8.6.3. Fuzzy Set, Its Definition and Arithmetic Operations
Definition 5. Given a universe of discourse W, a fuzzy set A is defined as:
for any w ΠW, there is a number m Π[0, 1] which is the degree of membership for w belonging to .
Projection m : W Æ [0, 1] is called the membership function of .
Example, Given W = {a, b, c, d}
if m = 1, m = 0.8 m = 0.4 m = 0 , then is a fuzzy set. If is used to represent the concept of "Circular shape", then m indicates the degrees of circularity of all elements in W.
This can be shown in a diagram
when W is composed of a finite number of elements, W is called a finite universe of discourse.
A fuzzy set defined on a finite W can be represented by a vector. For instance, the "circular shape" defined on W constitutes a fuzzy set which can be written as
= (1, 0.8, 0.4, 0) .
When there may be confusion between different elements, a fuzzy set may be represented as
= 1/a + 0.8/b + 0.4/c + 0/d ,
where denominators corresponds to elements in W and nominators represent the degrees of membership. "+" is only a separation mark. When the degree of membership is 0, that element can be omitted such as
= 1/a + 0.8/b + 0.4/c ,
we may also see in the following form
= {(1, a), (0.8, b), (0.4, c)} .
Example, If age is the universe of discourse, such as W = {0, 1, 2, ..., 200}, the fuzzy sets for "old" and "young" may be defined as
m =
m = .
Although W is a finite set, we can treat it as a continuous range between 0 to 200 to generate the curves for fuzzy sets and .
Definition 6. Given , Œ F(W), where F(W) is the set of all the fuzzy sets defined on W. The membership functions for » , « and are:
m » = max (m , m ) ,
m « = min (m , m ) , and
m = 1 - m , respectively .
If for W = {a, b, c, d}, two fuzzy sets are defined as:
= (1, 0.8, 0.4, 0) for circular shape
= (0.3, 0.4, 0.2, 0) for square shape,
then for circular or square we have
» = (1, 0.8, 0.4, 0)
for circular and square we have
« = (0.3, 0.4, 0.2, 0).
for not circular, we have
= (0, 0.2, 0.6, 1).
