# Fuzzy Operators

## Fuzzy intersection

Consider the intersection of two fuzzy sets. An intersection element that has a degree of membership in Set A and a different degree of membership in Set B has the lesser value as its membership in the intersection set. For example, a day that is 55 per cent warm and 75 per cent humid has only 55 percent membership in the set warm_and_humid days. This is an application of f-AND.

## Fuzzy union

For a union of fuzzy sets, the greater of the two degrees of membership is the degree of membership in the union set. The warm and humid day from above has a 75 percent membership in warm_or_humid days, and this applies f-OR.

## Fuzzy complement

Degree of membership in a complement set is 1 minus the degree of membership in the original set. A day that is 55 percent warm is 1 minus .55 or 45 percent cold if we are consistent in our definitions for warm and cold days. There is no difference from the NOT of predicate logic and thus no need for a f-NOT.

Since fuzzy logic is a subset of predicate logic, reasoning is the same. As would be expected, probability is often needed to clarify fuzzy, uncertain information. A degree of truth is the probability that a statement is true. Even with precise information, real-world problems may include uncertainty. There is a range of values over which real specimens fail, and a bridge builder will overdesign or use safety factors to make collapse highly unlikely. Unreliable data magnify the chances for erroneous conclusions, and fuzzy algorithms commonly estimate the level of confidence of the results.

Using f-OR is simplistic and tends to treat assigned memberships as if the cutoffs were sharp. This is contrary to the spirit of fuzzy reasoning where data are not sharply defined. Of more use is the probability-OR that doesn't select one of the two membership values but weighs both. Given two independent elements, A and B, the probability that one or the other belongs to a set is

p(A or B) = p(A) + p(B) - ( p(A) * p(B) )

Whereas the f-OR throws away the element with the lowest membership function, the p-OR keeps information from both and thus is more useful for assessing the contributions of many elements. For example, consider a battery of medical tests, each of which has a fuzzy range of values that discriminate between health and disease. Furthermore, extreme values may be highly indicative of a specific disease or of one of the diseases from a number of possibilities, but a measurement is not full and sufficient proof by itself. If we use the f-OR, the lab tests are considered one at at time with the least important discarded at each step. It is better to group them all with the p-OR that accumulates as the algorithm strives to assign health or disease as its truth.

When there are two independent events and each has its own probability of occurrence, the probability of simultaneous occurrence is

p(A and B) = p(A) * p(B)

This is the p-AND operation. Let's see an example of accumulating evidence with the p-OR. We will decide whether to perform a medical laboratory test based on the results of two previous tests. Our arbitrary index will be that a p-OR result of 0.65 from the first two tests is sufficient to justify spending the time and money to perform test Number~3. For Patient A, the test results are 0.5 and 0.4. The values are 0.2 and 0.3 for Patient B. The calculations are

A. 0.5 p-OR 0.4 = .5 + .4 - ( .5 × .4 ) = .7

B. 0.2 p-OR 0.3 = .2 + .3 - ( .2 × .3 ) = .44

This means that there is a basis for performing the third test (an experimental prefrontal lobotomy) on Patient A, but Patient B is allowed to escape.