Читать книгу Introduction to Fuzzy Logic - James K. Peckol - Страница 66

The following paragraphs are partially reused in Chapter 4.

The fuzzy property “close to 7” can be represented in several different ways. Who decides what that representation should be? That task falls upon the person doing the design.

To formulate a membership function for the fuzzy concept “close to 7,” one might hypothesize several desirable properties. These might include the following properties:

 Normality – It is desirable that the value of the membership function (grade of membership for 7 in the set F) for 7 be equal to 1, that is, μF(7) = 1.We are working with membership values 0 or 1.

 Monotonicity – The membership function should be monotonic. The closer r is to 7, the closer μF(r) should be to 1.0 and vice versa.We are working with membership values in the range 0.0–1.0.

 Symmetry – The membership function should be one such that numbers equally distant to the left and right of 7 have equal membership values.We are working with membership values in the range 0.0–1.0.

It is important that one realize that these criteria are relevant only to the fuzzy property “close to 7” and that other such concepts will have appropriate criteria for designing their membership functions.

Based on the criteria given, graphic expressions of the several possible membership functions may be designed. Three possible alternatives are given in Figure 1.3. Depending upon whether one is working in a crisp or fuzzy domain, the range of grade of membership (vertical) axis should be labeled either {0–1} or {0.0–1.0}.

Figure 1.3 Membership functions for “close to 7.”

Note that in graph c, every real number has some degree of membership in F although the numbers far from 7 have a much smaller degree. At this point, one might ask if representing the property “close to 7” in such a way makes sense.