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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 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 ].
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