UGC NET CSE | June 2016 | Part 3 | Question: 64

Question

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=		$z_1$	$z_2$	$z_3$
		$x_1$	$\begin{bmatrix} 0.7 & 0.6 & 0.5 \\ 0.8 & 0.8 & 0.8 \end{bmatrix}$
		$x_2$

Answer 1

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

Answer 2

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

₁z₁= max(min(0.7, 0.9), min(0.5, 0.1)) = max(0.7 0.1) = 0.7 x₁z₂= max(min(0.7, 0.6), min(0.5, 0.7)) = max(0.6, 0.5) = 0.6 x₁z₃= max(min(0.7, 0.2), min(0.5, 0.5)) = max(0.2, 0.5) = 0.5 x₂z₁= max(min(0.8, 0.9), min(0.4, 0.1)) = max(0.8, 0.1) = 0.8 x₂z₂= max(min(0.8, 0.6), min(0.4, 0.7)) = max(0.6, 0.4) = 0.6 x₃z₃= max(min(0.8, 0.2), min(0.4, 0.5)) = max(0.2, 0.4) = 0.4

So, option (C) is correct.