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()
()
1
μ
x
α
;
~
()
μ α
x
=
A
A
0
,
μ
x
<
α
.
The decomposition forms the basis for a method of fuzziness formalization.
According to this method, the fuzziness is expressed by means of a collection of
hierarchically ordered definite sets.
Let us define the basic set-theoretic operations at set
~
A
()
H
X
of all fuzzy subsets
of a definite set X .
Fuzzy sets ~ and ~ are equal, if for all
()
()
μ
x
μ
x
=
x
X
the condition
~
~
A
B
is satisfied.
The fuzzy set ~ belongs to fuzzy set ~ ,
~
~
()
()
μ
x
μ
x
A
B
, if
;
x
X
.
~
~
A
B
The fuzzy set ~ is referred to as complement of fuzzy set ~ , if
()
()
μ
x
=
1
μ
x
;
x
X
.
The fuzzy set ~ is referred to as intersection of fuzzy sets ~ and ~ ,
~
~
A
A
~
~
~
, if
C
=
A
B
()
()
()
μ
x
=
μ
x
μ
x
;
, where
is an operator from triangular norm class.
x
X
~
~
~
C
A
B
[][] []
Valid binary function
is referred to as triangular norm, this
T
:
0
×
0
0
function satisfies the following conditions:
()
(
)
(
)
1.
T
0
=
0
T
μ
,
=
T
1
μ
=
μ
(boundedness);
~
~
~
A
A
A
(
)
(
)
T
μ
,
μ
T
μ
,
μ
μ
μ
μ
μ
2.
, if
;
(monotonicity property);
~
~
~
~
~
~
~
~
A
B
C
D
A
C
B
D
(
)
(
)
T
μ
,
μ
=
T
μ
,
μ
3.
(commutativity);
~
~
~
~
A
B
B
A
[
(
)
]
[
(
)
]
T
μ
,
T
μ
,
μ
=
T
T
μ
,
μ
,
μ
4.
(associativity).
~
~
~
~
~
~
A
B
C
A
B
C
(
[]
)
The pair
0 forms a commutative semigroup with a unity.
Examples of triangular norms are:
,
T
(
)
(
)
T
μ
,
μ
=
min
μ
,
μ
;
1.
~
~
~
~
A
B
A
B
(
)
T
μ
,
μ
=
μ
×
μ
;
2.
~
~
~
~
B
B
A
A
(
)
(
)
The fuzzy set ~ is referred to as union of fuzzy sets ~ and ~ ,
3.
T
μ
,
μ
=
max
0
μ
+
μ
1
.
~
~
~
~
B
B
A
A
~
~
~
C
=
A
B
, if
()
()
()
μ
x
=
μ
x
μ
x
;
x
X
.
~
A real-valued binary function is referred to as triangular conorm,
[][] []
~
~
C
A
B
K
:
0
×
0
0
,
if this function satisfies to following conditions:
()
(
)
(
)
K
0
=
0
K
μ
,
=
K
1
μ
=
μ
1.
(boundedness);
~
~
~
A
A
A
(
)
(
)
2.
K
μ
,
μ
K
μ
,
μ
, if
μ
μ
,
μ
μ
(monotonicity property);
~
~
~
~
~
~
~
~
A
B
C
D
A
C
B
D
(
)
(
)
3.
(commutativity);
K
μ
,
μ
=
K
μ
,
μ
~
~
~
~
A
B
B
A
[
(
)
]
[
(
)
]
4.
K
μ
,
K
μ
,
μ
=
K
K
μ
,
μ
,
μ
(associativity).
~
~
~
~
~
~
A
B
C
A
B
C
 
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