Minimum about fuzzy logic
Fuzzy generalization AND and OR

As a generalization of logical product AND and logical sum OR, the most commonly used are operations of minimum and maximum (Zadeh’s or Gödel’s operations).

ANDm(a, b) = min(a, b)

ORm(a, b) = max(a, b),

ANDa(a, b) = ab

ORa(a, b) = a + b – ab

ANDb(a, b) = max(0, a + b - 1) ... the expression a + b – 1 bounded from below by 0

ORb(a, b) = min(a + b, 1) ... the expression a + b bounded from above by 1.

Behavior of the functions of generalized operators for two variables is generally visualized as a three-dimensional diagram. It can be understood as generalized logical maps (K-maps) depicted in space, where corners of the floor square represent the values assigned to the boundary (Boolean) values of operands 0 and 1 by the traditional Boolean operators.

Fig. 6.7: Logical graphs of two operators
Fig. 6.8: Demonstration of dissimilarities between minimized term function
Fig. 6.9: Demonstration of dissimilarities between minimized term function a

### Example 6.1

Let us give and illustrative example with logical values of the operands a = 0.3 and b = 0.8. It holds for the operations of maximum and minimum:

a ANDm b = 0.3 (the lesser of the two), a ORm b = 0.8 (the greater of the two)

for probability operations:

a ANDa b = 0.3 * 0.8 = 0.24 a ORa b = 0.3 + 0.8 – 0.24 = 0.86

and for Łukasiewicz’s operations:

a ANDb b = max (0; 0.3 + 0.8 – 1) = the greater of (0 and 0.1) = 0.1

a ORb b = min (0.3 + 0.8; 1) = the lesser of (1.1 and 1) = 1

It can be seen that the following inequalities hold for the results of generalized logical operators (not only for the abovementioned examples, but generally):

ANDb < ANDa < ANDm

ORm < ORa < ORb