Information Technology Reference
In-Depth Information
i
∈
K
[
x
i
,
y
i
] = [inf
i
x
i
,
sup
i
y
i
],ifinf
i
x
i
<
inf
i
y
i
≤
sup
i
y
i
= [sup
i
x
i
,
sup
i
y
i
],ifinf
i
x
i
= inf
i
y
i
≤
sup
i
y
i
.
Rest is straightforward as being aclosedset,[0
,
1] contains infimumandsupremumof
x
i
'sand
y
i
's.
ʣ
)
3.3
Graded Consequence: Form (
Now let us define an alternative definition for semantic graded consequence relation
which takes care of the common consensus zone.
Definition 3.3.
Given any collection of interval-valued fuzzy sets, say
{
T
i
}
i
∈
I
,
) =
r
([
i
∈
I
{
x
∈
X
T
i
(
x
)
ʣ
)
gr
(
X
|≈
ʱ
→
I
T
i
(
ʱ
)
}
]),where
r
([
x
1
,
x
2
]) =
x
2
.
...(
follows from
X
' is the right-hand end point of the common interval-value assigned to the sentence
'whenever every member of
X
is true
So, in this definition for graded semantic consequence the valueof'
ʱ
is also true' taking care of every expert's opin-
ion; that is, this method counts the maximumtruth-value assignment where all the ex-
perts agree.
ʱ
Lemma 3.5.
inf
i
r
([
x
i
,
y
i
]
i
∈
K
) =
r
(
i
∈
K
[
x
i
,
y
i
]).
Lemma 3.6.
l
k
I
lk
=
k
l
I
lk
for each
I
lk
∈
U
.
Lemma 3.7.
If
∗
I
is a t-representable t-norm with respect to an ordinary t-norm
∗
then
I
1
∗
I
I
2
≤
I
I
3
implies
r
(
I
1
)
∗
r
(
I
2
)
≤
r
(
I
3
).
[
x
i
,
x
i
]
[
y
i
,
y
i
]
Lemma 3.8.
If
{
}
K
and
{
}
K
are two collections of intervals such that
i
∈
i
∈
K
,then
i
∈
K
[
x
i
,
x
i
]
I
i
∈
K
[
y
i
,
y
i
].
[
x
i
,
x
i
]
I
[
y
i
,
y
i
] for each
i
≤
∈
≤
Proof.
x
i
≤
y
i
and
x
i
≤
y
i
∈
for each
i
K
.
x
i
}
i
∈
K
≤
y
i
}
i
∈
K
and inf
x
i
}
i
∈
K
≤
y
i
}
i
∈
K
.
Hence, sup
{
sup
{
{
inf
{
...(1)
As
x
i
≤
x
i
K
, there are two possibilities - (i) sup
i
x
i
≤
inf
i
x
i
, (ii) sup
i
x
i
>
inf
i
x
i
.
for
i
∈
(i) sup
i
x
i
≤
inf
i
x
i
≤
x
i
≤
y
i
K
.So,sup
i
x
i
≤
inf
i
x
i
≤
inf
i
y
i
.
for each
i
∈
Here again two subcases arise. (ia) sup
i
y
i
≤
inf
i
y
i
and (ib) sup
i
y
i
>
inf
i
y
i
.
inf
i
y
i
,then
i
∈
K
[
y
i
,
y
i
] = [sup
i
y
i
,
inf
i
y
i
].
(ia) If sup
i
y
i
≤
Hence inequalities of (1) ensure that
i
∈
K
[
x
i
,
x
i
]
≤
I
i
∈
K
[
y
i
,
y
i
].
(ib) If sup
i
y
i
>
inf
i
y
i
,then
i
∈
K
[
y
i
,
y
i
] = [inf
i
y
i
,
inf
i
y
i
].
Again from (i) sup
i
x
i
≤
inf
i
y
i
implies
i
∈
K
[
x
i
,
x
i
]
≤
I
i
∈
K
[
y
i
,
y
i
].
inf
i
x
i
≤
(ii) inf
i
x
i
<
sup
i
x
i
implies inf
i
x
i
<
sup
i
x
i
≤
sup
i
y
i
.
So,
i
∈
K
[
x
i
,
x
i
] = [inf
i
x
i
,
inf
i
x
i
]
≤
I
i
∈
K
[
y
i
,
y
i
] since inf
i
x
i
≤
sup
i
y
i
and
inf
i
x
i
≤
inf
i
y
i
.
I
j
}
j
∈
J
of intervals,
j
∈
J
I
j
≤
I
j
∈
J
I
j
.
Lemma 3.9.
For any collection
{
I
j
}
j
∈
J
of intervals,
j
∈
J
I
j
ↆ
e
j
∈
J
I
j
.
Lemma 3.10.
For any collection
{
Corollary 3.1.
r
[
j
∈
J
l
∈
L
I
lj
]
r
[
j
∈
J
l
∈
L
I
lj
].
≤