1. Union of fuzzy sets.

The union of two fuzzy sets A and B with respective membership functions is a fuzzy set C, written as , whose membership function is related to those A and B by

Note that has the associated property, that is, .As example, the union of fuzzy sets “about 50” and “about 57” is fuzzy set “about 50 or 57”. The graph of membership function is given below

Figure 4. Fuzzy set “about 50 or 57”.

2. The intersection of fuzzy sets.

The intersection of two fuzzy sets A and B is a fuzzy set C, written as , whose membership function is related to those of A and B by

or, in abbreviated form “A and B”.

It is easy to show that the intersection of A and B is the largest fuzzy set which is contained in both A and B. A and B are disjoint if is empty. Fig. 5 illustrates the membership function for the intersection set “not young and not old” for fuzzy sets “old” and “not old”.

Figure 5. Fuzzy set “not young and not old”

The membership function of given set is less than 1, then this set is subnormal. For the given set . To transform this set into nominal, is divided by .Fig. 6 illustrates the membership function of normalized set.

Figure 6. Normalized set.

3. Complement.

The complement of a fuzzy set A is denoted by and is defined by . The linguistic sense is defined as “not A”. Fig. 7 illustrates a fuzzy set “not old”, the complement of a fuzzy set “old”.

Figure 7. Fuzzy set “not old”.

We can also change fuzzy sets by using powered hedges: and

We can also change fuzzy sets by using powered hedges: and

4. Concentration.

The set C is the concentration for fuzzy set A. The membership function of the set C

.

Generic sense is following: for generic element “old” concentration is “very old” (fig. 8)

Figure 8. Concentration

5. Blur.

The set C is called the blur of fuzzy set A if the membership function

The generic sense of this operation is “not very”: for generic element “old” blur is “not very old (fig. 9)

Figure 9. Blur.

Last two operations are used only for fuzzy sets.

The choice of membership function is rather free because fuzzy theory methods formalize subjective model of outer world. So the result should not depend on chosen membership function. More objective evaluation of notion or event В needs membership functions based on expert polls.

There is several methods to define membership function:

1. heuristic method: subject himself defines the degree of membership; functions presented by different people for one generic element my differ;

2. statistic method: the membership function form and its values is defined by averaging;

3. partial representation with explaining examples;

4. interval definition: values intervals for pessimistic and optimistic edges of membership function.

Key questions

1. What property of objects is called fuzziness?

2. What is generic element?

3. Define fuzzy set notion. Explain the meaning of membership function.

4. What methods of membership function presentation do you know? Give examples

5. Define basic operation on fuzzy sets.

6. Define normal and subnormal fuzzy.