Information Technology Reference
In-Depth Information
The following are examples of fuzzy numbers.
6.2.1 Operations with Fuzzy Numbers
It can be proven that,
if
∗∈{+
,
−
,
×
,
:}
,
and if
μ, ˃
are fuzzy numbers, then
μ
˃
is also a fuzzy number
. For example, with the fuzzy number,
⊧
⊨
x
−
2
,
if x
∈
(
2
,
3
)
μ
3
(
x
)
=
4
−
x
,
if x
∈
(
3
,
4
)
⊩
,
,
0
otherwise
for which
[
a
,
b
]=[
3
,
3
]={
3
}
,
L
(
x
)
=
x
−
2,
R
(
x
)
=
4
−
x
, it results
⊧
⊨
t
−
4
,
if
t
∈[
4
,
6
]
2
8
−
t
(μ
3
↕
μ
3
)(
t
)
=
Sup
t
=
x
+
y
min
(μ
3
(
x
), μ
3
(
y
))
=
,
if
t
∈[
6
,
8
]
⊩
2
,
0
otherwise
since for
t
<
4, or
x
+
y
<
4, and for
t
>
8, or
x
+
y
>
8, it should be
(μ
3
↕
μ
3
)(
t
)
=
0;
for
t
=
6, or
x
+
y
=
6, it should be
(μ
3
↕
μ
3
)(
6
)
=
1, and:
•
For
t
∈[
4
,
6
]
,
L
is the segment joining
(
4
,
0
)
and
(
6
,
1
)
, that is
xy
1
401
611
x
−
4
0
=
=−
x
+
2
y
+
4
,
or
y
=
2
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