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In-Depth Information
ˀ
μ
(
∪
)
=
(μ(
), μ
A
∪
B
(
))
=
(μ(
),
(μ
A
(
), μ
B
(
)))
A
B
Sup
x
∈
X
min
x
x
Sup
x
∈
X
min
x
max
x
x
=
Sup
x
max
(
min
(μ(
x
), μ
A
(
x
)),
min
(μ(
x
), μ
B
(
x
)))
∈
X
=
(
(μ(
), μ
A
(
)),
(μ(
), μ
B
(
)))
max
Sup
x
∈
X
min
x
x
Sup
x
∈
X
min
x
x
=
max
(ˀ
μ
(
A
), ˀ
μ
(
B
)).
It is said that
μ
is the
possibility distribution
of
ˀ
μ
, a possibility measure that can
X
be considered as the one “conditioned” by
μ
. Thus, each fuzzy set
μ
∈[
0
,
1
]
such
that
Sup
μ
=
1(
μ
is never self-contradictory) induces the possibility measure
ˀ
μ
.
Provided
X
={
x
1
,...,
x
n
}
is a finite set, it results
ˀ
μ
(
A
)
=
Max
1
min
(μ(
x
i
), μ
A
(
x
i
)).
i
n
Hence, all non self-contradictory
fuzzy sets
, those
μ
such that
Sup
μ
=
1,
can be
viewed as possibility distributions
.
Examples 7.5.3
1. If
X
={
x
1
,
x
2
,
x
3
}
and
μ
=
1
|
x
1
+
0
.
6
|
x
2
+
0
.
8
|
x
3
, follows
ˀ
μ
(
X
)
=
Max
(
min
(
1
, μ
A
(
x
1
)),
min
(
0
.
6
, μ
A
(
x
2
)),
min
(
0
.
8
, μ
A
(
x
3
))
that, with
A
={
x
2
,
x
3
}
gives
ˀ
μ
(
A
)
=
max
(
0
,
0
.
6
,
0
.
8
)
=
0
.
8, and with
A
=
{
x
1
,
x
3
}
gives
ˀ
μ
(
A
)
=
max
(
min
(
1
,
1
),
min
(
0
.
6
,
0
),
min
(
0
.
8
,
1
))
=
max
(
1
,
0
,
0
.
8
)
=
1.
2. If
X
=[
0
,
10
]
,
A
=[
5
,
8
]
, with
μ
in the next figure, it is
ˀ
μ
(
[
5
,
8
]
)
=
0,
x
∈[
0
,
5
]∪[
8
,
10
]
Sup
min
(μ(
x
), μ
[
5
,
18
]
(
x
))
=
Sup
=
1.
μ(
x
)
,
x
∈[
5
,
18
]
x
∈[
0
,
10
]
x
∈[
0
,
10
]
A mapping
N
: F ₒ[
0
,
1
]
is a
measure of necessity
provided
•
N
(
∅
)
=
0
•
N
(
X
)
=
1
•
N
(
A
∩
B
)
=
min
(
N
(
A
),
N
(
B
))
, for all
A
,
B
in
F
.
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