Image Processing Reference
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with an increasing generating function as
1
λ
f x
(())
μ
=
ln[
1
−μ
( )(
x
1
−
e
)],
λ
>
0
and
μ
() ,
x
∈
⎣
0 1
⎦
λ
With Chaira's intuitionistic fuzzy generator, IFS may be written as
⎧
⎫
⎪
⎪
−
+−
1
μ
()
x
⎪
⎪
IFS
Ax
=
,
μ
( ),
x
x
xX
∈
λ
A
λ
1
(
e
1
) ()
μ
1.6 Intuitionistic Fuzzy Relations
Since 1965, when Zadeh introduced fuzzy set theory, researchers mod-
elled fuzzy relation (FR) on
X
, that is, a function
R
:
X
×
X
→ [0, 1], and
used FR in different fields. In 1983, when Atanassov introduced IFS, many
researchers extended FR using IFS and modelled intuitionistic fuzzy rela-
tions (IFRs) on
X
. Burillo and Bustince [2-4] had given the definition of
IFRs and their properties. Lei et al. [11] further explored IFRs and its com-
positional operations. To fulfil the properties of IFRs,
t
-norm and
t
-conorm
are required.
Let
X
and
Y
be two universes of discourse and IFS(
X
×
Y
) represent the
family of all IFSs in
X
×
Y
. Let
R
∈ IFS(
X
×
Y
) be the IFR, which is a subset in
X
×
Y
, given as
Rxy
=
{( , , (, , (,)|,
μ
x y
ν
x yxXy Y
∈
∈
}
R
R
where
μ
R
XY
:
×→
01
[ ,]
ν
R
XY
:
×→
01
[ ,]
denote the membership and non-membership functions of
R
, respectively,
that satisfy the condition 0 ≤ μ
R
(
x
,
y
) + ν
R
(
x
,
y
) ≤ 1 and π
R
(
x
,
y
) = 1
−
μ
R
(
x
,
y
)
−
ν
R
(
x
,
y
) is the hesitation index.
The complementary relation of
R
is
Rx y
=
{( , , (, , (, |( ,)
ν
x y
μ
x yxyXY
∈ ×
}
c
R
R
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