A fuzzy conjunction t(x, y) and disjunction s(x, y) form a pair if t(x, y) = 1 − s(1 − x, 1 − y). If t(x, y) = xy/(x + y − xy), then s(x, y) is

1. A(x + y)/(1 − xy)
2. B(x + y − 2xy)/(1 − xy)
3. C(x + y − xy)/(1 − xy)
4. D(x + y − xy)/(1 + xy)

Pick an option to check yourself, or open the answer below.

Show answer & explanation

Correct answer: B. (x + y − 2xy)/(1 − xy)

Explanation

s(x, y) = 1 − t(1 − x, 1 − y). With a = 1 − x and b = 1 − y, t = ab/(a + b − ab) and a + b − ab = 1 − xy, so s = 1 − (1 − x)(1 − y)/(1 − xy) = (x + y − 2xy)/(1 − xy).