Information Technology Reference
In-Depth Information
We call
μ
A
the membership function,
ν
A
the non membership function and
A
≤
B
⇐⇒
μ
A
≤
μ
B
,
ν
A
≥
ν
B
.
If
A
=(
μ
A
,
ν
A
)
,
B
=(
μ
B
,
ν
B
)
are two IF-sets, then we define
⊕
=((
μ
A
+
μ
B
)
∧
(
ν
A
+
ν
B
−
)
∨
)
A
B
1,
1
0
,
=((
μ
A
+
μ
B
−
)
∨
(
ν
A
+
ν
B
)
∧
)
A
B
1
0,
1
,
¬
=(
− μ
A
,1
− ν
A
)
A
1
.
Denote by
F
a family of IF sets such that
∈F
=
⇒
⊕
∈F
∈F
¬
∈F
A
,
B
A
B
,
A
B
,
A
.
F
Ω
→
[
]
Example 1.1.
Let
be the set of all fuzzy subsets of a set
Ω
.If
f
:
0, 1
then we define
=(
−
)
A
f
,1
f
,
i.e.
−
μ
A
.
Example 1.2.
ν
A
=
1
Let
(
Ω
S
)
S
σ
F
,
be a measurable space,
a
-algebra,
the family of all pairs such
Ω
→
[
]
Ω
→
[
]
F
that
μ
A
:
0, 1
,
ν
A
:
0, 1
are measurable. Then
is closed under the operations
⊕
¬
,
,
.
Example 1.3.
Let
(
Ω
,
T
)
be a topological space,
F
the family of all pairs such that
μ
A
:
Ω
→
[
0, 1
]
,
ν
A
:
Ω
→
[
0, 1
]
are continuous. Then
F
is closed under the operations
⊕
,
,
¬
.
Remark.
Of course, in any case
A
⊕
B
,
A
B
,
¬
A
are IF-sets, if
A
,
B
are IF-sets. E.g.
A
⊕
B
=((
μ
A
+
μ
B
)
∧
1,
(
ν
A
+
ν
B
−
1
)
∨
0
)
,
hence
(
μ
A
+
μ
B
)
∧
1
+(
ν
A
+
ν
B
−
1
)
∨
0
=
=((
μ
A
+
μ
B
)
∧
1
+(
ν
A
+
ν
B
−
1
))
∨
((
μ
A
+
μ
B
)
∧
1
)=
=((
μ
A
+
μ
B
+
ν
A
+
ν
B
−
)
∧
(
+
ν
A
+
ν
B
−
))
∨
((
μ
A
+
μ
B
)
∧
)
≤
1
1
1
1
≤
((
+
−
)
∧
(
ν
A
+
ν
B
))
∨
((
μ
A
+
μ
B
)
∧
)=
1
1
1
1
=(
1
∧
(
ν
A
+
ν
B
))
∨
((
μ
A
+
μ
B
)
∧
1
)
≤
≤
1
∨
1
=
1.
Probably the most important algebraic model of multi-valued logic is an MV-algebra
([48],[49]). MV-algebras play in multi-valued logic a role analogous to the role of Boolean
algebras in two-valued logic. Therefore we shall present a short information about MV-alegras
and after it we shall prove the main result of the section: a possibility to embed the family of
IF-sets to a suitable MV-algebra.
Let us start with a simple example.
