Enumerations

Describes a T-norm function, usually used for forming an intersection of fuzzy sets.

A function T must satisfy the following axioms in order to be T-normed:

1. Boundary condition: T(1, 1) = 1, T(1, 0) = 0, T(0, 1) = 0, T(0, 0) = 0
2. Commutativity: T(a, b) = T(b, a)
3. Monotonic: If a <= a' and b <= b' Then T(a, b) <= T(a', b')
4. Associativity: T(T(a, b), c) = T(a, T(b, c))

Describes a S-norm (T-conorm) function, usually used for forming a union of fuzzy sets.

A function S must satisfy the following axioms in order to be S-normed (T-conormed):

1. Boundary condition: S(1, 1) = 1, S(1, 0) = 1, S(0, 1) = 1, S(0, 0) = 0
2. Commutativity: S(a, b) = S(b, a)
3. Monotonic: If a <= a' and b <= b' Then S(a, b) <= S(a', b')
4. Associativity: S(T(a, b), c) = S(a, T(b, c))

Describes a complement (negation) function, usually used for forming a complement (negation) of a fuzzy set.

A function c must satisfy the following axioms in order to be capable of negating a fuzzy set:

1. Boundary condition: c(0) = 1, c(1) = 0
2. Monotonic nonincreasing: a < b -> c(a) >= c(b)
3. Involution: c(c(a)) = a

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Describes a function used for forming a fuzzy implication.

A function I must satisfy the following axioms in order to be called a fuzzy implication:

1. Boundary condition: S(1, 1) = 1, S(1, 0) = 0, S(0, 0) = 1
2. Non-increasing y: If x_1 <= x_2 Then I(x_1, y) >= I(x_2, y)
3. Non-decreasing x: If y_1 <= y_2 Then I(x, y_1) <= I(x, y_2)