Consider the following two fuzzy sets:

A = { $\frac{0.3}{x_1} + \frac{0.7}{x_2} + \frac{1}{x_3}$}

B = { $\frac{0.4}{y_1} + \frac{0.9}{y_2}$}

Perform the Cartesian product over three given fuzzy sets.

Solution: The fuzzy Cartesian product performed over fuzzy sets A and B results in fuzzy relation R given by,

R = A X B

The calculation of R is as follows:

$M_R (x_1 y_1 ) = min \ [ M_A (x_1) , M_B (y_1)]$

= min [0.3 , 0.4] = 0.3

$M_R (x_1 y_2) = min \ [ M_A (x_1), M_B (y_2)]$

= [0.3 , 0.9] = 0.3

$M_R (x_2 , y_1) min \ [M_A (x_2) , M_B (y_1)]$

= min [0.7, 0.4] = 0.4

$M_R (x_2, y_2) = min \ [ M_A (x_2) , M_B (y_2)]$

= min [0.7, 0.9]

= 0.7

$M_R (x_3 , y_1) = min \ [M_A (x_3), M_B (y_1)]$

= min [1, 0.4]

= 0.4

$M_R (x_3, y_2) = min \ [M_A (x_3), M_B (y_2)]$

= min [1, 0.9]

= 0.9

$R = \begin{bmatrix}0.3 & 0.3 \\ 0.4 & 0.7 \\ 0.4 & 0.9 \\\end{bmatrix}$