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In-Depth Information
verifies:
•
μ
0
(
μ
0
=
μ
1
x
)
=
1
−
μ
0
(
1
−
x
)
=
1:
•
μ
1
(
μ
1
=
μ
0
x
)
=
1
−
μ
1
(
1
−
x
)
=
1
−
1
=
0:
)
⃒
˃
∗
μ
∗
•
μ
˃
⃒
1
−
˃(
1
−
x
)
1
−
μ(
1
−
x
•
μ
∗∗
(
−
μ
∗
(
μ
∗∗
=
μ
x
)
=
1
1
−
x
)
=
1
−[
1
−
μ(
x
)
]=
μ(
x
)
:
.
μ
ₒ
μ
∗
can be taken as a “strong negation”
Hence,
it could seem
that the function
}
[
0
,
1
]
, then
for the fuzzy sets in
[
0
,
1
]
, but it is not the case. Notice that if
μ
∈{
0
,
1
μ
∗
∈{
}
[
0
,
1
]
, that is, if
it should be also
0
,
1
μ
represents a classical subset of
[
0
,
1
]
,
μ
∗
should represent not only a classical subset but precisely the complement of
also
1
μ
. But with
A
=[
0
,
2
]ↂ
X
,
1
,
0
x
0
.
5
,
μ
A
(
x
)
=
0
,
0
.
5
<
x
1
,
follows,
1
0
,
0
.
5
<
x
1
,
,
0
.
5
<
x
1
,
μ
A
(
x
)
=
1
−
μ
A
(
1
−
x
)
=
1
−
=
0
,
0
x
0
.
5
,
1
,
0
x
0
.
5
, but not
A
c
that represents the subset
∗
violates the preservation principle
, and hence it cannot be taken into account to
negate fuzzy sets.
[
0
,
0
.
5
]
=
(
0
.
5
,
1
]
.
The unary operation
2.1.4 Resolution
X
,
Let us denote by
μ
r
the constant fuzzy sets in
[
0
,
1
]
μ
r
(
x
)
=
r
,for
r
∈[
0
,
1
]
X
and all
x
∈
X
. Notice that in
{
0
,
1
}
there are only the “constants”
μ
0
and
μ
1
, that
correspond to the sets
∅
and
X
, respectively.
X
, let us denote by
μ
(
r
)
the fuzzy (crisp) set
Given
μ
∈[
0
,
1
]
1
,
if
r
μ(
x
),
μ
(
r
)
(
x
)
=
0
,
otherwise,
[
μ
(
r
)
]
for all
r
∈[
0
,
1
]
, and by
the corresponding classical subset
{
x
∈
X
;
r
μ(
x
)
}
.
[
μ
(
0
)
]=
μ
These sets are called the
r-cuts
of
X
.
For example, in the following figures are shown, respectively, the constant fuzzy
and it is always
set
μ
0
.
5
, and the 0
.
3-cut of two different fuzzy sets.
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