## Logical operations

The most important thing to realize about fuzzy logical reasoning is that it is a superset of standard Boolean logic, i.e., if the fuzzy values are kept at their extremes of 1 (completely true) and 0 (completely false), standard logical operations hold. In fuzzy logic, however, the truth of any statement is a matter of degree. The input values can be real numbers between 0 and 1. It should be noted that the results of the statement A AND B, where A and B are limited to the range (0, 1) can be resolved by using min (A, B). Similarly, an OR operation can be replaced with the max function so that A OR B becomes equivalent to max (A, B), and the operation NOT A is equivalent to the operation 1 - A. Given these three functions, any construction can be resolved using fuzzy sets and the fuzzy logical operations AND, OR, and NOT. An example of the operations on fuzzy sets is shown in Figure 11.26.

In Figure 11.26, only one particular correspondence between two-valued and multivalued logical operations for AND, OR, and NOT is defined. This correspondence is by no means unique. In more general terms, what are known as the fuzzy intersection or conjunction (AND), fuzzy union or disjunction (OR), and fuzzy complement (NOT) can be defined.

The intersection of two fuzzy sets, A and B, is specified in general by a binary mapping, T, which aggregates two membership functions as

The binary operator, T, may represent the multiplication of ^A(x) and \i.B(x). These fuzzy intersection operators are usually refined as T norm (triangular norm) operators. Similarly, in fuzzy intersection, the fuzzy union operator is specified in general by a binary mapping, S, as

Fast rev

Slow rev

Slow fwd

Fast fwd

Fast rev

Slow rev

Slow fwd

Fast fwd

FiGURE 11.25 Membership functions for linguistic variables describing motor operation.

20.1 0 0.1 Output variable, "motor"

Operations of fuzzy sets

Set A is a fuzzy interval between 5 and 6

Set B is a fuzzy number about 4

Fuzzy set between 5 and 6 AND about 4

VAn Operation equivalent to m/n(A, B)

Fuzzy set between 5 and 6 OR about 4

Operation equivalent to max(A, B)

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