Question

9. For a fuzzy system with double inputs and single output, x and y are the inputs, z is the output. Assume that the elements of the inputs and output in fuzzy domains are X-fa1,a2,a3), Y={b1,b2,b3}, Z-(c1,c2,c3}, respectively. The relation between inputs and output can be described by the following fuzzy rules: Ifx is A1 and y is B1, then z is C1, where A1 and C1 B2 0.7 0.5 0.2 + a3 0.3 0.4 0.9 0.6 0.8 0.1 b1 b2 b3' a2 c1 and y is B2 06+ 0.6 0.4 0.3 0.6 0.8 0.6 + are known, b3 Now the inputs thatx is A2 = + a2 a3 b1 b2 a1 find the fuzzy subset C2 for output z by the fuzzy Cartesian product. (15 scores)

Answer 1

Let the Cartesian product be  
Ax B
. Then the fuzzy Cartesian product is given by
LAx B
(X , Y )  = Min (
LAX), B(Y)
)

A2(X
={
X1, 0.6), (X2, 0.4), (X3, 0.3)
} , where X = { a1 , a2 ,a3 } . So ,
A2(X
= {
a1,0.6), (a2, 0.4), (a3, 0.3)
}

B2Y
= {
(Y1,0.6), (Y2,0.8), (Y3,0.6)
} , where Y = { b1 , b2 , b3 } . So ,
B2(Y(b1,0.6), (b2, 0.8), (b3, 0.6)
}   

LA2x B
=   
0.6 0.6 0.6 0.4 0.4 0.4 0.3 0.3 0.3
. =
C2(Z)