Information Technology Reference
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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
 
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