tconormstnorms

T-norms and t-conorms are fundamental tools in fuzzy logic and fuzzy set theory, used to model conjunction and disjunction of fuzzy truth values. They are binary operations defined on the unit interval [0,1] that generalize the logical operations of AND and OR to degrees of truth.

A t-norm is a commutative, associative, and monotone binary operation T: [0,1]^2 -> [0,1] with the boundary condition T(1, x) = x and T(0, x) = 0 for all x in [0,1]. These properties ensure that T behaves like a conjunction operator on fuzzy sets. A t-conorm, or S-norm, is its dual and models disjunction. It is commutative, associative, and monotone with the boundary conditions S(0, x) = x and S(1, x) = 1.

Duality between t-norms and t-conorms is captured by the relation S(x,y) = 1 - T(1 - x, 1 - y). Conversely, T(x,y) = 1 - S(1 - x, 1 - y). This duality links the two families and allows construction of a t-conorm from a given t-norm and vice versa.

Prominent examples include the Gödel (minimum) t-norm T(x,y) = min(x,y) with the t-conorm S(x,y) = max(x,y); the product t-norm T(x,y) = xy with the probabilistic sum S(x,y) = x + y − xy; and the Łukasiewicz t-norm T(x,y) = max(0, x + y − 1) with the Łukasiewicz t-conorm S(x,y) = min(1, x + y). There are other families, such as the Schweizer-Sklar t-norms, which provide parameterized forms.

These operators are used to aggregate fuzzy information in decision making, control systems, and knowledge representation, where they capture varying degrees of conjunction and disjunction beyond classical logic.