Information Technology Reference
In-Depth Information
By comparing Fig. 1.2.b and Fig. 1.3., one can notice the collapse of the
paradox highlighted when analyzing the example 1.1: being formalized by means
of fuzzy sets, the model recognizes values
x
and
x
are seen as close ones.
Thus, the computer model based on principles of the fuzzy logic, "perceives"
actually close situations as similar ones.
It is worth mentioning that there is a tendency of probability treatment of fuzzy
set [20]. Obviously there is no sense to compare concepts of probability and fuzzy
set at the same abstracting level. Let us give more details about this.
Let
(
)
X,
B,
P
is a probability space [3]:
is a field of Borel subsets of
B ⊆
P(X)
[]
set
X
,
B
→
0
is the probability measure
P
, which satisfies to the conditions:
A
⊆
X
⇔
P(A)
≥
0
1)
()
2)
P
φ
=
0
(
)
() () (
)
A
,
B
∈
B
P
A
∪
B
=
P
A
+
P
B
−
P
A
∩
B
3) if
, then
.
()
x
According to L. Zadeh, fuzzy set is described by membership function
taking its values in a point of the segment [0.1]. From the point of view of the
mapping theory,
μ
[]
[]
are absolutely different objects.
Probability
P
is defined with
σ
-algebra
B
, and
P
:
B
→
0
and
μ
:
X
→
0
()
x
μ
is a usual function with a
range of definition
X
.
So, it is possible to draw some analogies between the membership function of
fuzzy set and probability density function of a chance quantity, but not to identify
them.
A clear subset of universal set, defined as
()
[]
{
}
A
=
x
∈
X
:
μ
x
≥
α
;
α
∈
0
.
(1.1)
~
α
A
-cut) of fuzzy set
~
with membership function
is referred to as set of
-level (
α
α
()
μ
x
).
~
~
{
}
Example 1.3.
Definition of
-level sets. Let
A
=
0
/
x
;
0
7
/
x
;
0
/
x
;
0
/
x
,
α
1
2
3
4
{}
3
{
}
{
}
A
=
x
A
=
x
,
x
,
x
A
=
x
,
x
then
;
;
.
0
,
9
0
,
4
2
3
4
0
,
7
2
3
α ≥
α
A
≥
A
It is obvious, that with
condition
is satisfied.
1
2
α
α
1
2
The Theorem of Decomposition
[40]. Any fuzzy set
~
with membership
function
()
μ
x
can be decomposed by
-level sets
~
α
~
A
=
∈
∪
α
A
(1.2)
α
[]
α
0
,
or
[
]
()
()
μ
x
=
α ∈
∪
αμ
x
,
~
A
A
[]
0
,
α
where
