Image Processing Reference
In-Depth Information
The following definitions hold for all IFSs
A
and
B
of set
X
:
1.
A
∪
B
= {max(μ
A
, μ
B
), min
(,
ν
AB
}
)
2.
A
∩
B
= {min(μ
A
, μ
B
), max
(,
ν
AB
}
)
3.
A
≺
B
= {
x
, μ
A
(
x
) < μ
B
(
x
),
ν
A
(
x
) <
ν
B
(
x
)}
4.
Ax x
=
{, (),
ν μ
( )}
x
A
A
5.
A
≤
B
= {
x
, μ
A
(
x
) ≤ μ
B
(
x
),
ν
A
(
x
) ≥
ν
B
(
x
)}
6.
A
⋅
B
= {
x
, μ
A
(
x
) ⋅ μ
B
(
x
),
ν
A
(
x
) +
ν
B
(
x
)
−
ν
A
(
x
) ⋅
ν
B
(
x
)}
1.3 Some Operations on Intuitionistic Fuzzy Sets
Similar to fuzzy sets, IFSs have also undergone some operations. Operations
on IFS have been carried out by many authors [8,16,18]. The negation 'NOT';
the connectives 'AND' and 'OR'; hedges 'VERY', 'MORE OR LESS', 'VERY
VERY' and 'VERY HIGHLY'; and other terms are used to represent lin-
guistic variables. For two IFSs
A
and
B
, with μ(
x
) and
ν
(
x
as the member-
ship and non-membership degrees of the elements in two sets, the following
conditions hold:
ABAB x
and
=∧=
{,min( (), ()), max( (), ())}
μμ ν
x
x
x
ν
x
A
B
A
B
ABAB x
or
=∨=
{,max( (), ()), min( (), ())}
μμ ν
x
x
x
ν
x
A
B
A
B
The product of the two IFSs
A
and
B
is given by
AB x
⋅=
{( ,
μμν
( )
x
⋅
( ), () () () ( ))}
x
x
+
ν
x
−
ν
x
⋅
ν
x
A
B
A
B
A
B
2
2
2
AAAx x
=⋅
=
{,[()] ,
μ
11
− −
(
ν
())}
x
A
A
It is also called concentration of set
A
, CON(
A
):
12
/
12
/
12
/
AAAx x
=⋅
=
{,[()]
μ
,
11
−
(
−
ν
())}
x
A
A
or the dilation of set
A
, that is, DIL(
A
):
3
3
3
Ax x
=
{,[()] ,
μ
11
−
(
−
ν
())}
x
A
A
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