4 votes

Let R and S be two fuzzy relations defined as

R= | $y_1$ | $y_2$ | |

$x_1$ | $\begin{bmatrix} 0.7 & 0.5 \\ 0.8 & 0.4 \end{bmatrix}$ | ||

$x_2$ |

and

S= | $z_1$ | $z_2$ | $z_3$ | |

$y_1$ | $\begin{bmatrix} 0.9 & 0.6 & 0.2 \\ 0.1 & 0.7 & 0.5 \end{bmatrix}$ | |||

$y_2$ |

Then, the resulting relation T, which related elements of universe xto elements of universe z using max-min composition is given by

A. | T= | $z_1$ | $z_2$ | $z_3$ | |

$x_1$ | $\begin{bmatrix} .5 & .7 & .5 \\ .8 & .8 & .8 \end{bmatrix}$ | ||||

$x_2$ |

B. | T= | $z_1$ | $z_2$ | $z_3$ | |

$x_1$ | $\begin{bmatrix} .5 & .7 & .5 \\ .9 & .6 & .5 \end{bmatrix}$ | ||||

$x_2$ |

C. | T= | $z_1$ | $z_2$ | $z_3$ | |

$x_1$ | $\begin{bmatrix} 0.7 & 0.6 & 0.5 \\ 0.8 & 0.6 & 0.4 \end{bmatrix}$ | ||||

$x_2$ |

D. | T= | $z_1$ | $z_2$ | $z_3$ | |

$x_1$ | $\begin{bmatrix} 0.7 & 0.6 & 0.5 \\ 0.8 & 0.8 & 0.8 \end{bmatrix}$ | ||||

$x_2$ |

5 votes

Ans is C

Here to express relationship between x1, z1 , x1, z2 ..... etc we need to use y1 and y2 x1 is related to both y1 and y2 by first matrix and y1 and y2 are related to z1,z2,z3 by second matrix. so for relation between x1 to z1 (max min composition) first find the min of x1y1 and y1z1 & min of x1y2 and y2z1 and then their max that will be max(min(0.7,0.9),min(0.5,0.1))=max(0.7,0.1)=0.7

and so on , complete soln is as below

1 vote

Since x is related to y and y is related to z, to relate universe x and universe z we have to compute max-min composition:

x

So, option (C) is correct.