Information Technology Reference
In-Depth Information
Chapter 7
Fuzzy Measures
7.1 Introduction
Fuzzy sets not only appear by representing imprecise predicates, but also partial or
incomplete information. This is the case, for example, of a function
X
taking values
in
[
0
,
10
]
, of which it is only known that
5
X
7, 8
X
, and
X
2
.
]
,by
This information can be represented by means of a fuzzy set
μ
X
∈[
0
,
10
taking:
•
μ
X
(
x
)
=
1
⃔
X
takes the value
x
•
μ
X
(
)
=
⃔
x
0
X
don't takes the value
x
•
μ
X
(
)
=
·
+
⃔
It is unknown if
X
can take the value
x
,
and the above information resumed by:
x
a
x
b
⊧
⊨
0,
if
x
∈[
0
,
2
]∪[
8
,
10
]
x
−
2
,
if
x
∈[
2
,
5
]
3
μ
X
(
x
)
=
1,
if
x
∈[
5
,
7
]
⊩
8
−
x
, f
x
∈[
7
,
8
]
In these cases, the questions of the uncertainty concerning the questions,
•
X
∈
A
for
A
ↂ[
0
,
10
]
X
,
•
X
r
A
,for
μ
A
∈[
0
,
10
]
arise and suggests the problem of its measuring. In this section, several “measures”
of that kind of uncertainty, including probabilities when applicable, are introduced
and studied.
The area of a polygon is a measure of its extensional size, and the length of a
segment is a measure of its longitudinal size. Everybody knows several examples of
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