Information Technology Reference
In-Depth Information
<
d
or
d
≤
<
d
,
C
ʵ
([
c
,
d
])=[
c
+
,
d
].So,
Now either
c
+
ʵ
c
+
ʵ
.If
c
+
ʵ
ʵ
[
b
,
b
]
≤
,
d
].Andif
d
≤
,then
C
ʵ
([
c
,
d
])=[
d
,
d
] i.e., [
b
,
b
]
≤
,
d
].
I
[
c
+
ʵ
c
+
ʵ
I
[
c
+
ʵ
Hence [
a
,
b
]
I
C
ʵ
([
c
,
d
]).
Now we shall check the case for
C
ʵ
.Let[
a
,
b
]
≤
I
[
c
,
d
] implies
C
ʵ
([
a
,
b
])
≤
≤
I
[
c
,
d
].
There are two cases. (a)
C
ʵ
([
a
,
b
])=[
a
,
b
−
ʵ
] (b)
C
ʵ
([
a
,
b
])=[
a
,
a
].
(a) If
C
ʵ
([
a
,
b
])=[
a
,
b
−
ʵ
],then
a
<
b
−
ʵ
<
b
≤
d
and
b
−
ʵ
≤
d
−
ʵ
.
Now either
c
<
d
−
ʵ
or
d
−
ʵ
≤
c
.
If
c
<
d
−
ʵ
,
C
ʵ
([
c
,
d
])=[
c
,
d
−
ʵ
]. i.e., [
a
,
b
−
ʵ
]
≤
I
[
c
,
d
−
ʵ
] as
a
≤
c
,
b
≤
d
.
If
d
−
ʵ
≤
c
,
C
ʵ
([
c
,
d
])=[
c
,
c
],So,[
a
,
b
−
ʵ
]
≤
I
[
c
,
c
] as
a
≤
c
,
b
−
ʵ
≤
d
−
ʵ
≤
c
.
(b) Let
C
ʵ
([
a
,
b
])=[
a
,
a
].Then
b
−
ʵ
≤
a
≤
b
≤
d
,
a
≤
c
.
Now either
c
<
d
−
ʵ
or
d
−
ʵ
≤
c
.
If
c
<
d
−
ʵ
,
C
ʵ
([
c
,
d
])=[
c
,
d
−
ʵ
]. i.e., [
a
,
a
]
≤
I
[
c
,
d
−
ʵ
] since
a
≤
c
<
d
−
ʵ
.
And if
d
−
ʵ
≤
c
,
C
ʵ
([
c
,
d
])=[
c
,
c
].So,[
a
,
a
]
≤
I
[
c
,
c
].
Therefore, [
a
,
b
]
≤
I
C
ʵ
([
c
,
d
]).
Hence,
C
ʵ
([
a
,
b
])=
C
ʵ
([
a
,
b
])
∩
C
ʵ
([
a
,
b
])
≤
I
C
ʵ
([
c
,
d
])
∩
C
ʵ
([
c
,
d
])=
C
ʵ
([
c
,
d
]).
≤
I
[
c
,
d
] implies
C
ʵ
([
a
,
b
])
≤
I
[
c
,
d
] implies
C
I
(
I
ʵ
,
j
≤
I
C
I
(
I
ʵ
,
j
Corollary 3.2.
[
a
,
b
]
[
a
,
b
]
)
[
c
,
d
]
).
≤
I
[
c
,
d
] implies
C
I
(
I
[
a
,
b
]
)
≤
I
C
I
(
I
[
c
,
d
]
).
Theorem 3.6.
[
a
,
b
]
([
a
,
b
]),
C
j
Proof.
Let
C
i
([
c
,
d
])bedegenerate. Then either (i)
i
=
j
or (ii)
i
≤
j
, or (iii)
ʵ
ʵ
i
.(i)If
i
=
j
, by Theorem 3.5,
C
I
(
I
[
a
,
b
]
)=
C
i
≤
I
C
i
([
c
,
d
])=
C
I
(
I
[
c
,
d
]
). (ii) By
j
≤
([
a
,
b
])
ʵ
ʵ
l
(
C
j
Theorem 3.5,
l
(
C
I
(
I
[
a
,
b
]
)) =
l
(
C
i
([
c
,
d
])) =
l
(
C
I
(
I
[
c
,
d
]
)).
That is,
C
I
(
I
[
a
,
b
]
)
≤
I
C
I
(
I
[
c
,
d
]
). (iii) If
j
≤
i
,
r
(
C
I
(
I
[
c
,
d
]
)) =
r
(
C
j
l
(
C
i
([
a
,
b
]))
≤
([
c
,
d
]))
≤
ʵ
ʵ
ʵ
r
(
C
j
([
c
,
d
]))
≥
([
a
,
b
])).
ʵ
ʵ
By Note 3.2,
r
(
C
j
([
a
,
b
]))=
r
(
C
i
([
a
,
b
])) =
r
(
C
I
(
I
[
a
,
b
]
)) i.e.,
C
I
(
I
[
a
,
b
]
)
≤
I
C
I
(
I
[
c
,
d
]
).
ʵ
ʵ
Thus we have shown existence of a
C
I
which satisfies conditions of Definition 3.5.
Nowwecomebacktothegeneral context of the Definition 3.5 for
C
I
.
Note 3.4.
C
I
(
I
j
[
a
,
b
]
),
C
I
(
I
[
a
,
b
]
)
ↆ
e
[
a
,
b
],and
C
I
(
I
j
[
a
,
b
]
),
C
I
(
I
[
a
,
b
]
), [
a
,
b
] are mutually over-
lapping pairs of intervals. And for
I
ʵ
,
C
I
(
I
[
a
,
b
]
)
ↆ
e
C
I
(
I
ʵ
,
j
[
a
,
b
]
)
ↆ
e
[
a
,
b
]
We now propose different notions of
|≈
based on the notion of iterative revision.
Definition 3.9.
Given any collection of interval-valued fuzzy sets
{
T
i
}
i
∈
I
,
ʣ
1
)
C
I
(
I
j
ʣ
1
)
gr
(
X
(
(
|≈
ʱ
) =
r
(
∩
i
∈
I
{
)]
)
}
) for an arbitrarily fixed
j
≥
0,
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(
ʱ
ʣ
2
)
ʣ
2
)
gr
(
X
(
C
I
(
I
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(ʱ)]
)
}
) =
l
(
∩
i
∈
I
{
C
I
(
I
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(ʱ)]
)
}
).
(
|≈
ʱ
) =
r
(
∩
i
∈
I
{
Theorem 3.7.
For any graded consequence relation
|∼
, there is a collection interval-
ʣ
1
), (
ʣ
2
) coincide with
valued fuzzy sets such that
|≈
generated in the sense of (
|∼
.
Proof.
Given
|∼
,agraded consequence relation, let us consider
{
T
X
}
X
∈
P
(
F
)
such that
T
X
)=
r
[
Y
∈
P
(
F
)
{
C
I
(
I
[
x
∈
X
T
Y
(
x
)
→
I
T
Y
(ʱ)]
)
}
]
(
ʱ
)=[
gr
(
X
|∼
ʱ
)]
BIR
. We want to prove
gr
(
X
|∼
ʱ
