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In-Depth Information
Remark 7.2.2
1. For some specific problems, measures of fuzziness are selected
verifying the additional property of symmetry:
•
(μ
)
For some negation
N
,itis
m
(μ)
=
m
(
N
ⓦ
μ)
=
m
.
For example, with
N
=
1
−
id
, measures 1, 2, 3, 4 do verify this property of
symmetry
m
(μ)
=
m
(
1
−
μ)
.
2. With each measure of fuzziness
m
, it can be defined a
measure of booleanity
1
−
m
, that, obviously verifies:
•
the value of 1
−
m
is 1 for the crip sets.
•
the value of 1
−
m
is 0 for
μ
1
/
2
.
•
1
−
m
is non-increasing with respect to the order
S
.
X
, with the partial pointwise order '
Example 7.2.3
Take
F ∈[
0
,
1
]
μ
˃
⃔
μ(
x
)
X
˃(
x
)
, for all
x
∈
X
' (and such that
μ
0
, μ
1
∈ F
). A mapping
m
:[
0
,
1
]
ₒ[
0
,
1
]
is
a
fuzzy measure
provided
m
verifies:
1.
m
(μ
0
)
=
0
2.
m
(μ
1
)
=
1
3. If
μ
˃
, then
m
(μ)
m
(˃)
.
X
When
F ∈{
0
,
1
}
≈ P
(
X
)
, a fuzzy measure is defined by
(
∅
)
=
1.
m
0
(
)
=
2.
m
X
1
3. If
A
ↂ
B
, then
m
(
A
)
m
(
B
)
.
|
μ
|=
x
i
∈
X
μ(
For example, if
X
is a finite set
{
x
1
,...,
x
n
}
, and
x
i
)
, the crisp
(μ)
=
|
μ
|
n
cardinality of
μ
, then the function
m
, is a fuzzy measure.
Remember, that since
μ
·
˃
=
T
ⓦ
(μ
×
˃)
,
μ
+
˃
=
S
ⓦ
(μ
×
˃)
,itis
μ
·
˃
μ
,
μ
·
˃
˃
,
μ
μ
+
˃
,
˃
μ
+
˃
.
Provided
μ
·
˃
,
μ
+
˃
∈ F
, for all fuzzy measure
m
,is:
m
(μ
·
˃)
m
(μ)
,
m
(μ
·
˃)
m
(˃)
,
m
(μ)
m
(μ
+
˃)
,
m
(˃)
m
(μ
+
˃)
,
and
m
(μ
·
˃)
min
(
m
(μ),
m
(˃))
max
(
m
(μ),
m
(˃))
m
(μ
+
˃).
X
, it results
In the particular case in which
μ
,
˃
∈{
0
,
1
}
m
(
A
∩
B
)
min
(
m
(
A
),
m
(
B
))
max
(
m
(
A
),
m
(
B
))
m
(
A
∪
B
),
X
and
,
∈ P
(
)
[
,
]
for all
A
B
X
. Notice that fuzzy measures can be applied to both
0
1
P
(
)
X
.
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