X is a domain. If
is a fuzzy subset of
X, for any
, assigning a number
to
x,
is called a membership of
x with respect to
.
Mapping
,
is called a membership function of
.
It is noted that in the following discussion, domain
X is assumed to be infinite (not limited to finite). For simplicity, the membership function is denoted by
rather than
.
The main operations of fuzzy sets: given two fuzzy sets
and
, the union, intersection, and complement operations are defined as follows:
In practice such as fuzzy control, designers may choose the membership functions optionally in some degree, i.e., the membership functions of the same fuzzy variable may (slightly) be different, but the controllers designed from the different membership functions still have the same (or approximate) performances. The robustness of the fuzzy analysis method, based on (more or less) optionally chosen membership functions, has brought many people’s attention. Some researchers (
Liang and Song, 1996;
Lin and Tsumoto, 2000;
Mitsuishi et al., 2000;
Verkuilen, 2001;
Lin, 2001a) presented the probabilistic interpretation of membership functions. For example,
Lin (2001a) interpreted memberships as probabilities. Each sample space has a probability, and each point is associated with one sample space. So the total space is like a fiber space. Each fiber space is a probability space.
Liang and Song (1996) regarded the values of a membership function as independent and identically distributed random variables, and proved that the mean of the membership function exists for all the elements of the universe of discourse. He interpreted the meaning of a subjective concept of a group of people as the mean of a membership function for all people within the group.
Mitsuishi et al. (2000) introduced a new concept of empty fuzzy set, in order to define the membership functions in the probabilistic sense. That is, although different persons may assign (slightly) different membership functions to a fuzzy concept, when solving a real problem (fuzzy control or fuzzy reasoning), in average they can get an approximate result. Unfortunately, these results were obtained based on a strong assumption, i.e., the values of a membership function are assumed to be independent and identically distributed random variables.
Verkuilen (2001) introduced the concept of membership functions by the multi-scale method, and discussed its corresponding properties.
Lin (1988,
1992,
1997) presented a topological definition, topological rough set, of fuzzy sets by using neighborhood systems, discussed the properties of fuzzy sets from their structure, and then presented a definition of the equivalence between two fuzzy membership functions, and the necessary and sufficient conditions of the equivalence between two membership functions. He also discussed the concept of granular fuzzy sets in
Lin (1998,
2001b), and ‘elastic’ membership functions in
Lin (1996,
2000,
2001b). Lin’s works provide a structural interpretation of membership functions (fuzzy sets).
It can be seen that the membership function of a fuzzy set can be interpreted in two ways: one probabilistic, the other structural. We will show below that for a fuzzy set (concept), it may probably be described by different types of membership functions, as long as their structures (see the structural definition of fuzzy sets below) are the same, it still appears with the same characteristics. That is, although these membership functions are different in appearance, they are the same in essence. Therefore, the structural interpretation of fuzzy sets would be better than the probabilistic one. And it seems that in a given environment, most persons would have a similar structural interpretation for the same
fuzzy concept. We will introduce a structural definition of fuzzy sets, and discuss its properties below, since the structural description is more essential to a fuzzy set.