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then
μ
(
P
)
(
x
)
=
N
(μ
P
(
x
))
=
N
(
N
(μ
P
(
x
)))
=
(
N
ⓦ
N
)(μ
P
(
x
))
=
id
L
(μ
P
(
x
))
=
μ
P
(
x
),
for all
x
in
X
,or
μ
(
P
)
=
μ
P
.
Functions
N
verifying (1), (2), and (3) are called
strong negations
, and are almost
the only used with fuzzy sets, for the following reason. With
L
=[
0
,
1
]
, all strong
negations are, obviously, continuous, hence, if
μ
P
is continuous also
μ
P
is such,
and if
μ
P
has the same discontinuities. That is, strong
negations do not add discontinuities to those of
μ
P
has some discontinuities,
μ
P
.
, there is a family of strong negations widely used in fuzzy set theory,
the so-called Sugeno's negations:
If
L
=[
0
,
1
]
1
−
a
N
ʻ
(
a
)
=
a
,
with
ʻ >
−
1
,
for all
a
∈[
0
,
1
]
.
1
+
ʻ
1
−
a
1
−
a
1
−
a
For example,
N
0
(
a
)
=
1
−
a
,
N
1
(
a
)
=
a
,
N
−
0
.
5
=
5
a
,
N
2
(
a
)
=
2
a
,etc.
1
+
1
−
0
.
1
+
Since obviously,
N
ʻ
1
N
ʻ
2
⃔
ʻ
1
ʻ
2
,
it results:
•
If
ʻ
∈
(
−
1
,
0
]
, then
N
ʻ
N
0
•
If
ʻ
∈
(
0
,
+∞]
, then
N
0
<
N
ʻ
,
graphically
∈
N
,
ʻ
(−1, 0)
ʻ
μ
N
0
∈
(0,
+ ∞
)
N
,
ʻ
ʻ
Notice that, provided
N
is in the Sugeno's family of strong negations, it is enough
to know a concrete pair of numbers
(
a
,
N
ʻ
(
a
))
to compute the corresponding
ʻ
.For
example,
•
If
N
ʻ
(
0
.
5
)
=
0
.
5, it results
ʻ
=
0
5
14
•
If
N
ʻ
(
0
.
7
)
=
0
.
4, it results
ʻ
=−
1
•
If
N
ʻ
(
.
)
=
.
ʻ
=
2
.
0
4
0
5, it results
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