Two fuzzy relations are given by,

Obtain fuzzy relation T as a composition between the fuzzy relation.

Solution: The composition between two fuzzy relations is obtained by,

[a] Max – min composition.

[b] Max-product composition.

[a] Max – min composition.

$M_T (x_1 , z_1) = \ max \ [ min \ [ M_R (x_1, y_1) , M_S (y_1, z_1)]$

$min \ [M_R (x_1, y_2) , M_S (y_2, z_1)]$

= max [ min (0.6, 1), min (0.3,0.8)]

= max [0.6, 0.3]

= 0.6

$M_T (x_1 , z_2) = max \ [min \ [M_R (x_1, y_1), M_S (y_1, z_2)]$

$min \ [M_R (x_1, y_2), Ms (y_2, z_2)]$

$= max \ [min \ (0.6, 0.5), min \ (0.3, 0.4)]$

= max (0.5, 0.3) = 0.5

$M_T (x_1, z_3) = max \ [min \ (0.6, 0.3), min \ (0.3, 0.7)]$

= max [0.3, 0.3] = 0.3

$M_T (x_2, z_1) = max \ [min \ (0.2, 1) , min \ (0.9, 0.8)]$

= max [ 0.2, 0.8] = 0.8

$M_T (x_2 , z_2) = max \ [min \ (0.2, 0.5) , min \ (0.3, 0.4)]$

= max [0.2, 0.4] = 0.4

$M_T (x-2 , z_3) = max \ [min \ (0.2, 0.3) , min \ (0.9, 0.7)]$

= max (0.2, 0.7) = 0.7

$\therefore$ T = RoS = [0.6 0.5 0.3

0.8 0.4 0.7]

[b] Max product composition.

T = R . S

$M_T (x_1, z_1) \ = \ max \ [ M_R (x_1, y_1) . M_s (y_1, z_1)]$

$M_R (x_1, y_2). M_S (y_2, z_1)]$

= max (0.6, 0.24) = 0.6

$M_T (x_1 , z_2) = max \ [M_R (x_1, y_1) . Ms (y_1, z_2)]$

$[M_R (x_1, y_2). M_s (y_2 , z_2)]$

= max [0.3, 0.12] = 0.3

$M_T (x_1 , z_3) = max \ [0.18, 0.21] = 0.21$

$M_T (x_2, z_1) = max \ [0.2, 0.72] = 0.72$

$M_T (x_2, z_2) = max \ [0.10, 0.36] = 0.36$

$M_T (x_2, z_3) = max \ [0.06, 0.63] = 0.63$

$T = R . S = \begin{bmatrix}0.6 & 0.3 \\ 0.21 & 0.72 \\ 0.36 & 0.63 \\\end{bmatrix}$