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μ
(
u
)
μ
(
u
)
and
are relative membership degree (RMD) function that express
A
c
A
μ
(
u
)
μ
(
u
)
degrees of attractability and repellency respectively. We have
+
=1.
A
c
A
≤
μ
(
u
)
≤
≤
μ
(
u
)
≤
Here 0
1, 0
1.
A
c
A
Du
=
()
μ
(
u
)
μ
(
u
)
Du
is defined as relative difference
()
Let
-
, Where
A
c
A
degree of u to
A
. Mapping
Du D
→∈−
:
( )
[ 1,1]
is defined as relative difference
function of u to
A
. Then
Du
() 2 () 1
A
=
μ
u
−
μ
A
u u
() 1
=+
()2
,or
.Let
{
}
V
=
(, )
u Du UDu
∈
, ()
=
μ
()
u
−
μ
(),
u D
∈ −
[1,1]
0
A
c
A
{
}
Au u UDu
+
=
∈<
,0
( )
<
1
{
}
Au u U Du
−
=
∈−
,1
<
() 0
<
{
}
Au u UDu
=
∈
,
() 0
=
0
Here
V
is defined as variable fuzzy sets(VFS),
A
+
,
A
−
, and
A
are defined as at-
tracting sets, repelling sets and balance boundary or qualitative change boundary of
VFS
V
.
2.2 Methods of Relative Difference Function
X
= [a, b] are attracting sets of VFS
V
on real axis, i.e. interval of
We suppose that
,
X
′
X
′
μ
()
u
>
μ
()
u
X
, i.e.
X
⊂
= [c, d] is a certain interval containing
.
A
C
0
A
According to definition of VFS we know that interval [c, a] and [b, d] are all repel-
ling sets of VFS, i.e. interval of
μ
()
u
<
μ
()
u
. Suppose that M is point value of
D(u)=1 in attracting sets [a, b].
x
is a random value in interval
X
A
c
A
′
, then if
x
locates
at left side of
M
, its difference function is
xa
−
⎧
Dx
() (
=
)
β
x aM
∈
[,
]
⎪
⎪
Ma
xa
−
(1)
⎨
−
⎪
=−
β
∈
Dx
()
(
)
x ca
[, ]
⎪
ca
−
⎩
xa
−
⎧
β
μ
() 0.5[1 (
x
=
+
)]
x
∈
[,
a M
]
⎪
⎪
or
Ma
xa
−
(2)
⎨
−
⎪
β
μ
() 0.5[1 (
x
=
−
)]
x
∈
[, ]
c a
⎪
⎩
ca
−
And if
x
locates at right side of M, its difference function is
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