Discussion
If
\[ \begin{aligned} \underset{\sim}{P} = \Big\{ & \frac{0.1}{C_{1}} + \frac{0.5}{C_{2}} \\ & + \frac{10}{C_{3}} \Big\} \end{aligned} \]
and
\[ \begin{aligned} \underset{\sim}{S} = \Big\{ & \frac{0.3}{S_1} \\ & + \frac{0.8}{S_2} \Big\} . \end{aligned} \]
The relation between \(\underset{\sim}{P}\) and \(\underset{\sim}{S}\) in set theoretic form is :
Answer: A
The relation is the first matrix, with rows (0.1, 0.1), (0.3, 0.5) and (0.3, 0.8).
The relation between two fuzzy sets in set-theoretic form is their Cartesian product, whose membership is the minimum of the two memberships: \(\mu_R(C_i, S_j) = \min(\mu_P(C_i), \mu_S(S_j))\). Rows are \(C_1, C_2, C_3\) and columns \(S_1, S_2\).
- C1 (0.1): min(0.1, 0.3) = 0.1, min(0.1, 0.8) = 0.1
- C2 (0.5): min(0.5, 0.3) = 0.3, min(0.5, 0.8) = 0.5
- C3: min with 0.3 and 0.8 gives 0.3 and 0.8
The paper prints the grade of C3 as 10; a membership grade must lie between 0 and 1, so it stands for 1.0, and either way the minimums are 0.3 and 0.8. The other matrices contain values that no minimum of these grades can produce, such as 0.2, 0.9 or 1.
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