Soft Computing and Its Applications, Volume One

Fuzzy Sets, Fuzzy Operators, and Fuzzy Relations 99

A T-norm is Archimedean, if and only if,

1. T(x,y) is continuous,

2. T(x,x) x, for all x (0,1).

Archimedean T-norm is strict, if and only if,

3. T(x’, y’) T(x,y), if x’ < x,y’ < y, for all x’, y’, x,y (0,1).

Denition 2.31 Let T*: [0,1] [0,1] [0,1]. T* is a T-conorm, if and only if, for

all x,y,z (0,1);

1. T* (x,y) = T*(y,x),

2. T* (x,y) T* (x,z), if y z,

3. T* (x, T*(y,z)) = T* (T* (x,y),z),

4. T* (x,0) = x.

Some basic T-conorms are given in Table 4.

TABLE 4 Basic T-conorms

Maximum (Gödel) T*max (a, b) = max { a, b}

Lukasiewicz T*(a, b) = min { a + b, 1 }

Probabilistic T*(a, b) = ...