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In-Depth Information
Example 2.2.54
Describe in fuzzy terms, the statements
p
=
John is young and around forty,
q
=
John is old or around forty
.
Solution
. The solution will come after representing the predicates
P
=
young,
aP
=
Old, A40
=
around forty
, in a scale of 0-100 years. The general forms of
μ
P
,
μ
aP
, and
μ
A
40
,are
with
μ
P
is non-decreasing. Once these functions were
established accordingly with the current use of
P
and
A
40, it will be:
μ
aP
(
x
)
=
μ
P
(
100
−
x
)
since
Degree
(
p
)
=
T
(μ
P
(
x
), μ
A
40
(
x
)),
Degree
(
q
)
=
S
(μ
aP
(
x
), μ
A
40
(
x
)),
with convenient continuous t-norm
T
and t-conorm
S
. This formulas are the descrip-
tion of
p
and
q
in fuzzy terms.
For example, if
a
50
−
x
=
20
,
b
=
50
, μ
P
=
,
if 20
x
50, and 40
−
a
=
30
30
,
40
+
b
=
50, with
μ
A
40
piece-wise linear,
T
=
min
,
S
=
max, with
⊧
⊨
,
0
if 0
x
50
x
−
50
30
μ
aP
(
x
)
=
μ
P
(
100
−
x
)
=
,
if 50
x
80
⊩
1
,
if 80
x
100
,
the graphics is
The slashed function describes
p
, and the continuous one describes
q
. Of course,
Degree
(
p
)
min
(μ
P
(
x
), μ
A
40
(
x
))
μ
P
(
35
)
=
0
.
5.
Remark 2.2.55
To select
T
and
S
, the following points could be taken into account,
•
It could be perfectly the case that 'John is young and not young' with a positive
degree. Hence, the laws of Non-contradiction can be avoided, and
T
/
∈{
}
W
.
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