Читать книгу Introduction to Fuzzy Logic - James K. Peckol - Страница 65

In relation to crisp sets, as we noted, fuzzy sets are supersets (of crisp sets) whose members are composed of collections of objects that satisfy imprecise properties to varying degrees. As an example, we can write the statement that X is a real number close to 7 as:

 F = set of real numbers close to 7

But what do “set” and “close to” mean and how do we represent such a statement in mathematically correct terms?

Zadeh suggests that F is a fuzzy subset of the set of real numbers and proposes that it can be represented by its membership function, mF. The value of mF is the extent or grade of membership of each real number r in the subset of numbers close to 7. With such a construct, it is evident that fuzzy subsets correspond to a continuously valued logic and that any element can have various degrees of membership in the subset.

Let's look at another example. Consider that a car might be traveling on a freeway at a velocity between 20 and 90 mph. In the fuzzy world, we identify or define such a range as the universe of discourse. Within that range, we might also say that the range of 50–60 is the average velocity.

In the fuzzy context, the term average would be classed as a linguistic variable. A velocity below 50 or above 60 would not be considered a member of the average range. However, values within and equal to the two extrema would be considered members.