04-01-2013, 01:11 PM
Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems
[attachment=46461]
Fuzzy Set Theory
The classical set theory is built on the fundamental concept of “set” of
which an individual is either a member or not a member. A sharp, crisp, and
unambiguous distinction exists between a member and a nonmember for any
well-defined “set” of entities in this theory, and there is a very precise and
clear boundary to indicate if an entity belongs to the set. In other words, when
one asks the question “Is this entity a member of that set?” The answer is
either “yes” or “no.” This is true for both the deterministic and the stochastic
cases. In probability and statistics, one may ask a question like “What is the
probability of this entity being a member of that set?” In this case, although
an answer could be like “The probability for this entity to be a member of that
set is 90%,” the final outcome (i.e., conclusion) is still either “it is” or “it is
not” a member of the set. The chance for one to make a correct prediction as
“it is a member of the set” is 90%, which does not mean that it has 90%
membership in the set and in the meantime it possesses 10% non-membership.
Namely, in the classical set theory, it is not allowed that an element is in a set
and not in the set at the same time. Thus, many real-world application
problems cannot be described and handled by the classical set theory,
including all those involving elements with only partial membership of a set.
On the contrary, fuzzy set theory accepts partial memberships, and, therefore,
in a sense generalizes the classical set theory to some extent.
In order to introduce the concept of fuzzy sets, we first review the
elementary set theory of classical mathematics. It will be seen that the fuzzy
set theory is a very natural extension of the classical set theory, and is also a
rigorous mathematical notion.
I. CLASSICAL SET THEORY
A. Fundamental Concepts
Let S be a nonempty set, called the universe set below, consisting of all the
possible elements of concern in a particular context. Each of these elements is
called a member, or an element, of S. A union of several (finite or infinite)
members of S is called a subset of S. To indicate that a member s of S
belongs to a subset S of S, we write
Fuzzy Set Theory
describes the oldness is the one given in Figure 1.1(a),” to start with all the
rigorous mathematics in the rest of the investigation.
The fuzzy set theory is taking the same logical approach as what people
have been doing with the classical set theory: in the classical set theory, as
soon as the two-valued characteristic function has been defined and adopted,
rigorous mathematics follows; in the fuzzy set case, as soon as a multi-valued
characteristic function (the membership function) has been chosen and fixed, a
rigorous mathematical theory can be fully developed.
Now, we return to the subset Sf introduced above. Suppose that the
membership function associated with it, say the one shown in Figure 1.1(a),
has been chosen and fixed. Then, this subset Sf along with the membership
function used, which we will denote by μSf(s) with s ∈ Sf, is called a fuzzy
subset of the universe set S. A fuzzy subset thus consists of two components:
a subset and a membership function associated with it. This is different from
the classical set theory, where all sets and subsets share the same (and the
unique) membership function: the two-valued characteristic function
mentioned above.
Throughout this
[attachment=46461]
Fuzzy Set Theory
The classical set theory is built on the fundamental concept of “set” of
which an individual is either a member or not a member. A sharp, crisp, and
unambiguous distinction exists between a member and a nonmember for any
well-defined “set” of entities in this theory, and there is a very precise and
clear boundary to indicate if an entity belongs to the set. In other words, when
one asks the question “Is this entity a member of that set?” The answer is
either “yes” or “no.” This is true for both the deterministic and the stochastic
cases. In probability and statistics, one may ask a question like “What is the
probability of this entity being a member of that set?” In this case, although
an answer could be like “The probability for this entity to be a member of that
set is 90%,” the final outcome (i.e., conclusion) is still either “it is” or “it is
not” a member of the set. The chance for one to make a correct prediction as
“it is a member of the set” is 90%, which does not mean that it has 90%
membership in the set and in the meantime it possesses 10% non-membership.
Namely, in the classical set theory, it is not allowed that an element is in a set
and not in the set at the same time. Thus, many real-world application
problems cannot be described and handled by the classical set theory,
including all those involving elements with only partial membership of a set.
On the contrary, fuzzy set theory accepts partial memberships, and, therefore,
in a sense generalizes the classical set theory to some extent.
In order to introduce the concept of fuzzy sets, we first review the
elementary set theory of classical mathematics. It will be seen that the fuzzy
set theory is a very natural extension of the classical set theory, and is also a
rigorous mathematical notion.
I. CLASSICAL SET THEORY
A. Fundamental Concepts
Let S be a nonempty set, called the universe set below, consisting of all the
possible elements of concern in a particular context. Each of these elements is
called a member, or an element, of S. A union of several (finite or infinite)
members of S is called a subset of S. To indicate that a member s of S
belongs to a subset S of S, we write
Fuzzy Set Theory
describes the oldness is the one given in Figure 1.1(a),” to start with all the
rigorous mathematics in the rest of the investigation.
The fuzzy set theory is taking the same logical approach as what people
have been doing with the classical set theory: in the classical set theory, as
soon as the two-valued characteristic function has been defined and adopted,
rigorous mathematics follows; in the fuzzy set case, as soon as a multi-valued
characteristic function (the membership function) has been chosen and fixed, a
rigorous mathematical theory can be fully developed.
Now, we return to the subset Sf introduced above. Suppose that the
membership function associated with it, say the one shown in Figure 1.1(a),
has been chosen and fixed. Then, this subset Sf along with the membership
function used, which we will denote by μSf(s) with s ∈ Sf, is called a fuzzy
subset of the universe set S. A fuzzy subset thus consists of two components:
a subset and a membership function associated with it. This is different from
the classical set theory, where all sets and subsets share the same (and the
unique) membership function: the two-valued characteristic function
mentioned above.
Throughout this