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, multiplications and divisions, can evaluate the function to required accuracy. Such functions can be extended, using αcuts and interval arithmetic, to fuzzy functions. 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 αcuts 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)  


∗. 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 