Information Technology Reference
In-Depth Information
Let us consider properties of operations
. Let
~
,
~
,
~
be fuzzy sets,
∪
and
∩
then following conditions are satisfied.
1.
Commutativity
~
~
~
~
~
~
~
~
A
∪
B
=
B
∪
A
,
A
∩
B
=
B
∩
A
2.
Associativity
(
~
~
)
~
~
(
~
~
) (
~
~
)
~
~
(
~
~
)
A
∩
B
∩
C
=
A
∩
B
∩
C
,
A
∪
B
∪
C
=
A
∪
B
∩
C
.
3.
Idempotency
~
~
~
~
~
~
A
∩
A
=
A
,
A
∪
A
=
A
4.
Distributivity
(
~
~
~
) (
~
~
) (
~
~
)
~
(
~
~
) (
~
~
) (
~
~
)
A
∩
B
∪
C
=
A
∩
B
∪
A
∩
C
,
A
∪
B
∩
C
=
A
∪
B
∩
A
∪
C
.
~
~
A
∪ φ
=
A
5.
~
A
∩
φ =
φ
.
6.
~
7.
A
∪
X
=
X
.
~
~
8.
A
∩
X
=
A
9.
Laws of dualization
~
~
~
~
~
~
~
~
A
∩
B
=
A
∪
B
,
A
∪
B
=
A
∩
B
.
Unlike definite sets, for fuzzy ones, in the general case:
~
~
~
~
AA
There is a possibility to construct methods of processing and analysis of fuzzy
information with use of fuzzy sets and arithmetical operations [15. 41—44]. The
algebra of fuzzy numbers is the mathematical basis to construct such methods.
Fuzzy number (FN)
~
is referred to as fuzzy subset of set of real numbers
R
possessing membership function
∩
≠
φ
,
A
∩
A
≠
φ
.
[]
μ
:
R
→
0
.
~
A
is referred to as the support of FN
~
, if
The subset
S
A
⊂
R
[
]
()
S
=
supp
A
=
x
:
μ
x
>
0
.
~
A
A
FN
~
is referred to as positive, if
∀
∈
,
x
>
0
, and negative, if
x
S
,
∀
x
∈
S
A
A
x
<
0
.
A subset of the real line
R
, which defined in the form