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Elementary operators for fuzzy sets

The basic connective operations in classical set theory are those of intersection, union and complement. These operations on characteristic functions can be generalised to fuzzy sets in more than one way. However, one particular generalisation, which results in operations that are usually referred to us as standard fuzzy set operations, has a special significance in fuzzy set theory. In the following, only the standard operations are introduced. The following operations can be defined:

Whilst the operations according to Eqs. (15.15) and (15.16) are based on min/max operations, the complement is an algebraic one. Union and intersection can also be defined in an algebraic manner but giving different results as:

Demonstration Example 15.2   What is Fuzzy AND ?

Demonstration Example 15.3   What is Fuzzy OR ?

Demonstration Example 15.4   What is Fuzzy NOT ?

The standard connective operations for fuzzy sets are now defined. As one can easily see, these operations perform precisely as the corresponding operations for crisp sets when the range of membership grades is restricted to the set . That is, the standard fuzzy operations are generalisations of the corresponding classical set operations. However, they are not the only possible generalisation. As shown above, the fuzzy intersection, union and complement are not unique operations, contrary to their crisp counterparts. Different functions may be appropriate to represent these operations in different contexts. The capability to determine appropriate membership functions and meaningful fuzzy operations in the context of each particular application is crucial for making fuzzy set theory practically useful.

When fuzzy operators are later applied within more complex fuzzy logic operations for rules and fuzzy reasoning, one has to take care of the right combinations of fuzzy operations. For example, in classical set theory, the operations of intersection and union are dual with respect to the complement in the sense that they satisfy the De Morgan laws

   and    

It is desirable that this duality be satisfied for fuzzy sets as well. Other combinations need equivalences for commutativity, associativity and distributivity. From Table 15.1 the type of operations can be determined for which operations are valid. Only distributivity is not given in the arithmetic case. Therefore, one has to be careful in applications where arithmetic operations are performed.


Table 15.1: Validity of fuzzy equivalences


Interactive Questions 15.3   What do you know from fuzzy operators ?

Interactive Questions 15.4   The fuzzy NOT properties



Next: Fuzzy relations Up: Basics of fuzzy sets Previous: Membership functions   Contents
Christian Schmid 2005-05-09