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the classic moderate obesity set,
M
Mod
, are simultaneously activated by the fuzzy
evidence indicating the degree of evidence or belief associated to two
possible
crisp
sets. When dealing with fuzzy classes, there are partial membership values, as well.
Such a measure achieves unsharp borders of fuzzy sets yielding distinct degrees of
activation. In this example, the measure of fuzziness is obtained when a measure,
x
M
, generates two degrees of activation,
M
M
F
F
, respectively,
by achieving the fuzzy low obesity set,
M
LF
, and the fuzzy moderate obesity set,
M
Mod
, such that 1
=
μ
M
LF
(
)
and
μ
M
Mod
(
)
M
M
M
LF
(
M
Mod
(
0 (left measure in Fig. 16.4). The
upper index,
F
, refers to fuzzy set for representing the measure.
>
μ
)
>
μ
)
>
Fig. 16.4
Example of fuzziness in measure to a
fuzzy
set
16.2.3
Fuzziness in the Compositional Rule of Inference
An alternative to implement the fuzzy inference system is by using the
composi-
tional rule of inference
(CRI), leading to a
generalized modus ponens
. The CRI
embraces (
i
)the
projection principle
,(
ii
)the
conjunction principle
;and(
iii
)the
semantic global entailment
. This inference mechanism is a cornerstone in many ap-
plications of fuzzy systems, chiefly because it is regarded within a general category
of fuzzy reasoning.
Two important operations in fuzzy relations are
projection
and
conjunction
.
Moreover, the
extension principle
is used to transform fuzzy sets via functions into
their fuzzy-set counterparts by using the
cylindrical extension
. Such an operation
allows computing induced constraints into different universe of discourses.
The
outcome in
Y
is, thus, calculated as the
relational composition
:
M
N
=
◦
R
,
(16.7)
