Information Technology Reference
In-Depth Information
There are many possibilities for characterizations of operations with sets (union
A
∪
B
and
∩
intersection
A
B
). We shall use so called Lukasiewicz characterization:
χ
A
∪
B
=(
χ
A
+
χ
B
)
∧
1,
χ
A
∩
B
=(
χ
A
+
χ
B
−
1
)
∨
0.
(Here
(
f
∨
g
)(
ω
)=
max
(
f
(
ω
)
,
g
(
ω
))
,
(
f
∧
g
)(
ω
)=
min
(
f
(
ω
)
,
g
(
ω
))
.
)
Hence if
ϕ
A
,
ϕ
B
:
Ω
→
[
]
0, 1
are fuzzy sets, then the union (disjunction
ϕ
A
or
ϕ
B
of corresponding assertions) can be
defined by the formula
ϕ
A
⊕
ϕ
B
=(
ϕ
A
+
ϕ
B
−
1
)
∧
1,
the intersection (conjunction
ϕ
A
and
ϕ
B
of corresponding assertions) can be defined by the
formula
ϕ
A
ϕ
B
=(
ϕ
A
+
ϕ
B
−
)
∨
1
0.
In the chapter we shall work with a natural generalization of the notion of fuzzy set so-called
IF-set (see [1], [2]), what is a pair
=(
μ
A
,
ν
A
)
Ω
→
[
]
×
[
]
A
:
0, 1
0, 1
μ
A
,
ν
A
:
Ω
→
[
]
of fuzzy sets
0, 1
, where
μ
A
+
μ
A
≤
1.
Evidently a fuzzy set
ϕ
A
:
Ω
→
[
0, 1
]
can be considered as an IF-set, where
μ
A
=
ϕ
A
:
Ω
→
[
0, 1
]
,
ν
A
=
1
−
ϕ
A
:
Ω
→
[
0, 1
]
.
Here we have
μ
A
+
ν
A
=
1,
while generally it can be
μ
A
(
ω
)+
ν
A
(
ω
)
<
1 for some
ω
∈
Ω
. Geometrically an IF-set can be
regarded as a function
A
:
Ω
→
Δ
to the triangle
R
2
Δ
=
{
(
u
,
v
)
∈
:0
≤
u
,0
≤
v
,
u
+
v
≤
1
}
.
Fuzzy set can be considered as a mapping
ϕ
A
:
Ω
→
D
to the segment
R
2
;
u
=
{
(
)
∈
+
=
≤
≤
}
D
u
,
v
v
1, 0
u
1
Ω
→
and the classical set as a mapping
ψ
:
D
0
from
Ω
to two-point set
=
{
(
)
(
)
}
D
0
0, 1
,
1, 0
.
In the next definition we again use the Lukasiewicz operations.
Definition 1.1.
By an IF subset of a set
Ω
a pair
A
=(
μ
A
,
ν
A
)
of functions
μ
A
:
Ω
→
[
0, 1
]
,
ν
A
;
Ω
→
[
0, 1
]
is considered such that
μ
A
+
ν
A
≤
1.
