Information Technology Reference
In-Depth Information
(
,
)
= C
Remark 1.1.5
When
L
is not linear, as it is with
L
the unit complex
μ
P
can coincide with the primary meaning, but
being not linear there can exist more possibilities for the coincidence.
interval, it is also not sure that
Remark 1.1.6
Since in fuzzy logic it is always only considered the membership
function and not the primary meaning serving for its design, the working scientist
should not confuse the two relations
μ
P
can be called
the
working-meaning
of
P
in
X
, without forgetting that it can be bigger than the
primary meaning.
μ
P
and
P
. For this reason,
1.1.2 Fuzzy Sets
Once the use
(
P
,
μ
P
)
of
P
in
X
is given for the poset
(
L
,
)
, it will be said that
the new object
P
∼
defined by
•
x
∈
r
P
∼
, if and only if
r
=
μ
P
(
x
)
,for
r
∈
L
,
•
μ
P
=
μ
Q
is the
L
-set labeled
P
.Theset
L
X
P
∼
=
Q
∼
, if and only if
, is usually and abusively
called the set of all
L
-sets in
X
, since it contains all the possible degrees in
L
.
Notice that a predicate
P
could give many
L
-sets, in dependance on which poset
={
μ
;
μ
:
X
ₒ
L
}
L
X
, are chosen.
From now on, it will be supposed that
(
,
)
μ
P
∈
L
, and which function
(
,
)
ʱ (ʱ
l
has a
minimum
element
r
,
∀
r
∈
L
)
, and a
maximum
element
ˉ(
r
ˉ,
∀
r
∈
L
)
. With
L
0
={
ʱ, ˉ
}
,
(
L
0
,
)
is a poset isomorphic to
.
Since a classical subset
A
of
X
is characterized by its membership function
(
{
0
,
1
}
,
)
1
,
if
x
∈
A
μ
A
(
x
)
=
0
,
if
x
∈
A
,
the subset
A
can be viewed as an
L
-set, whose function only takes the values
ʱ
or
ˉ
,
that is,
ˉ,
if
x
∈
A
μ
A
(
x
)
=
ʱ,
if
x
∈
A
.
Hence, the classical subsets of
X
are nothing else than the functions in
L
0
,aset
included in
L
X
. Classical subsets are limiting or particular case of
L
-sets. They are
degenerate
L
-sets.
Notice that if the classical subset
A
is labeled by a crisp predicate
P
, this predicate
is semi-rigid since
X
/
=
P
has, at most, two classes. Crisp predicates are rigid.
μ
P
is perceptually designed, that is, designed from what is perceived
or known on the concrete use of
P
in
X
. It can be thought that such design is purely
subjective, in the sense of being made just by what the designer believes on the use
of
P
. But this would not be the case, except if the designer proceeds in a non rational
way. The
designer
should try to be as most sure as possible on the correction of his/
her/its perceptions.
Ve r y o f t e n
Search WWH ::
Custom Search