Information Technology Reference
In-Depth Information
) endorses the minimumtruth value assignment ensuring the limit, be-
low which none of the assignments for 'if every member of
X
is truethen
Definition (
ʣ
ʱ
is true' i.e.,
x
∈
X
T
i
(
x
)
→
), considering expert's opinion, lie. As an instance let, outoftwo
experts the value assignment to
x
∈
X
T
i
(
x
)
I
T
i
(
ʱ
→
I
T
i
(
ʱ
) following one's opinion is [
.
1
,.
3]
and that of the other is [
.
7
,.
9]. Then following (
follows from
X
' only
considers the left-hand end point of the least interval i.e. [
.
1
,.
3]. So, this value assign-
ment does not take care of the consensus of all. It only emphasises that the lower bound
of everyone's point of agreeing is
.
1, no matter whether someone really has marked a
high grade.
ʣ
)thevaluefor'
ʱ
3.2
Extension of
ↆ
as a Lattice Order Relation
Let us now explore a method of assigning the valueto'
follows from
X
' in such a way
that it takes care of every individual's opinion. That is, we are looking for an interval
which lie in the intersection of everyone's opinion. For this we need a complete lattice
structure with respect to the order relation
ʱ
on
U
.Letus extend the partially ordered
ↆ
relation
ↆ
into a lattice order by the following definition.
ↆ
e
is a binary relation on
U
defined as below.
Definition 3.2.
[
x
1
,
x
2
]
ↆ
e
[
y
1
,
y
2
] if
y
1
≤
x
1
<
x
2
≤
y
2
,
[
x
1
,
x
1
]
ↆ
e
[
y
1
,
y
2
] if
x
1
≤
y
2
.
Proposition 3.1.
If [
x
1
,
x
2
]
ↆ
[
y
1
,
y
2
] then [
x
1
,
x
2
]
ↆ
e
[
y
1
,
y
2
] .
Note 3.1.
The converse of the above proposition does not hold. For the intervals [
.
2
,.
2]
and [
.
3
,.
7], [
.
2
,.
2]
ↆ
e
[
.
3
,.
7],but [
.
2
,.
2]
[
.
3
,.
7]. Also to be noted that for two intervals
[
x
1
,
x
2
] and [
y
1
,
y
2
], [
x
1
,
x
2
]
ↆ
e
[
y
1
,
y
2
] does not hold if
x
1
<
x
2
and
y
1
=
y
2
.Let[
x
1
,
x
2
]
ↆ
e
[
y
1
,
y
2
] be such that
x
1
=
x
2
<
y
1
≤
y
2
holds. This pair of intervals are called non-
overlapping intervals under the relation
ↆ
e
; other pairs of intervals under the relation
ↆ
e
are known as overlapping intervals under the relation
ↆ
e
.
Proposition 3.2.
(
U
,ↆ
e
) forms a poset.
Proposition 3.3.
(
U
,ↆ
e
) is a lattice where the greatest lower bound, say
,andthe
least upper bound, say
are defined as follows.
[
x
1
,
x
2
]
[
y
1
y
2
] = [max(
x
1
,
y
1
)
,
min(
x
2
,
y
2
)] if max(
x
1
,
y
1
)
≤
min(
x
2
,
y
2
)
= [min(
x
2
,
y
2
)
,
min(
x
2
,
y
2
)],otherwise.
[
x
1
,
x
2
]
[
y
1
y
2
] = [max(
x
2
,
y
2
)
,
max(
x
2
,
y
2
)],if
x
1
=
x
2
,
y
1
=
y
2
.
= [max(
x
1
,
y
1
)
,
max(
x
2
,
y
2
)],if
x
1
=
x
2
<
y
1
<
y
2
= [min(
x
1
,
y
1
)
,
max(
x
2
,
y
2
)],otherwise.
(i.e. either
x
1
=
x
2
,
y
1
<
y
2
,
y
1
≤
x
1
,or
x
1
<
x
2
,
y
1
<
y
2
)
Proposition 3.4.
(
U
,ↆ
e
) forms a complete lattice.
Proof
For arbitrary collection
}
i
∈
K
,
i
∈
K
[
x
i
,
y
i
] = [sup
i
x
i
,
inf
i
y
i
] if sup
i
x
i
≤
{
[
x
i
,
y
i
]
inf
i
y
i
= [inf
i
y
i
,
inf
i
y
i
],otherwise.
