Information Technology Reference
In-Depth Information
Chapter 2
Algebras of Fuzzy Sets
2.1 Introduction
From now on it will be only considered the case in which
(
L
,
)
=
(
[
0
,
1
]
,
)
, that
is, of Zadeh's fuzzy sets, with predicates
P
in
X
known through a degree
μ
P
:
X
ₒ
[
0
,
1
]
, and without knowing, necessarily, its primary use
P
. The set of all fuzzy sets
X
, will be also denoted by
F
in
X
,
[
0
,
1
]
(
X
)
. In this case, the preorder
μ
P
is linear,
or total, since for all
x
,
y
in
X
it is either
μ
P
(
x
)
μ
P
(
y
)
,or
μ
P
(
y
)
μ
P
(
x
)
, that
is, it is either
x
μ
P
y
or
y
μ
P
x
for all
x
,
y
in
X
. Hence,
μ
P
rarely will perfectly
reflect the primary use of
P
in
X
, since
P
is usually not linear.
X
,
In the case in which
X
is finite,
X
={
x
1
,...,
x
n
}
, the fuzzy sets
μ
∈[
0
,
1
]
will be represented by
μ
=
μ(
x
1
)/
x
1
+
μ(
x
2
)/
x
2
+···+
μ(
x
n
)/
x
n
,
with the convention that if some term
μ(
x
j
)/
x
j
does not appear, is that it is
μ(
x
j
)
=
0.
For example, with
X
={
1
,
2
,
3
,
4
}
, the expression
μ
=
0
.
5
/
x
1
+
0
.
7
/
x
2
+
1
/
x
4
,
refers to the fuzzy set in
X
given by
μ(
x
1
)
=
0
.
5,
μ(
x
2
)
=
0
.
7,
μ(
x
3
)
=
0,
μ(
x
4
)
=
1.
μ
=
Analogously, the fuzzy set
N
0
ⓦ
μ (
N
0
=
1
−
id) is
μ
=
.
/
x
1
+
.
/
x
2
+
/
x
3
,
0
5
0
3
1
2.1.1 Cartesian Product
If
A
,
B
are crisp subsets in
X
and
Y
, respectively, that is,
A
∈ P
(
X
)
and
B
∈ P
(
X
)
,
its cartesian product
A
×
B
={
(
a
,
b
)
;
a
∈
A
,
b
∈
B
}ↂ
X
×
Y
, is with the
Search WWH ::
Custom Search