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(
)
=
.
(
)
=
.
(
)
=
.
For instance, with
p
1
0
4,
p
2
0
5,
p
3
0
1, we have a consistent
(
)
=
.
(
)
=
.
(
)
=
.
probability, as well as with
p
1
0
5,
p
2
0
3,
p
3
0
2. But the probability
given by
p
(
1
)
=
0
.
6,
p
(
2
)
=
0
.
2,
p
(
3
)
=
0
.
2 is not consistent, because of
p
(
2
)<
0
3. With it there is an element whose probability is smaller than its necessity. In the
same vein, the probability given by the triplet
p
.
6, is
also non-consistent because one of the probabilities is greater than the corresponding
possibility.
(
1
)
=
0
.
1,
p
(
2
)
=
0
.
3,
p
(
3
)
=
0
.
7.8 Probability of Fuzzy Sets
Let's shortly formalize the classical concept of probability. In a universe
X
,let
F ↂ P
(
X
)
be a Boolean algebra of parts of
X
, that is,
B
,
A
C
,
B
ↂ
∈ F
•
If
A
,
B
∈ F ⇒
A
∩
B
,
A
∪
.
• ∅ ∈ F
,
X
∈ F
.
It is said that
p
: F ₒ[
0
,
1
]
is a probability in
(
X
,
F
)
, provided
•
p
(
∅
)
=
0
•
If
A
∩
B
=
0, then
p
(
A
∪
B
)
=
p
(
A
)
+
p
(
B
)
.
A
C
Theorem 7.8.1
1.
p
(
)
=
1
−
p
(
A
)
, for all A
∈ F
.
ↂ
(
)
(
)
2.
If A
B, then p
A
p
B
(that is p is a measure)
(
)
=
3.
p
X
1
.
4.
p
(
A
∪
B
)
+
p
(
A
∩
B
)
=
p
(
A
)
+
p
(
B
)
, for all A, B
∈ F
.
Proof
Items (1) and (2) just follow from the fact that
p
is a 0-measure.
p
(
X
)
=
(
∅
ↂ
)
=
p
1
−
p
(
∅
)
=
1. Finally, since
A
∪
B
=
(
A
∩
B
)
∪
(
B
−
A
)
∪
(
A
−
B
),
with
(
A
∩
B
)
∩
(
B
−
A
)
= ∅
,
(
A
∩
B
)
∩
(
A
−
B
)
= ∅
, and
(
B
−
A
)
∩
(
A
−
B
)
= ∅
,
follows
p
(
A
∪
B
)
=
p
(
A
∩
B
)
+
p
(
B
−
A
)
+
p
(
A
−
B
).
But, from
A
=
(
A
−
B
)
∪
(
A
∩
B
)
, and
B
=
(
B
−
A
)
∪
(
A
∩
B
)
, it also follows
(since the unions are disjunct):
•
p
(
A
)
=
p
(
A
−
B
)
+
p
(
A
∩
B
)
⇒
p
(
A
−
B
)
=
p
(
A
)
−
p
(
A
∩
B
)
•
p
(
B
)
=
p
(
B
−
A
)
+
p
(
A
∩
B
)
⇒
p
(
B
−
B
)
=
p
(
B
)
−
p
(
A
∩
B
)
.
Hence,
p
(
A
∪
B
)
=
p
(
A
∩
B
)
+
p
(
A
)
−
p
(
A
∩
B
)
+
p
(
B
)
−
p
(
A
∩
B
)
=
p
(
A
)
+
p
(
B
)
−
p
(
A
∩
B
)
.
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