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[
]
1
~
~
2
~
~
θ
2
,
θ
The proposition 6.5.
Boundaries of the weighed segment
of product
2
a
X
a
X
X
of fuzzy numbers
~
and
look like
2
()
()
⎛
q
⎞
⎛
q
⎞
−
1
1
1
−
1
1
1
~
~
⎜
⎜
⎝
2
2
⎟
⎟
⎠
⎜
⎜
⎝
2
2
⎟
⎟
⎠
θ
=
b
x
+
x
x
+
x
−
b
x
+
x
x
+
x
;
2
q
q
M
M
L
q
q
M
M
a
X
3
12
6
6
20
q
q
q
q
()
()
r
r
⎛
⎞
⎛
⎞
−
1
1
1
−
1
1
⎜
⎜
⎝
⎟
⎟
⎠
⎜
⎜
⎝
⎟
⎟
⎠
θ
2
~
~
=
b
x
2
+
x
x
+
x
2
+
b
x
2
+
x
2
x
2
+
x
2
,
2
r
r
M
M
R
r
r
M
M
a
X
3
12
6
6
20
r
r
r
r
where
1
b
−
b
≥
0
L
,
q
=
1
⎩
⎨
⎧
⎩
⎨
⎧
L
q
=
M
=
q
2
b
+
b
<
0
R
,
q
=
2
R
2
b
−
b
≥
0
L
,
r
=
1
⎩
⎨
⎧
⎩
⎨
⎧
L
r
=
M
=
r
1
b
+
b
<
0
R
,
r
=
2
The proof
. Let us consider a fuzzy number which is multiplication of the fuzzy
(
R
~
)
X
≡
x
,
x
,
x
,
x
,
x
x
0
by itself, and let us denote it with
−
≥
1
2
L
R
1
L
. Let us write out
α
-level set
~
~
~
~
~
2
D
=
X
×
X
=
X
[
]
[
(
)
(
)
]
1
2
==
According to multiplication operation for fuzzy numbers, the
α
-level set
~
looks
like
X
X
,
X
x
−
1
−
α
x
,
x
+
1
−
α
x
.
α
α
α
1
L
2
R
[
]
(
)
(
)
(
)
(
)
2
2
D
=
x
2
1
−
2
1
−
α
x
x
+
1
−
α
x
2
,
x
2
2
+
2
1
−
α
x
x
+
1
−
α
x
2
.
α
1
L
L
2
R
R
a
~
[
]
~
~
(
)
1
,
2
C
C
If
a
≡
b
,
b
L
b
,
is a nonnegative number, the
α
-level set
looks like
,
α
α
R
where
[
]
−
(
)
(
)
2
C
1
=
b
x
2
1
−
2
1
−
α
x
x
+
1
−
α
x
2
α
1
L
L
[
]
(
)
(
)
(
)
2
3
2
1
2
−
b
1
−
α
x
−
2
1
−
α
x
x
+
1
−
α
x
;
L
1
L
L
[
]
+
(
)
(
)
2
C
2
=
b
x
2
2
+
2
1
−
α
x
x
+
1
−
α
x
2
α
2
R
R
[
]
(
)
(
)
(
)
2
3
2
2
2
+
b
1
−
α
x
+
2
1
−
α
x
x
+
1
−
α
x
.
R
2
R
R
Then
1
(
)
∫
θ
1
~
~
=
bx
2
1
+
C
1
α
d
α
=
2
α
a
X
0
1
1
1
1
1
⎛
⎞
⎛
⎞
2
1
2
2
1
2
=
b
x
−
x
x
+
x
−
b
x
−
x
x
+
x
;
⎜
⎝
⎟
⎠
⎜
⎝
⎟
⎠
1
L
L
L
1
L
L
3
12
6
6
20
