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˃
⃔
|
min
(μ, ˃)
|
|
μ
|
μ
r
r
,
(μ, ˃)
|=
i
=
1
min
with
|
min
(μ(
x
i
), ˃(
x
i
))
.
In last example, it is
|
min
(μ, ˃)
|=
0
.
7
+
0
.
7
+
1
+
0
.
6
=
3
,
|
μ
|=
0
.
7
+
0
.
8
+
1
+
0
.
7
=
3
.
2, and
r
=
3
/
3
.
2
=
0
.
9375
≈
0
.
94. That is,
μ
9375
˃
:
μ
'is almost
0
.
included in'
˃
.
|
min
(μ,˃)
|
|
˃
|
Since
|
˃
|=
0
.
70001
+
0
.
7
+
1
+
0
.
6
=
3
.
00001, it is
=
0
.
9999, or
˃
0
.
9999
μ
. That is,
˃
is more included in
μ
, than
μ
is included in
˃
!
Remark 2.2.2
Of course, if
μ
˃
,itismin
(μ, ˃)
=
μ
, and
r
=
1, that is,
μ
˃
⃒
μ
1
˃.
Nevertheless, since it is only
min
(μ(
x
i
), ˃(
x
i
))
min
(
μ(
x
i
),
˃(
x
i
)),
from
μ
1
˃
(or
|
min
(μ, ˃)
| |
μ
|
) it does not necessarily follow
μ
˃
.
Let us show an example with crisp subsets. If
X
={
1
,
2
,
3
,
4
,
5
,
6
,
7
}
, and
A
=
{
1
,
3
,
5
,
7
}
,
B
={
1
,
3
,
5
,
6
}
,itis
7
7
7
min
(μ
A
(
i
), μ
B
(
i
))
=
3
,
1
μ
B
(
i
)
=
4
,
1
μ
A
(
i
)
=
4
.
i
=
1
i
=
i
=
hence
μ
A
μ
B
,
or
A
ↂ
B
3
4
3
4
μ
B
μ
A
,
or
B
ↂ
A
3
4
3
4
2.2.2 Algebras of Fuzzy Sets
X
Once
F
(
X
)
=
(
[
0
,
1
]
; ;=
)
is taken, a
general algebra of fuzzy sets
comes from
endowing
F
(
X
)
with three operations:
X
X
,
1.
:[
0
,
1
]
ₒ[
0
,
1
]
X
X
X
2.
·:[
0
,
1
]
×[
0
,
1
]
ₒ[
0
,
1
]
,
X
X
X
3.
+:[
0
,
1
]
×[
0
,
1
]
ₒ[
0
,
1
]
,
μ
of
respectively called the
complement
μ
,the
intersection
μ
·
˃
of '
μ
and
˃
', and
X
; ;=; ·;+;
)
the
union
μ
+
˃
of '
μ
or
˃
'. Then
(
[
0
,
1
]
, is called an
algebra of
fuzzy sets
, provided the following laws do hold:
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