A fuzzy conjunction operator denoted as \(𝑡(𝑥,𝑦)\) and a fuzzy disjunction…

Correct relation: t(x,y) = 1 − s(1 − x, 1 − y)

Reasoning:

• Define the standard fuzzy negation n(x) = 1 − x. This negation is involutive, so n^{-1} = n.

• The general duality condition links a t-norm and s-norm by applying negation to inputs and output: t(x,y) = n( s( n(x), n(y) ) )

• Because n is its own inverse, substitute n(x)=1−x to obtain t(x,y) = 1 − s(1 − x, 1 − y).

Why the other forms fail:

• Expression 1 − s(x,y) only complements the output; it does not complement the inputs, so it does not satisfy the negation-input-output structure required for duality.

• Expression s(1 − x, 1 − y) complements only the inputs but omits the outer complement; the final negation is required to yield the dual t.

• Expression s(1 + x, 1 + y) uses inputs outside the [0,1] domain and does not follow the negation-based formula, so it is invalid as a dual relation.