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forms are called protoforms; for instance, the protoform of the Kleene-Diennes fuzzy
conditional is
a
+
·
b
, and that of Mamdani fuzzy conditional is
a
b
.
μ
P
ₒ
μ
Q
does not correspond with a protoform, one can try to represent it
by means of an R-implication, that is, by
J
min
or by some
J
pr od
˕
, since all the
J
W
˕
do correspond to a protoform
a
+
If
μ
P
+
μ
Q
, with + represented by
W
˕
and
b
,or
by
N
.
In addition, there is a problem that should be taken into account when representing
μ
P
˕
μ
P
ₒ
μ
Q
should be represented by a function
J
that is a min-conditional, but that we are
not able to decide a protoform and we take
J
min
. Since
J
ₒ
μ
Q
. The problem is the following. Suppose that we know
J
min
, we will reach
the biggest possible output. This should be known. Analogously, if
J
should be a
prod-conditional, from
J
J
pr od
, follows the same comment.
Example 3.2.14
Let's stop for a while in the above mentioned concept of implication
function, a concept that comes directly from the properties shown by the Boolean
conditional
a
a
+
ₒ
b
=
b
, whose truth value is usually represented by
v(
a
ₒ
b
)
=
max
(
1
−
v(
a
), v(
b
)).
b
2
, it follows
a
+
a
+
Look that, if
b
1
b
1
b
2
,or
a
ₒ
b
1
a
ₒ
b
2
,if
a
1
a
2
,
is
a
2
a
1
, and
a
2
ₒ
b
a
1
ₒ
b
. Since
v(
a
)
and
v(
b
)
only take the values
{
0
,
1
}
,
v(
a
ₒ
b
)
shows the truth-table.
v(
a
)
v(
b
)
v(
a
ₒ
b
)
0
0
1
0
1
1
1
0
0
1
1
1
Because of that, a function
J
:[
0
,
1
]×[
0
,
1
]ₒ[
0
,
1
]
is called a fuzzy implication
function provided:
1.
J
is decreasing in its first variable, and non-decreasing in its second one
(
,
)
=
(
,
)
=
(
,
)
=
(
,
)
=
2.
J
0
0
J
0
1
J
1
1
1,
J
1
0
0
Obviously, S-implications and R-implications are fuzzy implication functions, but
Q and D operators are not always so, since, for instance
J
S
(
0
.
4
,
0
.
3
)
=
max
(
1
−
0
.
4
,
min
(
0
.
4
,
0
.
3
))
=
0
.
6
J
D
(
0
.
3
,
0
.
1
)
=
max
(
0
.
1
,
min
(
0
.
7
,
0
.
9
))
=
0
.
7
J
D
(
0
.
3
,
0
.
4
)
=
max
(
0
.
4
,
min
(
0
.
7
,
0
.
6
))
=
0
.
6
,
that is, for instance 0
.
1
<
0
.
4but
J
D
(
0
.
3
,
0
.
1
)>
J
D
(
0
.
3
,
0
.
4
)
.
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