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In-Depth Information
2.2 The Concept of an 'Algebra of Fuzzy Sets'
2.2.1 Introduction
Functions
X
μ
∈
F
(
X
)
=[
0
,
1
]
,
will be labeled only when it is some predicate P in X such that
μ
P
=
μ
, and it is
obvious that it could be the fact of having
μ
P
=
μ
Q
=
μ
R
··· =
μ
, in which case the
predicates
P
,
Q
,
R
,...
are exact synonyms in
X
. Notwithstanding there are much
X
more functions in
[
0
,
1
]
than predicates in
X
, and given a not previously labeled
X
, it can be 'artificially' introduced the predicate
M
μ
∈[
0
,
1
]
(
=
μ)
such that,
Degree up to which
x
is
M
=
μ(
x
),
for all
x
in
X
.
•
Notice that
F
(
X
)
will be taken as 'ordered' (partially) by means of the binary
pointwise relation
μ
˃
⃔
μ(
x
)
˃(
x
),
for all
x
∈
X
,
that induces the pointwise identity
μ
=
˃
⃔
μ
˃
˃
μ
⃔
μ(
)
=
˃(
),
∈
.
and
x
x
for all
x
X
The pointwise relation
is also called the 'inclusion', and
μ
˃
denoted by
'
μ
is included in
'. It enjoys the laws reflexive, antisymmetric and transitive.
It will be always considered that
F
˃
(
X
)
denotes, at least, the structure
X
X
, that is,
A
and
B
are in
(
[
0
,
1
]
; ;=
)
. Observe that if
μ
A
, μ
B
∈{
0
,
1
}
P
(
X
)
,
then it follows
μ
A
μ
B
⃔
A
ↂ
B
;
μ
A
=
μ
B
⃔
A
=
B
,
and
x
∈
A
⃔
μ
A
(
x
)
=
1
;
x
∈
A
⃔
μ
A
(
x
)
=
0
.
The classical symbol
∈
is the fuzzy symbol
1
, and
∈
is
0
.
For example, with the fuzzy set
P
∼
given by function
μ
in the next figure it is
x
∈
P
∼
if 0
x
3, and 7
x
10, but
x
∈
P
∼
if 4
x
6, and
x
μ(
x
)
P
∼
,if
x
∈
(
3
,
4
)
∪
(
6
,
7
)
with 0
< μ(
x
)<
1. If
x
=
3
.
5, since it is
μ(
x
)
=
x
−
3, when
x
∈
(
3
,
4
)
,itis3
.
5
0
.
5
P
∼
.
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