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μ
small
μ
not big
.
whose graphics show that the pair (
big, small
) is coherent, since
x
x
Example 1.2.5
In
[
0
,
10
]
take
μ
big
(
x
)
=
10
and
μ
not big
(
x
)
=
1
−
μ
big
(
x
)
=
1
−
10
.
Which symmetries
A
:[
0
,
10
]ₒ[
0
,
10
]
can be taken for having
μ
small
=
μ
big
ⓦ
A
?
From the coherence's condition with
N
0
, follows
A
(
x
)
x
10
μ
small
(
x
)
=
μ
big
(
A
(
x
))
=
μ
not big
(
x
)
=
1
−
10
hence,
A
(
x
)
10
−
x
is the condition
A
must satisfy. For example,
x
10
•
If
A
1
(
x
)
=
10
−
x
, it results
μ
small
(
x
)
=
1
−
=
μ
not big
(
x
)
, a non-regular
case.
10
−
x
•
If
A
2
(
)
=
·
x
,forwhich
A
2
(
)
−
μ
small
(
)
=
μ
big
(
·
x
10
x
10
x
, it results
x
10
10
+
10
−
x
10
−
x
x
)
=
x
,
with graphics
10
+
10
+
also showing coherence.
x
10
Example 1.2.6
With the same
μ
big
(
x
)
=
in the previous example, and
μ
small
=
μ
big
(
10
−
x
)
, which Sugeno's strong negation
N
can be used for having
μ
not big
(
x
)
=
ʻ
N
ʻ
(μ
big
(
?
It should be,
x
))
10
−
x
x
10
)
=
10
−
x
μ
small
(
x
)
=
μ
big
(
10
−
x
)
=
N
ʻ
(μ
big
(
x
))
=
N
ʻ
(
10
10
+
ʻ
x
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