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2.9 Direct Cluster Analysis Based on Intuitionistic
Fuzzy Implication
2.9.1 The Intuitionistic Fuzzy Implication Operator
and Intuitionistic Fuzzy Products
Definition 2.35
(Kohout and Bandler 1980, 1984) Let
U
i
(
i
=
1
,
2
)
be two ordinary
subsets, and
L
⊂
U
1
×
U
2
an ordinary relation. Then for any
a
,
b
∈
U
2
,
Lb
={
a
|
aLb
}
and
aL
={
b
|
aLb
}
are respectively called a former set and a latter set.
Definition 2.36
(Kohout and Bandler 1980, 1984) Let
U
i
(
i
=
1
,
2
,
3
)
be ordinary
subsets,
L
1
⊂
U
1
×
U
2
and
L
2
⊂
U
2
×
U
3
, then a triangle product
L
1
L
2
⊂
U
1
×
U
3
of
L
1
and
L
2
can be defined as:
aL
1
L
2
c
⇔
aL
1
⊂
L
2
c
,
for any
(
a
,
c
)
∈
U
1
×
U
2
(2.196)
Similarly, a square product
L
1
L
2
is defined as:
aL
1
⇔
aL
1
=
,
(
,
)
∈
×
L
2
c
L
2
c
for any
a
c
U
W
(2.197)
where
aL
1
=
L
2
c
if and only if
aL
1
⊂
L
2
c
and
aL
1
⊃
L
2
c
.
Wang and Liu (1999) introduced a fuzzy implication operator as follows:
Definition 2.37
(Wang and Liu 1999) Let
I
1
be a binary operation on
[
0
,
1
]
,if
I
1
(
0
,
0
)
=
I
1
(
0
,
1
)
=
I
1
(
1
,
1
)
=
1 and
I
(
1
,
0
)
=
0
(2.198)
then
I
1
is called a fuzzy implication operator.
For any
a
,
b
∈[
0
,
1
]
,
I
1
(
a
,
b
)
is a fuzzy implication operator, which can also be
denoted as
a
→
b
. Especially, the well-known Lukasiewicz implication operator is
given as
ϕ(
a
,
b
)
=
min
(
1
−
a
+
b
,
1
)
, which means that the result of “
a
imply
b
”is
min
.
Motivated by the idea of Definition 2.37, Wang et al. (2012) defined the concept
of intuitionistic fuzzy implication operator:
(
1
−
a
+
b
,
1
)
Definition 2.38
(Wang et al. 2012) Let
I
1
be a binary operation on the set of all
IFVs,
V
,if
I
1
((
0
,
1
), (
0
,
1
))
=
I
1
((
0
,
1
), (
1
,
0
))
=
I
1
((
1
,
0
), (
1
,
0
))
=
(
1
,
0
),
I
1
((
1
,
0
), (
0
,
1
))
=
(
0
,
1
)
then
I
1
is called an intuitionistic fuzzy implication operator.
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