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approximate reasoning such an input value is a subset in
X
and works as a singleton
restriction on
x
or a fuzzy restriction
imposed
by
M
.
A linguistic term (value) can be interpreted as a label for a
fuzzy restriction
upon
the linguistic variable being characterized by a
compatibility function
, as well. The
calculus of the restriction is one of the most important concepts in the theory of
fuzzy systems and approximate reasoning because it can be related to human cog-
nition. In particular, it is related to situations that involve the construction of con-
cepts, patter recognition, and decision- making processes in fuzzy environments or
in the presence of uncertainties [37]. The area underneath a fuzzy set corresponds
to an
elastic restriction
of the possible (also similar) values of elements
x
X
,also
denominated
possibility distribution function
. One of the most important roles as-
signed to the calculus of fuzzy constraints is to supply a kind of reasoning that is
neither exact, nor inexact. Known as the main element to build up the approximate
reasoning, the fuzzy restriction assumes a basic role not only in the process to form
the approximate inference mechanism but the measurement, as well [36].
∈
Fig. 16.2
Fuzzy graph concerning a set of fuzzy rules [2]
Observe, however, that the type of imperfection and, consequently, the uncer-
tainty and imprecision represented by fuzzy sets is not the same as a
fuzzy measure
defined by the function:
g
:
℘
(
X
)
→
[
0
,
1
]
.
(16.6)
A fuzzy measure assigns to each
crisp set
of
X
a number in the unit interval
[
0
,
1
]
,
where
℘
(
X
)
is the power set of crisp set and not the power of fuzzy sets on
X
, i.e.,
˜
℘
(
. A fuzzy measure concerns the degree of evidence or belief associated to a
fuzzy set that a particular element belongs to a crisp set, a set with sharp boundaries.
It indicates the degree of evidence or subjective certainty that an element belongs to
a subset is not known with certainty, even though the classes are disjoint intervals.
In the presence of perfect evidence, a full membership is assigned to one and only
one of the available crisp sets. The fuzzy measure is employed due to the fact that
X
)