The notion central to fuzzy systems is that truth values (in fuzzy logic) or membership values (in fuzzy sets) are indicated by a value on the range [0.0, 1.0], with 0.0 representing absolute Falseness and 1.0 representing absolute Truth. For example, let us take the statement: "Jane is old."

If Jane's age was 75, we might assign the statement the truth value of 0.80. The statement could be translated into set terminology as follows:

"Jane is a member of the set of old people."

This statement would be rendered symbolically with fuzzy sets as:

mOLD(Jane) = 0.80

where m is the membership function, operating in this case on the fuzzy set of old people, which returns a value between 0.0 and 1.0.

At this juncture it is important to point out the distinction between fuzzy systems and probability. Both operate over the same numeric range, and at first glance both have similar values: 0.0 representing False (or non- membership), and 1.0 representing True (or membership). However, there is a distinction to be made between the two statements:

The probabilistic approach yields the natural-language statement, "There is an 80% chance that Jane is old," while the fuzzy terminology corresponds to "Jane's degree of membership within the set of old people is 0.80." The semantic difference is significant:

the first view supposes that Jane is or is not old (still caught in the Law of the Excluded Middle); it is just that we only have an 80% chance of knowing which set she is in. By contrast, fuzzy terminology supposes that Jane is "more or less" old, or some other term corresponding to the value of 0.80. Further distinctions arising out of the operations will be noted below.

The next step in establishing a complete system of fuzzy logic is to define the operations of EMPTY, EQUAL, COMPLEMENT (NOT), CONTAINMENT, UNION (OR), and INTERSECTION (AND). Before we can do this rigorously, we must state some formal definitions:

Definition 1

: Let X be some set of objects, with elements noted as x. Thus,

X = {x}.

Definition 2

: A fuzzy set A in X is characterized by a membership function mA(x) which maps each point in X onto the real interval [0.0, 1.0]. As mA(x) approaches 1.0, the "grade of membership" of x in A increases.

Definition 3

: A is EMPTY iff for all x, mA(x) = 0.0.

Definition 4

: A = B iff for all x: mA(x) = mB(x) [or, mA = mB].

Definition 5

: mA' = 1 - mA.

Definition 6

: A is CONTAINED in B iff mA <= mB.

Definition 7

: C = A UNION B, where: mC(x) = MAX(mA(x), mB(x)).

Definition 8:

C = A INTERSECTION B where: mC(x) = MIN(mA(x), mB(x)).

It is important to note the last two operations, UNION (OR) and INTERSECTION

(AND), which represent the clearest point of departure from a probabilistic theory for sets to fuzzy sets. Operationally, the differences are as follows:

For independent events, the probabilistic operation for AND is multiplication,

which (it can be argued) is counterintuitive for fuzzy systems. For example, let us

presume that x = Bob, S is the fuzzy set of smart people, and T is the fuzzy set of

tall people. Then, if mS(x) = 0.90 and uT(x) = 0.90, the probabilistic result would

be: mS(x) * mT(x) = 0.81 whereas the fuzzy result would be:

MIN(uS(x), uT(x)) = 0.90

The probabilistic calculation yields a result that is lower than either of the two

initial values, which when viewed as "the chance of knowing" makes good sense.

However, in fuzzy terms the two membership functions would read something

like "Bob is very smart" and "Bob is very tall." If we presume for the sake of

argument that "very" is a stronger term than "quite," and that we would correlate

"quite" with the value 0.81, then the semantic difference becomes obvious. The

probabilistic calculation would yield the statement If Bob is very smart, and Bob

is very tall, then Bob is a quite tall, smart person.The fuzzy calculation, however,

would yield If Bob is very smart, and Bob is very tall, then Bob is a very tall,

smart person.Another problem arises as we incorporate more factors into our

equations (such as the fuzzy set of heavy people, etc.). We find that the ultimate

result of a series of AND's approaches 0.0, even if all factors are initially high.

Fuzzy theorists argue that this is wrong: that five factors of the value 0.90 (let us

say, "very") AND'ed together, should yield a value of 0.90 (again, "very"), not

0.59 (perhaps equivalent to "somewhat").

Similarly, the probabilistic version of A OR B is (A+B - A*B), which approaches

1.0 as additional factors are considered. Fuzzy theorists argue that a sting of low

membership grades should not produce a high membership grade instead, the limit

of the resulting membership grade should be the strongest membership value in

the collection. The skeptical observer will note that the assignment of values to

linguistic meanings (such as 0.90 to "very") and vice versa, is a most imprecise

operation. Fuzzy systems, it should be noted, lay no claim to establishing a formal

procedure for assignments at this level; in fact, the only argument for a particular

assignment is its intuitive strength. What fuzzy logic does propose is to establish a

formal method of operating on these values, once the primitives have been

established.

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