Information Technology Reference
In-Depth Information
μ
A
×
B
:
×
ₒ{
,
}
membership function
X
Y
0
1
, given by
μ
A
×
B
(
x
,
y
)
=
min
(μ
A
(
x
), μ
B
(
y
)
for all
x
∈
X
,
y
∈
Y
.Itis
μ
A
×
B
(
x
,
y
)
=
1
⃔
μ
A
(
x
)
=
μ
B
(
y
)
=
1.
In the same vein, if
μ
∈
F
(
X
),
and
˃
∈
F
(
Y
)
, the cartesian product
μ
×
˃
is
defined by directly generalizing the classical case:
μ
×
˃
=
min
ⓦ
(μ, ˃).
For example, with
X
={
x
1
,
x
2
,
x
3
}
,
Y
={
y
1
,
y
2
}
, and
μ
=
1
/
x
1
+
0
.
8
/
x
2
, ˃
=
0
.
9
/
y
1
+
0
.
7
/
y
2
,
it is
μ
×
˃
=
0
.
9
/(
x
1
,
y
1
)
+
0
.
7
/(
x
1
,
y
2
)
+
0
.
8
/(
x
2
,
y
1
)
+
0
.
7
/(
x
2
,
y
2
),
with
(μ
×
˃)(
x
3
,
y
1
)
=
(μ
×
˃)(
x
3
,
y
2
)
=
0.
y
7
x
5
With
μ
bi
g
(
x
)
=
if
x
∈[
0
,
5
]
and
μ
small
(
y
)
=
1
−
if
y
∈[
0
,
7
]
,itis
x
5
,
y
7
),
(μ
bi
g
×
μ
small
)(
x
,
y
)
=
min
(
1
−
[
,
]×
the representation of the cartesian product as a surface contained in the cube
0
5
[
,
]×[
,
]
0
.
Of course, if
7
0
1
X
,
X
, it is not only
X
μ
=
μ
A
∈{
0
,
1
}
˃
=
μ
B
∈{
0
,
1
}
μ
×
˃
∈{
0
,
1
}
but
μ
×
˃
=
μ
A
×
B
.
2.1.2 Extension Principle
If
f
:
X
ₒ
Y
is a mapping and
A
is a crisp subset of
X
,
A
ↂ
X
,itis
f
(
A
)
={
y
∈
Y
;
f
(
a
)
=
y
,
a
∈
A
}
the f-image of A in Y. Notice that
1
,
if it exists
x
∈
A
such that
f
(
x
)
=
y
,
μ
f
(
A
)
(
y
)
=
sup
{
μ
A
(
x
)
;
f
(
x
)
=
y
}=
0
,
otherwise
.
With these f-image, the mapping
f
:
X
ₒ
Y
is extended to the respective power
sets by
f
: P
(
X
)
ₒ P
(
Y
),
A
ₒ
f
(
A
).
Search WWH ::
Custom Search