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For example,
•[
7
,
8
]↕[−
1
,
9
]=[
6
,
17
]
•[
7
,
8
][−
1
,
9
]=[−
2
,
9
]
•[
3
,
4
][
2
,
2
]=[
6
,
8
]
•[
4
,
10
]
÷
[
1
,
2
]=[
2
,
10
]
•
2
↕[
7
,
8
]=[
2
,
2
]↕[
7
,
8
]=[
9
,
10
]
•
2
[
7
,
8
]=[
2
,
2
][
7
,
8
]=[
min
(
14
,
16
,
14
,
16
),
max
(
14
,
16
)
]=[
14
,
16
]
]=
min
2
,
2
,
max
2
,
2
=
2
,
4
8
8
•[
7
,
8
]
÷
2
=[
7
,
8
]
÷
[
2
,
2
In short, and accordingly with the 'preservation of the classical case', through
the extension principle both the numerical and the interval arithmetics are preserved.
What follows is a yet larger arithmetic with fuzzy concepts.
For all
]
R
, and all
r
μ
∈[
0
,
1
∈[
0
,
1
]
, it can be computed that:
(
r
↕
μ)(
t
)
=
(μ
r
↕
μ)(
t
)
=
Sup
min
(μ
r
(
x
), μ(
y
))
•
t
=
x
+
y
min
1
μ(
, μ(
y
),
if
x
=
r
y
),
x
=
r
=
Sup
=
Sup
0
, μ(
y
),
if
x
=
r
0
,
x
=
r
t
=
x
+
y
t
=
x
+
y
=
μ(
t
−
r
).
((
−
)
↕
μ)(
)
=
μ(
−
)
(μ
1
↕
μ)(
)
=
μ(
−
)
(μ
0
↕
μ)(
)
=
μ(
)
Hence
1
t
t
1
,
t
t
1
,
t
t
.
))
=
μ
r
,if
r
•
(
r
μ)(
t
)
=
(μ
r
μ)(
t
)
=
Sup
t
min
(μ
r
(
x
), μ(
t
=
0. Hence,
=
x
·
y
r
μ
(
t
)
=
μ(
rt
)
,
(
1
μ)(
t
)
=
μ(
t
)
,
(μ
1
μ)(
t
)
=
μ(
t
)
.
•
(μ
0
μ)(
t
)
=
Sup
t
=
x
·
y
min
(μ
0
(
x
), μ(
t
))
=
0
=
μ
0
(
t
)
.
μ
t
,
if
t
=
0
1
•
μ
(
t
)
=
Sup
t
=
min
(
1
, μ(
y
))
=
Sup
t
=
y
μ(
y
)
=
0
,
if
t
=
0
x
y
x
Example 6.1.1
Given the fuzzy set
compute 2
μ
and 2
↕
μ
.
It is
(
2
μ)(
t
)
=
μ(
t
/
2
)
, and
(
2
↕
μ)(
t
)
=
μ(
t
−
2
)
. Hence,
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