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In-Depth Information
Fig. 15.4
x
5. Function
10
is not, of course, the only that can represent the pred-
icate 'small', it is just a
linear model
for it, like
μ
S
(
x
)=
1
−
⎧
⎨
≤
≤
1
,if
0
x
2
−
x
3
5
μ
S
(
x
)=
,if
2
≤
x
≤
5
⎩
0
,if
5
≤
x
≤
10
=
−
is a picewise linear model for 'small' (see figure 15.5).
Taking N
1
id,
⎧
⎨
0
,if
0
≤
x
≤
2
5
−
x
3
x
−
2
3
x
−
2
3
is
μ
not S
(
x
)=
1
−
μ
S
(
x
)=
,if
2
≤
x
≤
5
, thus, since
=
,itis
⎩
1
,if
5
≤
x
≤
10
s
5
.
If instead of considering 'not small' it is considered the antonym 'big', and it is
defined by
=
3
.
5
, and the kernel is K
1
−
id
(
μ
S
)=[
0
,
3
.
5
)
with s
=
3
.
5
and B
=
α
(
x
)=
10
−
x, from
⎧
⎨
≤
≤
0
,if
0
x
5
x
−
5
μ
big
(
x
)=
μ
S
(
10
−
x
)=
,if
5
≤
x
≤
8
⎩
3
1
,if
8
≤
x
≤
10
in the figure 15.5 is obvious that it is s
α
=
B
=
5
, and hence K
α
(
μ
S
)=[
0
,
5
]
.
Thus, K
1
−
id
(
μ
S
)
⊂
K
α
(
μ
S
)
.
Fig. 15.5
