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Now we extend Lukasiewicz implication operator to intuitionistic fuzzy environ-
ment. For any two IFVs
α
=
(μ
α
,
v
α
)
β
=
(μ
β
,
v
β
)
and
, if we only consider the
membership degrees
cannot reflect
the superiority of IFVs, so we should consider the non-memberships
v
μ
α
and
μ
β
of
α
and
β
, then min
{
1
−
μ
α
+
μ
β
,
1
}
as
well. Then based on the components of IFVs and the form of Lukasiewicz implica-
tion operator, Wang et al. (2012) defined an intuitionistic fuzzy Lukasiewicz impli-
cation operator
and
v
α
β
ϕ(α, β)
, whose membership degree and non-membership degree are
expressed as:
min
{
1
,
min
{
1
−
μ
α
+
μ
β
,
1
−
v
β
+
v
α
}} =
min
{
1
,
1
−
μ
α
+
μ
β
,
1
−
v
β
+
v
α
}
and
max
{
0
,
min
{
1
−
(
1
−
μ
α
+
μ
β
),
1
−
(
1
−
v
β
+
v
α
)
}} =
max
{
0
,
min
{
μ
α
−
μ
β
,
v
β
−
v
α
}}
respectively, i.e.,
ϕ(α, β)
=
min
v
α
}}
(2.199)
{
1
,
1
−
μ
α
+
μ
β
,
1
−
v
β
+
v
α
}
,
max
{
0
,
min
{
μ
α
−
μ
β
,
v
β
−
Clearly, we need to prove that the value of
ϕ(α, β)
should satisfy all the conditions
of an IFV. In fact, from Eq. (
2.199
), we have
{
,
−
μ
α
+
μ
β
,
−
v
β
+
v
α
}≥
,
{
,
{
μ
α
−
μ
β
,
v
β
−
v
α
}} ≥
min
1
1
1
0
max
0
min
0 (2.200)
and since
max
{
0
,
min
{
μ
α
−
μ
β
,
v
β
−
v
α
}} =
1
−
min
{
1
,
max
{
1
−
μ
α
+
μ
β
,
1
−
v
β
+
α
}}
(2.201)
v
min
{
1
,
max
{
1
−
μ
α
+
μ
β
,
1
−
v
β
+
v
α
}} ≥
min
{
1
,
1
−
μ
α
+
μ
β
,
1
−
v
β
+
v
α
}
(2.202)
then
1
−
min
{
1
,
max
{
1
−
μ
α
+
μ
β
,
1
−
v
β
+
v
α
}} +
min
{
1
,
1
−
μ
α
+
μ
β
,
1
−
v
β
+
v
α
}≤
1
which indicates that the value of
ϕ(α, β)
derived by Eq. (
2.201
)isanIFV.
Example 2.14
(Wang et al. 2012) Let
α
=
(
0
,
1
)
and
β
=
(
1
,
0
)
, then byEq. (
2.198
),
we have
ϕ(α, α)
=
(
min
{
1
,
1
−
0
+
0
,
1
−
1
+
1
}
,
max
{
0
,
min
{
0
−
0
,
1
−
1
}}
)
=
(
1
,
0
)
ϕ(β, β)
=
(
{
,
−
+
,
−
+
}
,
{
,
{
−
,
−
}}
)
=
(
,
)
min
1
1
1
1
1
0
0
max
0
min
1
1
0
0
1
0
ϕ(α, β)
=
(
min
{
1
,
1
−
0
+
1
,
1
−
0
+
1
}
,
max
{
0
,
min
{
0
−
1
,
0
−
1
}}
)
=
(
1
,
0
)
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