27)
= (2.28)
z2(α)

All the functions we usually use in engineering and science have a computer algorithm which, using a finite number of additions, subtractions, multipli-cations and divisions, can evaluate the function to required accuracy. Such functions can be extended, using α-cuts and interval arithmetic, to fuzzy func-tions. Let h : [a, b] IR be such a function. Then its extension H(X) = Z, X in [a, b] is done, via interval arithmetic, in computing h(X[α]) = Z[α], α in [0, 1]. We input the interval X[α], perform the arithmetic operations needed to evaluate h on this interval, and obtain the interval Z[α]. Then put these α-cuts together to obtain the value Z. The extension to more independent variables is straightforward.

For example, consider the fuzzy function

Z = H(X) =A X + B C X + D, (2.29)

Z = H(X) = 2X + 10 3X + 4, (2.31)

would be the extension of

h(x) = 2x + 10 3x + 4. (2.32)

We know that Z can be different from Z

. But for basic fuzzy arithmetic

in Section 2.3 the two methods give the same results. In the example below we show that for h(x) = x(1 − x), x in [0, 1], we can get Z X in [0, 1]. What is known ([3],[7]) is that for usual functions in science and∗ ̸= Z for some

Z = (1 − X) X, (2.33)

for X a triangular fuzzy number in [0, 1]. Let X[α] = [x1(α), x2(α)]. Using interval arithmetic we obtain

z1(α) =
(2.34)
= (2.35)
z2(α)
Z (z) = sup X(x)|(1 − x)x = z, 0 ≤ x ≤ 1�
(2.36)
x
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