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:
ₒ
In the same vein, given a mapping
f
X
Y
, it can be extended to the fuzzy
(
),
(
)
power sets
F
X
F
Y
by
f
:
F
(
X
)
ₒ
F
(
Y
)
f
(μ)(
y
)
=
sup
{
μ(
x
)
;
f
(
x
)
=
y
}
,
for all
y
∈
Y
,
and
f
is known as the 'extension' of
f
to the fuzzy parts, and the definition as the
Zadeh's
Extension Principle
.
For example, if
x
f
:[
0
,
10
]ₒ[
0
,
1
]
, is given by
f
(
x
)
=
1
−
10
,the
f
x
10
]
[
0
,
10
]
fuzzy set
μ(
x
)
=
in
[
0
,
1
extends to the fuzzy set in
[
0
,
1
]
,
(μ)(
y
)
=
x
x
10
sup
{
μ(
x
)
;
f
(
x
)
=
y
}=
sup
{
10
;
1
−
=
y
}=
1
−
y
, for all
y
∈[
0
,
1
]
.
If
X
={
1
,
2
,
3
,
4
}
,
Y
={
a
,
b
,
c
}
, the mapping
f
:
X
ₒ
Y
such that
f
(
1
)
=
f
(
2
)
=
a
,
f
(
3
)
=
f
(
4
)
=
b
,
, to the fuzzy set
f
extends the fuzzy set
μ
=
1
/
1
+
0
.
4
/
2
+
1
/
3
+
0
.
7
/
4in
F
(
X
)
(μ)
in
F
(
Y
)
with values,
f
f
−
1
(μ)(
a
)
=
max
{
μ(
x
)
;
x
∈
(
a
)
}=
max
{
μ(
1
), μ(
2
)
}=
max
(
1
,
0
.
4
)
=
1
f
(μ)(
b
)
=
max
{
μ(
3
), μ(
4
)
}=
1
f
0, since
f
−
1
(μ)(
c
)
=
(
c
)
= ∅
.
Hence,
f
(μ)
=
1
/
a
+
1
/
b
,
that corresponds to the crisp subset
{
a
,
b
}
of
Y
.
X
,itis
f
(μ
A
)
=
μ
f
(
A
)
, that is not only a crisp sub-
set of
Y
, but coincides with the classical extension
f
Notice that if
μ
=
μ
A
∈{
0
,
1
}
(
A
)
of
A
. Nevertheless, as it is
shown by the above example, it can happen that
f
(μ)
∈ P
(
Y
)
with
μ
∈
F
(
X
)
−P
(
X
)
.
2.1.3 Preservation of the Classical Case
Like with the cartesian product and with the extension principle, all operations with
fuzzy sets must reproduce, when the data are crisp, the corresponding result obtained
in the crisp theory. This is the
principle of preservation
of the classical case, that is
forced by the will, and the necessity, of including all 'the classical' as a particular
case of the algebras of fuzzy sets.
To illustrate this preservation's principle, let us show a negative example. With
]
[
0
,
1
]
, the function
X
=[
0
,
1
]
, and all
μ
∈[
0
,
1
μ
∗
(
x
)
=
1
−
μ(
1
−
x
),
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