Information Technology Reference
In-Depth Information
In [13], definitions of 138 operations “implication” and 34 operations “negation”
are given. In Table
2.1
the currently exist
in
g 45 negations are given. In some of these
definitions, we use the functions sg and sg that are defined by:
⎧
⎨
⎧
⎨
1if
x
>
0
0if
x
>
0
sg
(
x
)
=
0
,
sg
(
x
)
=
⎩
⎩
0if
x
≤
1if
x
≤
0
Let a set
E
be fixed. An Intuitionistic Fuzzy Set (IFS)
A
in
E
is an object of the
following form (see, e.g., [7, 13]):
A
={
x
,μ
A
(
x
), ν
A
(
x
)
|
x
∈
E
}
,
where functions
define the degree of member-
ship and the degree of non-membership of the element
x
μ
A
:
E
→[
0
,
1
]
and
ν
A
:
E
→[
0
,
1
]
∈
E
, respectively, and for
every
x
∈
E
:
≤
μ
A
(
)
+
ν
A
(
)
≤
.
0
x
x
1
2.2 IFIMs and EIFIMs
Let
I
be a fixed set. By IFIM with index sets
K
and
L
(
K
,
L
⊂
I
)
, we denote the
object:
[
K
,
L
,
{
μ
k
i
,
l
j
,ν
k
i
,
l
j
}]
l
1
...
l
j
...
l
n
k
1
μ
k
1
,
l
1
,ν
k
1
,
l
1
...
μ
k
1
,
l
j
,ν
k
1
,
l
j
...
μ
k
1
,
l
n
,ν
k
1
,
l
n
.
.
.
.
. . .
. . .
≡
,
k
i
μ
k
i
,
l
1
,ν
k
i
,
l
1
...
μ
k
i
,
l
j
,ν
k
i
,
l
j
...
μ
k
i
,
l
n
,ν
k
i
,
l
n
.
.
.
.
. . .
. . .
k
m
μ
k
m
,
l
1
,ν
k
m
,
l
1
...
μ
k
m
,
l
j
,ν
k
m
,
l
j
...
μ
k
m
,
l
n
,ν
k
m
,
l
n
where for every 1
≤
i
≤
m
,
1
≤
j
≤
n
:0
≤
μ
k
i
,
l
j
,ν
k
i
,
l
j
,μ
k
i
,
l
j
+
ν
k
i
,
l
j
≤
1
.
For brevity, we can mention the above object by
[
K
,
L
,
{
μ
k
i
,
l
j
,ν
k
i
,
l
j
}]
,
where
K
={
k
1
,
k
2
,...,
k
m
}
,
L
={
l
1
,
l
2
,...,
l
n
}
,
for 1
≤
i
≤
m
,
and 1
≤
j
≤
n
:
μ
k
i
,
l
j
,ν
k
i
,
l
j
,μ
k
i
,
l
j
+
ν
k
i
,
l
j
∈[
0
,
1
]
.
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