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not compatible to
M
at all. Otherwise, the unitary membership degree states that an
element is fully represented in the set, being completely compatible to
M
.When
the element is mapped into this interval, 0
<
μ
(
)
<
1, there is a partial degree
of membership. The
j
-th approximate (fuzzy) proposition is, thus, associated to
information (knowledge) that is simultaneously
uncertain
and
imprecise
,i.e.,
vague
,
being understood as the mechanism employed to build up the approximate (fuzzy)
human reasoning. The antecedent in the expression (16.1) form a space within the
premise space,
P
1
× ...×
x
M
X
P
n
.
A compound proposition connected by conjunctive
logic operators,
, form a fuzzy region, expressing flexibility (or in doubt) in making
decisions, judgment or analysis. In so doing, it represents the fuzziness in reasoning.
An argument employed to make deductive inferences is known as an
inference
rule
and can be built up by diverse manners, limited to be tautological. Arguments
in the form of fuzzy IF-THEN rules may also be defined as
fuzzy relations
,
R
, with
the membership grade:
∧
R
([
x
1
,...,
x
n
]
,
y
)=
f
([
M
(
x
1
)
,...,
M
(
x
n
)]
,
N
(
y
))
,
(16.3)
n
in
X
n
×
gives birth to a
relational
fuzzy inference system
. A functional input-output mapping as given in (16.1) can
be understood as the union of Cartesian products obtained by the association of
fuzzy linguistic terms. The resulting fuzzy graph,
f
∗
, concerning a set of fuzzy
rules as presented in (16.2) is inherent to the fuzzy relation (16.3). A fuzzy graph,
f
∗
, describes a functional mapping,
f
:
U
Y
.Sucha
nonlinear
mapping,
f
:
[
0
,
1
]
→
[
0
,
1
]
→
V
,
X
∈
U
,
Y
∈
V
from the linguistic
variables,
X
i
for
i
m
, in the universes of discourse,
U
, to the linguistic
variable,
Y
, in the output universe of discourse,
V
, given as:
=
1
,
2
,...,
f
∗
=[
M
1
(
x
1
)
,...,
M
1
(
x
n
)]
×
N
(
y
)
1
+[
M
2
(
x
1
)
,...,
M
2
(
x
n
)]
×
N
(
y
)
2
+
...
+[
M
m
(
x
1
)
,...,
M
m
(
x
n
)]
×
N
(
y
)
m
(16.4)
where the operations
+
and
×
denote, respectively, the operations of disjunction and
Cartesian product,
M
1
j
(
. When understood as the combination
of fuzzy relations or, particularly, the union of the Cartesian products concerning
pairs of input,
x
1
)
× ...×
M
nj
(
x
n
)
, the fuzzy graph,
f
∗
, from
X
n
[
M
(
x
1
)
,...,
M
(
x
n
)]
, and output,
N
(
y
)
to
Y
can be given as:
i
[
f
∗
=
M
(
x
1
)
,...,
M
(
x
n
)]
i
×
N
(
y
)
i
(16.5)
and illustrated in Fig. 16.2.
16.2.2
Conditional Restriction as a Fuzziness Mechanism
of Measure
Once the approximate (fuzzy) reasoning is available (16.1), it is required fact(s) to
infer (induce) new facts,
,or
previous decisions (Fig. 16.2). The primary fact represents, thus, a
conditional re-
striction
on the values of
x
imposed by the measure or observation. In the context of
(
y
=
b
,
μ
([
b
1
,
b
2
]))
, from current facts,
(
x
=
a
,
μ
(
a
))
