Let R and S be two fuzzy relations: R = [[0.6, 0.4], [0.7, 0.3]] with rows x1,…

2017

Let R and S be two fuzzy relations:

R = [[0.6, 0.4], [0.7, 0.3]] with rows x1, x2 and columns y1, y2.

S = [[0.8, 0.5, 0.1], [0.0, 0.6, 0.4]] with rows y1, y2 and columns z1, z2, z3.

Using max-min composition, what is the resulting relation T from universe X to universe Z?

  1. A.

    \(\begin{matrix} && && z_1& &z_2&z_3\end{matrix}\\T=\begin{matrix}x_1\\x_2\end{matrix}\begin{bmatrix} 0.4 &0.6&0.4 \\ 0.7&0.7&0.7 \end{bmatrix} \\\)

  2. B.

    \(\begin{matrix} && && z_1& &z_2&z_3\end{matrix}\\T=\begin{matrix}x_1\\x_2\end{matrix}\begin{bmatrix} 0.4 &0.6&0.4 \\ 0.8&0.5&0.4 \end{bmatrix} \\\)

  3. C.

    \(\begin{matrix} && && z_1& &z_2&z_3\end{matrix}\\T=\begin{matrix}x_1\\x_2\end{matrix}\begin{bmatrix} 0.6&0.5&0.4 \\ 0.7&0.5&0.3 \end{bmatrix} \\\)

  4. D.

    \(\begin{matrix} && && z_1& &z_2&z_3\end{matrix}\\T=\begin{matrix}x_1\\x_2\end{matrix}\begin{bmatrix} 0.6 &0.5&0.5 \\ 0.7&0.7&0.7\end{bmatrix}\)

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Correct answer: C

Rule: T(xi,zj) = max over yk of min(R(xi,yk), S(yk,zj)).

  • T(x1,z1) = max(min(0.6,0.8), min(0.4,0.0)) = 0.6

  • T(x1,z2) = max(min(0.6,0.5), min(0.4,0.6)) = 0.5

  • T(x1,z3) = max(min(0.6,0.1), min(0.4,0.4)) = 0.4

  • T(x2,z1) = max(min(0.7,0.8), min(0.3,0.0)) = 0.7

  • T(x2,z2) = max(min(0.7,0.5), min(0.3,0.6)) = 0.5

  • T(x2,z3) = max(min(0.7,0.1), min(0.3,0.4)) = 0.3

Therefore, T = [[0.6, 0.5, 0.4], [0.7, 0.5, 0.3]].

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