Information Technology Reference
In-Depth Information
(
,
(
,
−
(
,
)))
=
(
,
(
+
−
,
−
(
,
)))
=
W
a
max
b
1
max
a
b
max
0
max
a
b
1
a
max
a
b
W
1
2
(
a
,
b
),
if
b
a
or
b
>
a
and
b
>
b
. It is a W-conditional.
1
2
0
,
if
b
>
a
and
b
Example 3.2.11
The protoform
μ
ₒ
˃
=
μ
·
˃
(coming from the classical con-
junctive conditional
a
ₒ
b
=
a
·
b
)
,gives
J
(
a
,
b
)
=
T
(
a
,
b
),
functions with the inconvenience of the property
J
(
a
,
b
)
=
J
(
b
,
a
)
, but verifying,
T
0
(
a
,
J
(
a
,
b
))
=
T
0
(
a
,
T
(
a
,
b
))
T
(
a
,
b
)
min
(
a
,
b
)
b
that is, all of them are conditionals for any t-norm
T
0
and in particular, for the greatest
of them. They are always taken as min-conditionals. For example,
•
If
T
=
min,
J
(
a
,
b
)
=
min
(
a
,
b
)
, is called the Mamdani conditional
)
=
˕
−
1
•
If
T
=
prod
,
J
(
a
,
b
(˕(
a
)
·
˕(
b
))
, are called Larsen conditionals
˕
•
T
=
W
˕
is never used, since it can be
J
(
a
,
b
)
=
0 with
a
>
0 and
b
>
0.
x
(
1
+
x
)
˕
−
1
For
example, with
˕(
x
)
=
(an order automorphism), it is
(
x
)
=
2
√
8
x
+
1
−
1
, and
2
a
(
1
+
a
)
b
(
1
+
b
)
ab
(
1
+
a
)(
1
+
b
)
)
=
˕
−
1
))
=
˕
−
1
)
=
˕
−
1
J
(
a
,
b
(˕(
a
)
·
˕(
b
(
.
(
)
2
2
4
√
ab
(
1
+
a
)(
1
+
b
)
+
1
−
1
2
=
that, of course, is a min-conditional.
Remark 3.2.12
A way of avoiding the undesirable symmetry
J
(
a
,
b
)
=
J
(
b
,
a
)
in
the case of Mamdani-Larsen min-conditionals, is taking
a
r
b
s
J
(
a
,
b
)
=
T
(
,
),
a
r
b
s
a
r
b
s
b
s
with real numbers
r
,
s
with 1
>
s
. Then
T
0
(
a
,
T
(
,
))
min
(
min
(
,
),
)
b
s
b
, and
J
is a min-conditional.
Example 3.2.13
Once given a conditional statement 'If x is P, then y is Q', and
represented 'x is P' by
μ
P
(
x
)
, and 'y is Q' by
μ
Q
(
y
)
, it remains to be understood
what it is meant by the 'statement'
μ
P
(
x
)
ₒ
μ
Q
(
y
)
. It could be, or not to be,
μ
P
(
x
)
ₒ
μ
Q
(
y
)
=
(μ
P
ₒ
μ
Q
)(
x
,
y
)
, with
μ
P
ₒ
μ
Q
a fuzzy set in
X
×
Y
,
(
)
identified with some expression involving the connectives
and
(
·
)
,
or
(
+
)
,
not
.
μ
P
ₒ
μ
Q
is expressible in material form, for
In the affirmative case, it is said that
μ
P
ₒ
μ
Q
=
μ
P
+
μ
Q
,or
μ
P
ₒ
μ
Q
=
μ
P
+
μ
P
·
μ
Q
, etc. These material
example,
Search WWH ::
Custom Search