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Previous Year Question (PYQs)

A fuzzy conjunction operator $t(x,y)$ and a fuzzy disjunction operator $s(x,y)$ form a pair if they satisfy:

$t(x,y)=1-s(1-x,1-y)$

If $t(x,y)=\dfrac{xy}{x+y-xy}$, then $s(x,y)$ is given by:





Solution

Given:

$t(x,y)=1-s(1-x,1-y)$

So,

$s(x,y)=1-t(1-x,1-y)$

Now,

$t(1-x,1-y)=\dfrac{(1-x)(1-y)}{(1-x)+(1-y)-(1-x)(1-y)}$

Numerator:

$(1-x)(1-y)=1-x-y+xy$

Denominator:

$(1-x)+(1-y)-(1-x)(1-y)$

$=2-x-y-(1-x-y+xy)$

$=1-xy$

So,

$t(1-x,1-y)=\dfrac{1-x-y+xy}{1-xy}$

Now,

$s(x,y)=1-\dfrac{1-x-y+xy}{1-xy}$

$s(x,y)=\dfrac{1-xy-1+x+y-xy}{1-xy}$

$s(x,y)=\dfrac{x+y-2xy}{1-xy}$


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