logic is reasoning with revision and that if a default election is made from a number of possible alternatives based on the alternatives not being believed, then the concept or argument under debate or consideration is not extensible.
1.8 Sets and Logic
1.8.1 Classical Sets
Classical sets are considered crisp because their members satisfy precise properties. For example, for illustration, let H be the set of integer real numbers from 6 to 8. Using set notation, one can express H as:
(1.1)
One can also define a function μH(r) called a membership function to specify the membership of r in the set H,
(1.2)
The expression states:
r is a member of the set H (membership in H = 1) if its value is 6, 7, or 8. Otherwise, it is not a member of the set (membership in H = 0).
One can also present the same information graphically as in Figure 1.2.
Whichever representation is chosen, it remains clear that every real number, r, is surrounded by crisp boundaries and is either in the set H or not in the set H.
Moreover, because the membership function μ maps the associated universe of discourse of every classical set onto the set {0, 1}, it should be evident that crisp sets correspond to a two‐valued logic. An element is either in the set or it is not in the set, and it is either TRUE or it is FALSE.
Figure 1.2 Membership in subset H.
1.8.2 Fuzzy Subsets
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.
1.8.3 Fuzzy Membership Functions
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.
Example 1.1
Consider a shopping trip with a friend in Paris who poses the question:
How much does that cost?
To which you answer:
About 7 euros.
which certainly can be represented graphically.
As a further illustration, let the crisp set H and the fuzzy subset F represent the heights of players on a basketball team. If, for an arbitrary player p, we know that the membership in the set H is given as μH(p) = 1, then all we know is that the player's height is somewhere between 6 and 8 ft. On the other hand, if we know that the membership in the set P is given as μF(p) = 0.85, we know that the player's height is close to 7 ft. Which information is more useful?
Consider the phrase:
Etienne is old.
The phrase could also be expressed as:
Etienne is a member of the set of old people.
If