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In-Depth Information
3
W
∗
(
•
(
x
1
,
x
2
)
=
.
7
x
1
.
x
2
+
.
x
1
,
x
2
)
A
0
0
•
(
x
1
,
x
2
)
=
.
(
x
1
,
x
2
)
+
.
(
x
1
+
x
2
−
x
1
.
x
2
)
A
0
6min
0
4
•
A
(
x
1
,
x
2
)
=
0
.
6
W
(
x
1
,
x
2
)
+
0
.
4max
(
x
1
,
x
2
)
,
are aggregation functions.
2.3.4 Examples
The pointwise aggregation of classical sets is not, in general, a classical set, but a
fuzzy one. For example, the arithmetic mean verifies
1
2
,
M
(
0
,
0
)
=
0
,
M
(
0
,
1
)
=
M
(
1
,
0
)
=
M
(
1
,
1
)
=
1
and, if
A
,
B
are crisp subsets,
M
(
A
,
B
)
is not a crisp subset if given by
M
(μ
A
, μ
B
)
(
x
)
=
M
(μ
A
(
x
), μ
B
(
x
))
. On the contrary, with the geometric mean
G
,itis
(
,
)
=
(
,
)
=
(
,
)
=
,
(
,
)
=
,
G
0
0
G
0
1
G
1
0
0
G
1
1
1
and
G
is a crisp set.
In all cases, if
(
A
,
B
)
X
Y
, and A is an aggregation function, then
μ
∈[
0
,
1
]
, ˃
∈[
0
,
1
]
A
(μ, ˃)(
x
,
y
)
=
A
(μ(
x
), ˃(
y
)),
X
×
Y
for all
x
∈
X
,
y
∈
Y
, is a fuzzy set
A
(μ, ˃)
∈[
0
,
1
]
called the aggregation of
μ
X
,
and
˃
. When
X
=
Y
it could be defined the fuzzy set
A
(μ, ˃)
∈[
0
,
1
]
A
(μ, ˃)(
x
)
=
A
(μ(
x
), ˃(
x
)),
for all
x
∈
X
.
Example 2.3.1
If
X
={
1
,
2
,
3
,
4
,
5
}
, and
μ
=
0
.
6
/
1
+
0
.
7
/
2
+
0
.
5
/
3
+
1
/
4,
˃
=
0
.
9
/
1
+
0
.
5
/
3
+
0
.
7
/
4
+
0
.
8
/
5, compute
M
(μ, ˃)
,
G
(μ, ˃)
, and
O
(μ, ˃)
with
O
the OWA with weights
p
1
=
0
.
4
,
p
2
=
0
.
6.
Solution.
M
(μ, ˃)
=
0
.
75
/
1
+
0
.
35
/
2
+
0
.
5
/
3
+
0
.
85
/
4
+
0
.
4
/
5
G
(μ, ˃)
=
0
.
735
/
1
+
0
/
2
+
0
.
5
/
3
+
0
.
837
/
4
+
0
/
5
O
(μ, ˃)
=
(
0
.
4
×
0
.
6
+
0
.
6
×
0
.
9
)/
1
+
(
0
.
4
×
0
+
0
.
6
×
0
.
7
)/
2
+
(
0
.
4
×
0
.
5
+
0
.
6
×
0
.
5
)/
3
+
(
0
.
4
×
0
.
7
+
0
.
6
×
1
)/
4
+
(
0
.
4
×
0
+
0
.
6
×
0
.
8
)/
5
=
.
/
+
.
/
+
.
/
+
.
/
+
.
/
0
72
1
0
42
2
0
5
3
0
88
4
0
48
5.
Notice that
G
(μ, ˃)
M
(μ, ˃)
, but that neither
G
(μ, ˃)
and
O
(μ, ˃)
, nor
M
(μ, ˃)
and
O
(μ, ˃)
, are order-comparable.
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