Discussion
If the fuzzy relations \(\underset{\sim}{R}\) and \(\underset{\sim}{S}\) are defined as follows :
\[ \underset{\sim}{R} = \begin{bmatrix} 0.7 & 0.5 \\ 0.8 & 0.4 \end{bmatrix} \]
and
\[ \underset{\sim}{S} = \begin{bmatrix} 0.9 & 0.6 & 0.2 \\ 0.1 & 0.7 & 0.5 \end{bmatrix} \]
then using max-min composition T = RoS is represented as :
Answer: A
The composition is the first matrix, with rows (0.7, 0.6, 0.5) and (0.8, 0.6, 0.4).
In max-min composition, \(T(x, z) = \max_y \min(R(x, y), S(y, z))\): take the minimum of each pair along a row of R and a column of S, then the maximum of those minimums.
- Row 1: max(min(0.7, 0.9), min(0.5, 0.1)) = 0.7; max(0.6, 0.5) = 0.6; max(0.2, 0.5) = 0.5
- Row 2: max(min(0.8, 0.9), min(0.4, 0.1)) = 0.8; max(0.6, 0.4) = 0.6; max(0.2, 0.4) = 0.4
The second option simply repeats S, and the last two begin with 0.2 where the composition gives 0.7, so they do not follow the max-min rule.
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