Information Technology Reference
In-Depth Information
The solution of the problem was offered by L. Zadeh in 1965. In [14] he
introduced the concept of a fuzzy set.
The basic idea of Zadeh consisted in "allowing" a characteristic function to accept
not only value 0 (complete non-membership) or 1 (complete membership), but also
intermediate values of a membership from a segment [0. 1]. Thus, he has substituted
the concept of characteristic function with the concept of membership function
()
[]
μ
According to [14], the set of pairs of the following form is referred to as a fuzzy set
~
()
x
:
X
→
0
.
~
A
{
[
]
}
μ
From the definition one can understand that specification of a fuzzy subset
~
in
X
is equivalent to specification of its membership function
x
,
x
:
x
X
.
~
A
()
. Following the
traditional way, we will use the term “fuzzy set” instead of more correct term ”fuzzy
subset”.
Value of membership function
μ
x
~
μ
for an element
x
to fuzzy set
~
is
referred to as grade of membership
x
to
~
. This value can be interpreted as the
level of an element correspondence to the concept formalized by a fuzzy set
~
,
with the correspondence level being determined by an expert (group of experts).
Domain of a membership function
()
x
~
()
μ
x
is referred to as universal set
X
of
~
fuzzy set
~
.
A set which membership function is equal to zero for all elements of universal
set
X
is referred to as empty set Ø).
Using fuzzy sets, it is possible to define various concepts in more natural
manner of perception and description of objects. Let us explain with an example.
Example 1.2.
Formalization of the concept of “normal functioning of an object” on
the basis of the fuzzy set theory. As in the example 1.1., let a parameter
x
be defined
at universal set
X
and take values from
X
to
X
. The concept of “normal
functioning of an object” can be defined as fuzzy set
~
which membership function
is shown in Fig. 1.3.
Fig. 1.3
Membership function of a fuzzy set
~
formalizing the concept of “normal
functioning of an object”
