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Let
us
consider
arithmetical
operations
for
Λ
-tolerance
numbers
~
~
(
)
(
)
A
≡
a
,
a
,
a
1
,
a
B
≡
b
,
b
,
b
2
,
b
1
2
L
R
,
1
2
L
R
.
1
2
The Proposition 2.1.
[121] Sum of
-tolerance numbers is a
-tolerance number.
Λ
Λ
The poof
. By definition of the expanded union operation for tolerance
(
)
-
L
−
R
~
~
()
()
numbers
with membership functions
μ
x
and
μ
x
, accordingly, we'll
A
,
B
~
~
~
~
obtain tolerance
(
)
-number
with membership function
A
⊕
B
L
−
R
[
]
()
()
()
μ
z
=
max
min
μ
x
,
μ
y
,
~
~
A
B
z
=
x
+
y
which
is
symbolically
noted
by
parameters
~
~
(
)
A
⊕
B
≡
a
+
b
,
a
+
b
,
a
+
b
,
a
+
b
. Let us show that the function which is
the left boundary of membership function of this number increases monotonically.
Let us write out two sets of
1
1
2
2
L
L
R
R
1
2
1
2
~
~
-level of number
,
according to (2.1):
α
α
>
α
A
+
B
2
1
[
]
(
)
()
()
()
()
A
+
B
=
a
+
b
−
L
−
1
α
a
−
L
−
1
α
b
;
a
+
b
+
R
−
1
α
a
+
R
−
1
α
b
;
α
1
1
1
1
L
2
1
L
2
2
1
1
R
2
1
R
1
1
2
1
2
[
]
(
)
()
()
()
()
A
+
B
=
a
+
b
−
L
−
1
α
a
−
L
−
1
α
b
;
a
+
b
+
R
−
1
α
a
+
R
−
1
α
b
.
α
1
1
1
1
L
2
1
L
2
2
1
1
R
2
1
R
2
1
2
1
2
L
,
α
>
α
As functions
L
,
R
,
R
are monotonically decreasing, then at
2
1
()
()
()
()
−
1
−
1
−
1
−
1
a
+
b
−
L
α
a
−
L
α
b
>
a
+
b
−
L
α
a
−
L
α
b
;
1
1
1
2
L
2
2
L
1
1
1
1
L
2
1
L
1
2
1
2
()
()
()
()
−
1
−
1
−
1
−
1
a
+
b
+
R
α
a
+
R
α
b
<
a
+
b
+
R
α
a
+
R
α
b
.
2
2
1
2
R
2
2
R
2
2
1
1
R
2
1
R
1
2
1
2
~
~
A
⊕
B
Hence, the left boundary of membership function
monotonically increases,
~
~
and the right boundary monotonically decreases, and
A
⊕
B
belongs to
Λ
. The
proposition 2.1 is proved.
Similarly, it is possible to show that sum of
-unimodal numbers is a
Λ
-unimodal number, and the sum of
Λ
-tolerance and
Λ
-unimodal numbers is
Λ
-tolerance number. If
, and
R
=
R
=
R
, then sum of these numbers is
L
=
L
=
L
Λ
1
2
1
2
(
)
-number and belongs to
Λ
.
L
−
R
The Proposition 2.2.
[121] Product (result of multiplication) of
Λ
-tolerance
numbers is a
Λ
-tolerance number.
The proof
. Let us consider a fuzzy number which is the product of
-tolerance
Λ
~
~
(
)
(
)
A
≡
a
,
a
,
a
1
,
a
B
≡
b
,
b
,
b
2
,
b
numbers
,
, and let us denote it through
1
2
L
R
1
2
L
R
1
2
-level of numbers
~
and
~
according to (2.1)
~
~
~
D
=
A
⊗
B
. Let write out sets of
α
[
]
[
]
()
()
1
2
−
1
−
1
A
=
A
,
A
=
a
−
L
α
a
,
a
+
R
α
a
;
α
α
α
1
1
L
2
1
R
1
1
