A fuzzy conjuction operators \(t(x, y)\)and a fuzzy disjunction operator,…
2019
A fuzzy conjuction operators \(t(x, y)\)and a fuzzy disjunction operator, \(s(x, y)\), form a pair if they satisfy:
\(f(x, y) = 1 - s(1 - x, 1 - y)\)
if \(f(x , y) = {xy \over (x + y -xy)}\), then \(s(x, y)\) is given by
- A.
\(x + y \over 1 - xy\) - B.
\(x + y - 2xy \over 1 - xy\) - C.
\(x + y - xy \over 1 - xy\) - D.
\(x + y - xy \over 1 + xy\)
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Correct answer: B
Answer: s(x, y) = (x + y - 2xy) / (1 - xy)
Derivation:
Start from the pairing relation: s(x,y) = 1 - f(1 - x, 1 - y).
Compute f(1 - x, 1 - y):
f(1 - x, 1 - y) = (1 - x)(1 - y) / [(1 - x) + (1 - y) - (1 - x)(1 - y)].
Simplify numerator: (1 - x)(1 - y) = 1 - x - y + xy.
Simplify denominator: (1 - x) + (1 - y) - (1 - x)(1 - y) = 1 - xy.
Thus f(1 - x, 1 - y) = (1 - x - y + xy) / (1 - xy).
Now compute s(x,y) = 1 - f(1 - x, 1 - y) = 1 - (1 - x - y + xy)/(1 - xy).
Bring to a common denominator: s(x,y) = [(1 - xy) - (1 - x - y + xy)]/(1 - xy) = (x + y - 2xy)/(1 - xy).
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