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ↂ
⃔
∩
=
(
)
=
(
(
),
(
))
(
)
Since
A
B
A
B
A
,itis
N
A
min
N
A
N
B
N
B
.
(
(
),
(
))
Hence, all these mappings are actually fuzzy measures. From
min
N
A
N
B
A
C
ˀ(
A
)
+
ˀ(
B
)
,itfollows
N
(
A
∩
B
)
ˀ(
A
)
+
ˀ(
B
)
, Then 0
=
N
(
∅
)
=
N
(
A
∩
)
=
A
C
min
(
N
(
A
),
N
(
))
, that implies:
A
C
A
C
Either
N
(
A
)
=
0
,
or
N
(
)
=
0
,
and
N
(
A
)
+
N
(
)
1
,
or
A
C
N
(
)
1
−
N
(
A
).
Obviously, if
A
1
,
A
2
,...,
A
n
∈ F :
N
(
A
1
∩
A
2
∩
...
∩
A
n
)
=
min
(
N
(
A
1
),
N
(
A
2
),...,
N
(
A
n
))
. Then if
X
={
x
1
,...,
x
n
}
, to define a necessity measure it is
needed to take
N
(
x
1
),
N
(
x
2
),...,
N
(
x
3
)
such that: 0
=
min
(
N
(
x
1
),
N
(
x
2
),...,
N
(
x
n
))
, an equality that forces some of the values
N
(
x
i
)
to be 0.
Theorem 7.5.4
Given a possibility measure
ˀ
: F ₒ[
0
,
1
]
, the function N
ˀ
(
A
)
=
A
C
1
−
ˀ(
)
, for all A
∈ F
, is a necessity measure.
Proof N
ˀ
(
∅
)
=
1
−
ˀ(
X
)
=
0.
N
ˀ
(
X
)
=
1
−
ˀ(
∅
)
=
1.
N
ˀ
(
A
∩
B
)
=
A
C
B
C
A
C
B
C
A
C
B
C
1
−
ˀ(
∪
)
=
1
−
max
(ˀ(
), ˀ(
))
=
min
(
1
−
ˀ(
),
1
−
ˀ(
))
=
(
N
ˀ
(
),
N
ˀ
(
))
min
A
B
.
Theorem 7.5.5
Given a necessity measure N
: F ₒ[
0
,
1
]
, the function
ˀ
N
(
A
)
=
A
C
1
−
N
(
)
, for all A
∈ F
is a possibility measure.
Proof
ˀ
N
(
∅
)
=
1
−
N
(
X
)
=
0.
ˀ
N
(
X
)
=
1
−
N
(
∅
)
=
1.
ˀ
N
(
A
∪
B
)
=
1
−
A
C
B
C
A
C
B
C
N
(
∩
)
=
1
−
min
(
N
(
),
N
(
))
=
max
(ˀ
N
(
A
), ˀ
N
(
B
))
.
are called dual-pairs of possibility/necessity mea-
sures. Notice that with them it can be read:
The pairs
(ˀ,
N
ˀ
)
and
(
N
, ˀ
N
)
N ecessit y o f A
=
“
not the possibility of not A
”
Possibility of A
=
“
not the necessit y o f not A
”
.
Remark 7.5.6
If
ˀ
=
ˀ
μ
, the corresponding
N
is given by
ˀ
μ
N
ˀ
μ
(
A
)
=
1
−
ˀ
μ
(
A
)
=
1
−
Sup
x
min
(μ(
x
), μ
A
(
x
))
=
Inf
x
max
(
1
−
μ(
x
), μ
A
C
(
x
)),
∈
X
∈
X
that, in the finite case
X
={
x
1
,...,
x
n
}
,is
N
ˀ
μ
(
)
=
(
−
μ(
x
i
), μ
A
C
(
x
i
)).
A
min
1
i
n
max
1
Theorem 7.5.7
For all dual pair
(ˀ,
N
)
is:
1.
If N
(
A
)>
0
, then
ˀ(
A
)
=
1
2.
If
ˀ(
A
)<
1
, then N
(
A
)
=
0
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