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A
B
=((
μ
A
+
μ
B
)
∧
1,
(
ν
A
+
ν
B
)
∧
1
)
;
=((
μ
A
+
μ
B
−
)
∨
(
ν
A
+
ν
B
−
)
∨
)
A
B
1
0,
1
0
.
V
V→
[
]
Denote by
the family of all IV-events. By an IV-state a map
m
:
0, 1
is considered such
that the following properties are satisfied:
(i)
m
((
0, 0
)) =
0,
m
((
1, 1
)) =
1;
(ii)
A
B
=(
0, 0
)=
⇒
m
(
A
B
)=
m
(
A
)+
m
(
B
)
;
(iii)
A
n
A
=
⇒
m
(
A
n
)
m
(
A
)
.
Theorem 6.1.
V
Ω
S
))
→
[
]
Let
be the family of all IV-events (with respect to
,
,
m
:
0, 1
be an
IV-state. Define
F
=
{
(
μ
A
,1
− ν
A
)
(
μ
A
,
ν
A
)
∈V}
;
,
m
:
F→
[
0, 1
]
,
m
((
μ
A
,
ν
A
)) =
1
−
m
(
μ
A
,1
−
ν
A
))
,
ϕ
:
V→F
,
ϕ
((
μ
A
,
ν
A
)) = (
μ
A
,1
−
ν
A
)
.
F
(
Ω
S
)
ϕ
Then
is the family of all IF-events (with respect to
,
,
m
is an IF-state and
is an
isomorphism such that
ϕ
((
)) = (
)
ϕ
((
)) = (
)
0, 0
0, 1
,
1, 0
1, 1
,
ϕ
(
A
B
)=
ϕ
(
A
)
ϕ
(
B
)
,
ϕ
(
A
B
)=
ϕ
(
A
)
⊕
ϕ
(
B
)
,
ϕ
(
¬
A
)=
¬
ϕ
(
A
)
,
m
(
A
)) =
m
(
ϕ
(
A
))
,
A
∈V
.
Proof. It is almost straightforward. Of course, the using of the family
V
is more natural and
F
the results can be applied immediately to probability theory on
.
8. Conclusion
The structures studied in this chapter have two aspects: the first one is practical, the second
theoretical one. Fuzzy sets and their generalization - Atanassov intuitionistic fuzzy sets - in
both directions new possibilities give.
From the practical point of view we can recommend e. g. [1], [9], [69]. Of course, the whole IF
- theory can be motivated by practical problems and applications (see[10],[44 - 46], [53]).
The main contribution of the presented theory is a new point of view on human thinking
and creation. We consider algebraic models for multi valued logic: IF-events, and more
generally MV-algebras, D-posets, and effect algebras. They are important for many valued
logic as Boolean algebras for two valued logic. Of course, we presented also some results
about entropy ([11], [12], [40 - 42], [59]), or inclusion - exclusion principle ([6], [26], [30])for an
illustration. But the more important idea is in building the probability theory on IF-events.
The theoretical description of uncertainty has two parts in the present time :
objective -
probability and statistics, and subjective - fuzzy sets.
We show that both parts can be
considered together.
