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7.5 Measures of Possibility and Necessity
Letitbe
F ↂ P
(
X
)
a Boolean algebra of subsets of
X
. A mapping
ˀ
: F ₒ[
0
,
1
]
is called a
measure of possibility
, provided:
•
ˀ(
∅
)
= ∅
•
ˀ(
X
)
=
1
•
ˀ(
A
∪
B
)
=
max
(ˀ(
A
), ˀ(
B
))
, for all
A
,
B
in
F
.
Notice that the last property does not require
A
∩
B
= ∅
. Actually, any of these
mappings are fuzzy measures, since:
A
ↂ
B
⃔
A
∪
B
=
B
:
ˀ(
B
)
=
max
(ˀ(
A
), ˀ(
B
))
ˀ(
A
)
,or
ˀ(
A
)
ˀ(
B
).
From
max
(ˀ(
A
), ˀ(
B
))
ˀ(
A
)
+
ˀ(
B
)
,itfollows
ˀ(
A
∪
B
)
ˀ(
A
)
+
ˀ(
B
)
even if
A
∩
B
= ∅
. Hence, possibility measures are sub-additive.
A
C
A
C
A
C
From
A
∪
=
X
,itis1
=
max
(ˀ(
A
), ˀ(
))
ˀ(
A
)
+
ˀ(
)
,or1
−
ˀ(
A
)
A
C
ˀ(
.
Obviously,
)
ˀ(
A
1
∪
A
2
∪
...
∪
A
n
)
=
max
(ˀ(
A
1
), ˀ(
A
2
),...,ˀ(
A
n
))
,
for all
A
1
,...,
A
n
in
F
.
Hence, if
X
={
x
1
,...,
x
n
}
is a finite set, to have a possibility measure
ˀ
, its
ˀ(
x
i
)
values
do verify:
1
=
ˀ(
X
)
=
max
(ˀ(
x
1
), ˀ(
x
2
), . . . , ˀ(
x
n
)),
forcing that some of the values
ˀ(
x
i
)
should equal 1. For example, if
X
={
x
1
,
x
2
,
x
3
}
,
the three values
ˀ(
x
1
)
=
0,
ˀ(
x
2
)
=
0
.
5,
ˀ(
x
3
)
=
1, define a particular mea-
sure of possibility on
P
(
X
)
. It is, for example,
ˀ(
{
x
1
,
x
2
}
)
=
max
(
0
,
0
.
5
)
=
0
.
5,
ˀ(
{
x
1
,
x
3
}
)
=
max
(
0
,
1
)
=
1, etc.
Remark 7.5.1
Instead of a family
of crisp sets no problem arises in considering
a family of fuzzy sets. Measures of possibility can be applied to fuzzy sets with the
only changes of
A
F
μ
˃
. The only caution
is to use the connectives
min
,
max
to preserve distributivity.
μ
˃
, and
A
∩
B
= ∅
by
∩
B
= ∅
by
X
Theorem 7.5.2
For each
μ
∈[
0
,
1
]
such that Sup
μ
=
1
, the mapping
ˀ
μ
: F ₒ
[
0
,
1
]
defined by
ˀ
μ
(
A
)
=
Sup
x
min
(μ(
x
), μ
A
(
x
))
,A
∈ F
,
∈
X
is a possibility measure.
Proof
ˀ
μ
(
∅
)
=
Sup
x
min
(μ(
x
),
0
)
=
0.
ˀ
μ
(
X
)
=
Sup
x
min
(μ(
x
),
1
)
=
∈
X
∈
X
Sup
x
X
μ(
x
)
=
1. Finally, since
μ
A
∪
B
(
x
)
=
max
(μ
A
(
x
), μ
B
(
x
))
for all
x
∈
X
:
∈
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