Information Technology Reference
In-Depth Information
Chapter 6
Fuzzy Arithmetic
6.1 Introduction
As it was explained before, any operation
∗:R × R ₒ R
, can be extended to one
:[
0
,
1
]
R
×[
0
,
1
]
R
ₒ[
0
,
1
]
R
, by means of the extension principle:
(μ
˃)(
t
)
=
Sup
t
=
x
∗
y
min
(μ(
x
), ˃(
y
)).
}
R
ↂ[
]
R
.
This extension includes the crisp subsets
A
ↂ R
, since
μ
A
∈{
0
,
1
0
,
1
For example, with
A
1
={
1
,
2
,
3
}
, and
A
2
={
1
,
3
,
5
}
, and only taking into account
the numbers in
N ↂ R
,itis
(μ
A
1
↕
μ
A
2
)(
t
)
=
Sup
min
(μ
A
1
(
x
), μ
A
2
(
y
)),
with
t
,
x
,
y
in
N
,
t
=
x
+
y
and
+
the addition of natural numbers. Since
x
+
y
∈{
2
,
4
,
5
,
3
,
5
,
7
,
8
}
, it results
μ
A
2
↕
μ
A
2
=
μ
{
2
,
3
,
4
,
5
,
6
,
7
,
8
}
, that is
A
1
↕
A
2
={
2
,
3
,
4
,
5
,
6
,
7
,
8
}
.
In the same vein, if
A
1
=[
a
,
b
]
, and
A
2
=[
c
,
d
]
are intervals of the real line
R
,
it results
(μ
A
1
↕
μ
A
2
)(
t
)
=
Sup
min
(μ
[
a
,
b
]
(
x
), μ
[
c
,
d
]
(
y
))
t
=
x
+
y
=
μ
[
a
+
c
,
b
+
d
]
,or
[
a
,
b
]↕[
c
,
d
]=[
a
+
c
,
b
+
d
]
.
Analogously, it results
•[
a
,
b
][
c
,
d
]=[
a
−
d
,
b
−
c
]
•[
a
,
b
]⊗[
c
,
d
]=[
min
(
ad
,
ac
,
bd
,
bc
),
max
(
ad
,
ac
,
bd
,
bc
)
]
min
c
,
d
max
c
,
d
.
a
b
b
a
b
b
•
If 0
/
∈[
c
,
d
]
,
[
a
,
b
]
÷
[
c
,
d
]=
d
,
c
,
,
d
,
c
,
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