Information Technology Reference
In-Depth Information
It should be pointed out that Zadeh's CRI is not a “result” but a meta-rule. It is a
“directive” allowing to reach a solution to our problem, and it should be noticed
that when
P
∗
P
it is not in general
Q
∗
=
=
Q
. For example, in the case of
ML
-
implications it is:
μ
Q
∗
(
y
)
=
Sup
x
Min
(μ
P
(
x
),
T
(μ
P
(
x
), μ
Q
(
y
)))
∈
X
Sup
x
T
(μ
P
(
x
), μ
Q
(
y
))
=
T
(
Sup
x
X
μ
P
(
x
), μ
Q
(
y
))
=
μ
Q
(
y
),
∈
X
∈
provided that
Sup
μ
P
=
1, and because of
T
(μ
P
(
x
), μ
Q
(
y
))
μ
Q
(
y
)
and
T
is
continuous. But, for example, if
T
=
Min
,
Sup
μ
P
=
0
.
9 and
Sup
μ
Q
=
1, then
μ
Q
∗
(
y
)
=
Min
(
0
.
9
, μ
Q
(
y
))
=
μ
Q
(
y
)
. Notice that for all the cases in which
μ
P
is
normalized (
μ
P
(
x
0
)
=
1forsome
x
0
∈
X
), Mamdani-Larsen implications do verify
μ
Q
∗
=
μ
Q
whenever
μ
P
∗
=
μ
P
.
Fourth Step: Numerical Input
This is the case in which
. That is “
x
is
P
∗
”isthe
μ
P
∗
=
∈{
x
0
}
is exactly
x
x
0
or
x
statement “
x
is
x
0
” and hence
1f
x
=
x
0
μ
P
∗
(
x
)
=
0f
x
=
x
0
In that case,
μ
Q
∗
(
y
)
=
Sup
x
T
1
(μ
P
j
(
x
),
J
(μ
P
(
x
), μ
Q
(
y
)))
=
J
(μ
P
(
x
0
), μ
Q
(
y
)),
∈
X
∈
for all
y
Y
.
For example, let
J
be a
ML
-implication, then
μ
Q
∗
(
)
=
(μ
P
(
x
0
), μ
Q
(
))
y
T
y
.
If
X
=[
0
,
10
]
,
Y
=[
0
,
1
]
,
P
=
close to
4,
Q
=
big
, with uses as shown in the
following figure and moreover
x
0
=
3
.
5, with
J
(
a
,
b
)
=
Min
(
a
,
b
)
, then
μ
Q
∗
(
y
)
=
Min
(μ
P
(
3
.
5
), μ
Q
(
y
))
=
Min
(
0
.
5
, μ
Q
(
y
))
, with
μ
P
(
x
)
=
x
−
3 between 3 and 4.
1
1
μ
μ
Q
P
0.5
0
0
0.5
1
3
4
5
10
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