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In-Depth Information
()
[]
{
}
A
=
x
∈
R
:
μ
x
≥
α
,
α
∈
0
.
(1.3)
α
A
-cut) of FN
~
with membership function
is referred to as set of
-level (
α
α
()
μ
x
, by analogy with (1.1).
FN
~
with membership function
~
()
μ
x
is referred to as normal, if
~
()
max
μ
x
=
1
x
∈
R
.
~
A
x
FN
~
with membership function
()
μ
x
is referred to as unimodal, if there is a
~
()
x
∈
R
unique point
, for which the equality
μ
x
=
0
is satisfied.
~
A
FN
~
with membership function
()
μ
x
is referred to as multimodal, if a point
~
()
x
∈
R
:
μ
x
=
1
is not unique.
FN
~
with membership function
~
A
()
μ
x
is referred to as tolerant, if there is an
~
()
μ
x
=
1
interval for all points of which the equality
is satisfied. This interval is
~
A
referred to as an interval of tolerance of FN
~
.
The expanded binary arithmetical operation denoted as
~
[15], for fuzzy
∇
numbers
~
,
~
with membership functions
()
()
μ
x
μ
x
,
, accordingly, is
~
~
defined as follows:
~
~
~
~
[
]
()
()
( )
μμ
(1.4)
Based on (1.4), one can define such arithmetical operations as expanded addition,
subtraction, multiplication and division of FN
~
,
~
with membership functions
()
C
=
A
∇
B
⇔
z
=
∨
z
∧
μ
y
;
∀
x
,
y
,
z
∈
R
.
~
~
~
C
A
B
z
=
x
∇
y
()
μ
x
μ
x
,
, accordingly, for special cases of an intersection operator
∧
in a
~
~
()
()
(
)
μ
x
∧
μ
x
=
min
μ
,
μ
class of triangular norms
, and also an union
~
~
~
~
A
B
A
B
()
()
∨
in a class of triangular conorms
(
)
μ
x
∨
μ
x
=
min
μ
,
μ
operator
:
~
~
~
~
A
B
A
B
~
~
~
[
]
()
()
()
⎫
C
=
A
⊕
B
⇔
μ
z
=
max
min
μ
x
,
μ
x
;
~
~
~
C
A
B
⎪
z
=
x
+
y
~
~
~
[
]
()
()
()
C
=
A
−
B
⇔
μ
z
=
max
min
μ
x
,
μ
x
;
⎪
⎪
~
~
~
C
A
B
z
=
x
−
y
~
~
~
⎬
[
]
()
()
()
(1.5)
C
=
A
⊗
B
⇔
μ
z
=
max
min
μ
x
,
μ
x
;
~
~
~
⎪
B
C
A
z
=
x
⋅
y
~
~
~
[
]
⎪
⎪
()
()
()
C
=
A
:
B
⇔
μ
z
=
max
min
μ
x
,
μ
x
,
y
≠
0
~
~
~
C
A
B
⎭
z
=
x
:
y
(
)
FN of
-type [15] are frequently used to solve problems in the various areas.
The following conditions are superimposed on functions
L
and
R
:
L
−
R
()
()
()
()
1.
L
0
=
R
0
=
1
L
1
=
R
1
=
1
[]
()
()
2.
and
are nonincreasing functions at
.
L
x
∀
x
∈
0
R
x
