Discussion
Let X = [x1, x2, x3], Y = [y1, y2] and z = [z1, z2, z3]. Let R be fuzzy relation
| y1 | y2 | |
|---|---|---|
| x1 | 0.5 | 0.1 |
| x2 | 0.2 | 0.9 |
| x3 | 0.8 | 0.6 |
and S be a fuzzy relation
| z1 | z2 | z3 | |
|---|---|---|---|
| y1 | 0.6 | 0.4 | 0.7 |
| y2 | 0.5 | 0.8 | 0.9 |
then RoS, by max-min composition yields (rows x1, x2, x3; columns z1, z2, z3) :
Answer: C
The max-min composition gives [0.5 0.4 0.5; 0.5 0.8 0.9; 0.6 0.6 0.7].
Each entry is \(\max_{k} \min(R(x_{i}, y_{k}), S(y_{k}, z_{j}))\). For row x1 = (0.5, 0.1):
\[ \begin{aligned} z_{1} &: \max(\min(0.5, 0.6), \min(0.1, 0.5)) = 0.5 \\ z_{2} &: \max(\min(0.5, 0.4), \min(0.1, 0.8)) = 0.4 \\ z_{3} &: \max(\min(0.5, 0.7), \min(0.1, 0.9)) = 0.5 \end{aligned} \]
The same steps give 0.5, 0.8, 0.9 for x2 = (0.2, 0.9) and 0.6, 0.6, 0.7 for x3 = (0.8, 0.6). Option A already fails at x1z2 (0.5 instead of 0.4); option B takes maxima without the min step, and option D takes minima throughout.
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