Information Technology Reference
In-Depth Information
Examples of triangular conorms are:
(
)
(
)
1.
K
μ
,
μ
=
max
μ
,
μ
;
~
~
~
~
A
B
A
B
2.
(
)
K
μ
,
μ
=
μ
+
μ
−
μ
×
μ
;
~
~
~
~
~
~
A
B
A
B
A
B
(
)
(
)
3.
K
μ
,
μ
=
min
1
μ
,
μ
~
~
~
~
A
B
A
B
(
)
Let us obtain following definitions for triangular norm
min , and for triangular
(
)
conorm
max .
The fuzzy set
~
is referred to as intersection of fuzzy sets
~
and
~
,
B
~
~
~
C
=
A
∩
, if
[
]
()
()
()
μμ
The fuzzy set
~
is referred to as union of fuzzy sets
~
and
~
,
x
=
min
x
,
μ
x
;
∀
x
∈
X
.
~
~
~
C
A
B
~
~
~
C
=
A
∪
B
, if
[
]
()
()
()
μμ
The fuzzy set
~
is referred to as a difference of fuzzy sets
~
and
~
,
B
x
=
max
x
,
μ
x
;
∀
x
∈
X
.
~
~
~
C
A
B
~
~
~
~
~
, if
C
=
A
−
B
=
A
∪
[
]
()
()
()
μμ
The fuzzy set
~
is referred to as a disjunctive sum of fuzzy sets
~
and
~
,
(
x
=
min
x
,
−
μ
x
;
∀
x
∈
X
.
~
C
A
B
~
~
~
)
(
)
, if
C
=
A
⊕
B
=
A
−
B
∪
B
−
A
[
(
)
(
)
]
()
()
()
()
()
μ
x
=
max
min
μ
x
,
−
μ
x
,
min
1
−
μ
x
,
μ
x
;
~
~
~
~
~
C
A
B
A
B
∀
x
∈
X
Example 1.4.
Operations with fuzzy sets. Let
{
~
}
A
=
0
/
x
;
0
/
x
;
0
/
x
;
0
/
x
;
1
2
3
4
~
{
}
B
=
0
/
x
;
0
/
x
;
1
/
x
;
0
2
/
x
.
1
2
3
4
Then
~
{
}
A
=
0
/
x
;
0
/
x
;
0
/
x
;
0
/
x
;
1
2
3
4
~
{
}
B
=
0
/
x
;
0
2
/
x
;
0
/
x
;
0
/
x
;
1
2
3
4
~
~
{
}
A
∩
B
=
0
,
/
x
;
0
7
/
x
;
0
,
/
x
;
0
,
2
/
x
;
1
2
3
4
~
{
}
A
∪
B
=
0
/
x
;
0
/
x
;
1
/
x
;
0
/
x
;
1
2
3
4
~
~
~
~
{
}
A
−
B
=
A
∩
B
=
0
/
x
;
0
2
/
x
;
0
/
x
;
0
/
x
;
1
2
3
4
~
~
~
~
{
}
B
−
A
=
B
∩
A
=
0
/
x
;
0
/
x
;
0
/
x
;
0
2
/
x
;
1
2
3
4
~
~
{
}
A
⊕
B
=
0
/
x
;
0
/
x
;
0
/
x
;
0
/
x
.
1
2
3
4
