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μ
+
˃
=
(
μ
·
˃
=
(μ, ˃)
=
˃, μ
+
˃
=
(μ, ˃)
=
μ
−
μ)
+
˃
it follows
min
max
, and
1
is the pointed curve. That is,
μ
(
x
),
if
x
∈[
0
,
6
.
87
]
(μ
+
˃)(
x
)
=
˃(
x
),
if
x
∈[
6
.
87
,
10
]
.
x
10
x
−
5
Notice that
x
=
6
.
87 comes from the equation 1
−
=
.
6
Example 2.2.51
Consider the Sugeno's family of strong negations
N
ʻ
(
x
)
=
1
−
x
1
1
x
(ʻ >
−
)
−
< ʻ
1
< ʻ
2
, it follows 1
+
ʻ
1
x
<
+
ʻ
2
x
,
<
1
.If
1
1
or
+
ʻ
1
x
,
1
+
ʻ
1
+
ʻ
2
x
1
that is,
N
ʻ
2
(
)<
N
ʻ
1
(
)
x
x
. Hence,
If
ʻ
0
:
N
0
N
ʻ
If 0
ʻ
:
N
ʻ
N
0
N
,
ʻ
∈
ʻ
N
0
N
,
ʻ
∈
ʻ
μ
=
μ
=
Compare the graphics of
N
0
ⓦ
μ
and
N
1
ⓦ
μ
, in a figure, if
⊧
⊨
0
,
if
x
∈[
0
,
3
]∪[
7
,
10
]
1
,
if
x
∈[
4
,
6
]
μ(
x
)
=
x
−
3
,
if
x
∈[
3
,
4
]
⊩
7
−
x
,
if
x
∈[
6
,
7
]
Solution
.Itis
⊧
⊨
1
,
if
x
∈[
0
,
3
]∪[
0
,
4
]
0
,
if
x
∈[
4
,
6
]
μ
(
x
)
=
4
−
x
⊩
,
if
x
∈[
3
,
4
]
x
x
−
6
,
if
x
∈[
6
,
7
]
,
x
1
−
μ(
x
)
μ
1
(
and
x
)
=
N
1
(μ(
x
))
=
,or
1
+
μ(
x
)
⊧
⊨
0
,
if
x
∈[
0
,
3
]∪[
7
,
10
]
1
,
if
x
∈[
4
,
5
]
μ
1
(
x
)
=
4
−
x
⊩
2
,
if
x
∈[
3
,
4
]
x
−
x
−
6
x
,
if
x
∈[
6
,
7
]
8
−
Hence,
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