Information Technology Reference
In-Depth Information
Remark 2.2.20
Let us show an example of a 'union' that is not-decomposable. Define
max
X
(μ(
x
), ˃(
x
)),
if
μ
or
˃
are in
{
0
,
1
}
(crisp)
(μ
+
˃)(
x
)
=
max
(
H
(μ),
H
(˃)),
otherwise
.
It is easy to show that this operation verifies the laws b, d and e in Sect.
2.1.1
. Hence,
it is a union for fuzzy sets that, in addition, is commutative. It is not idempotent,
since if
X
X
,itis
μ
∈[
0
,
1
]
−{
0
,
1
}
(μ
+
μ)(
x
)
=
H
(μ)
=
μ(
x
)
. It does not exist a
function
G
:[
0
,
1
]×[
0
,
1
]ₒ[
0
,
1
]
such that
(μ
+
˃)(
x
)
=
G
(μ(
x
), ˃(
x
))
X
. Indeed, let us suppose that such a
G
does exist,
for all
x
∈
X
and all
μ, ˃
[
0
,
1
]
and take
μ
=
μ
0
.
5
. Then
1
1
1
1
•
With
˃
=
μ
0
,itis
(μ
+
˃)(
x
)
=
max
(
2
,
0
)
=
2
. Hence
G
(
2
,
0
)
=
2
.
1
•
With
˃(
x
)
=
x
,is
(μ
+
˃)(
x
)
=
max
(
H
(μ),
H
(˃))
=
max
(
2
,
1
)
=
1, and
1
(μ
+
˃)(
, that is absurd.
To have a not-decomposable 'intersection', it is enough to define, with
0
)
=
1
=
G
(
2
,
0
)
μ
=
1
−
μ
,
the dual operation,
)
=[
(μ
+
˃
)
]
(
−
(μ
+
˃
)(
(μ
·
˃)(
x
x
)
=
1
x
)
min
X
(μ(
x
), ˃(
x
)),
if
μ
or
˃
are in
{
0
,
1
}
=
(μ
),
(˃
)),
max
(
H
H
otherwise
.
2.2.5 Standard Algebras of Fuzzy Sets
An standard algebra of fuzzy sets is a decomposable algebra of fuzzy sets such that:
X
1.
μ
·
˃
=
˃
·
μ
, for all
μ, ˃
in
[
0
,
1
]
(
·
is commutative)
X
2.
μ
+
˃
=
˃
+
μ
, for all
μ, ˃
in
[
0
,
1
]
(
+
is commutative)
X
3.
μ
·
(˃
·
ʻ)
=
(μ
·
˃)
·
ʻ
, for all
μ, ˃, ʻ
in
[
0
,
1
]
(
·
is associative)
X
4.
μ
+
(˃
+
ʻ)
=
(μ
+
˃)
+
ʻ
, for all
μ, ˃, ʻ
in
[
0
,
1
]
(
+
is associative)
μ
=
μ
X
(
is involutive).
5.
, for all
μ
in
[
0
,
1
]
Hence, writing
ⓦ
(μ
×
˃), μ
=
μ
·
˃
=
ⓦ
(μ
×
˃), μ
+
˃
=
ⓦ
μ,
T
S
N
functions
T
,
S
:[
0
,
1
]×[
0
,
1
]ₒ[
0
,
1
]
and
N
:[
0
,
1
]ₒ[
0
,
1
]
, in addition to the
corresponding general properties stated before, must verify
Search WWH ::
Custom Search