I
n
t
e
r
n
at
io
n
al
Jou
r
n
al
of
A
d
van
c
e
s
i
n
A
p
p
li
e
d
S
c
ie
n
c
e
s
(
I
JA
A
S
)
V
ol
.
14
, N
o.
4
,
D
e
c
e
m
be
r
20
25
, pp.
1083
~
1088
I
S
S
N
:
2252
-
8814
,
D
O
I
:
10.11591/
ij
a
a
s
.
v14.
i
4
.
pp1083
-
108
8
1083
Jou
r
n
al
h
om
e
page
:
ht
tp
:
//
ij
aas
.i
ae
s
c
or
e
.c
om
N
e
w
ap
p
r
oac
h
of
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z
a A
lm
yal
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1
,
Jw
n
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ar
M
o
s
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ah
ar
y
2
1
D
e
pa
r
t
m
e
nt
of
M
a
t
he
m
a
t
i
c
s
a
nd
C
om
put
e
r
A
ppl
i
c
a
t
i
ons
, C
ol
l
e
ge
of
S
c
i
e
nc
e
,
A
L
-
M
ut
ha
nna
U
ni
ve
r
s
i
t
y, A
l
-
M
ut
ha
nn
a
, I
r
a
q
2
D
e
pa
r
t
m
e
nt
of
A
gr
i
c
ul
t
ur
a
l
S
t
a
t
i
s
t
i
c
s
, S
C
S
C
ol
l
e
ge
of
A
gr
i
c
ul
t
ur
e
, A
s
s
a
m
A
gr
i
c
ul
t
ur
a
l
U
ni
ve
r
s
i
t
y, D
hubr
i
, I
ndi
a
A
r
t
ic
le
I
n
f
o
A
B
S
T
R
A
C
T
A
r
ti
c
le
h
is
to
r
y
:
R
e
c
e
iv
e
d
J
a
n
05
,
2025
R
e
vi
s
e
d
J
un
20
,
2025
A
c
c
e
pt
e
d
J
ul
14
,
2025
This paper provides a comparative analysis of the fuzzy Q
-
neighborhood and
the
fuzzy
neighborhood
system
of
a
fuzzy
point.
Specifically,
we
investigate
the
relationship
between
the
elements
of
these
systems
when
bo
th
are
defined at the same fuzz
y point. We address questions such as: how are
these
elements
interconn
ected,
and
which
system
contains
the
other?
Furtherm
ore,
we
give
the
dual
of
the
fuzzy
Q
-
neighborhood
system,
which
is
nam
ed
the
fuzzy
DQ
-
neighborhood
system,
and
prove
that
these
two
systems
are
not
equivalen
t.
Finall
y,
we
examine
the
properties
of
these
systems
to
determin
e
whether
they
satisfy
the
conditions
of
fuzzy
topology,
Supra
topolo
gy,
or
filter theo
ry.
K
e
y
w
o
r
d
s
:
C
om
pa
r
a
ti
ve
a
na
ly
s
i
s
F
uz
z
y f
il
te
r
F
uz
z
y ne
ig
hbor
hood s
ys
te
m
F
uz
z
y
s
upr
a
to
pol
ogy
Q
ua
s
i
-
c
oi
nc
id
e
nt
This is an
open
acce
ss artic
le unde
r the
CC BY
-
SA
license.
C
or
r
e
s
pon
di
n
g A
u
th
or
:
A
m
e
r
H
im
z
a
A
lm
ya
ly
D
e
pa
r
tm
e
nt
of
M
a
th
e
m
a
ti
c
s
a
nd C
om
put
e
r
A
ppl
ic
a
ti
ons
, C
ol
le
ge
of
S
c
ie
nc
e
, A
L
-
M
ut
ha
nna
U
ni
ve
r
s
it
y
S
a
m
a
w
a
h, A
l
M
ut
ha
nna
G
ove
r
nor
a
te
, I
r
a
q
E
m
a
il
:
a
m
e
r
hi
m
z
i@mu.e
du.i
q
1.
I
N
T
R
O
D
U
C
T
I
O
N
T
he
pr
in
c
ip
le
s
of
f
uz
z
y
s
ys
t
e
m
s
w
e
r
e
in
tr
oduc
e
d
by
Z
a
de
h
[
1]
,
s
ubs
e
que
nt
ly
in
f
lu
e
nc
in
g
va
r
io
us
m
a
th
e
m
a
ti
c
a
l
di
s
c
ip
li
ne
s
s
uc
h
a
s
a
lg
e
br
a
[
2]
,
to
pol
ogy
[
3]
,
a
nd
ge
om
e
tr
y
[
4]
.
A
m
ong
th
e
s
e
,
to
pol
ogy
h
a
s
s
e
e
n
th
e
m
os
t
s
ig
ni
f
ic
a
nt
d
e
ve
lo
pm
e
nt
,
w
hi
c
h
w
a
s
la
te
r
c
a
ll
e
d
f
uz
z
y
to
pol
ogy
.
I
t
is
d
e
f
in
e
d
th
r
ough
two
pr
im
a
r
y
a
ppr
oa
c
he
s
:
th
os
e
of
C
ha
ng
[
3]
a
nd
L
ow
e
n
[
5]
.
I
n
th
is
w
or
k,
w
e
a
dopt
C
ha
ng'
s
f
r
a
m
e
w
or
k
[
3]
,
w
hi
c
h
de
f
in
e
s
a
f
uz
z
y
to
pol
ogy
on
th
e
s
e
t
X
a
s
a
f
a
m
il
y
T
⊆
I
X
,
w
he
r
e
I
=
[
0
,
1
]
,
s
a
ti
s
f
yi
ng
th
e
w
e
ll
-
known
a
xi
om
s
.
T
hi
s
a
ppr
oa
c
h
ha
s
b
e
e
n
e
xt
e
ns
iv
e
ly
s
tu
di
e
d
by
r
e
s
e
a
r
c
he
r
s
s
uc
h
a
s
Š
os
ta
k
[
6]
,
R
a
m
a
da
n
[
7]
,
a
nd
V
a
di
ve
l
[
8]
.
R
e
s
e
a
r
c
h
in
th
is
f
ie
ld
ha
s
pr
im
a
r
il
y
a
im
e
d
to
e
xpl
or
e
n
e
w
s
tr
uc
tu
r
e
s
th
a
t
e
xpa
nd
e
xi
s
ti
ng
m
a
th
e
m
a
ti
c
a
l
hor
iz
ons
by
ge
ne
r
a
li
z
in
g
f
ounda
ti
ona
l
c
onc
e
pt
s
or
unc
ove
r
in
g
nove
l
r
e
la
ti
ons
hi
ps
.
G
iv
e
n
th
a
t
f
uz
z
y
to
pol
ogy,
a
s
a
di
s
c
ip
li
ne
,
is
r
e
la
ti
ve
ly
young
,
or
ig
in
a
ti
ng
in
1965
,
it
s
de
v
e
lo
pm
e
nt
ha
s
pr
oc
e
e
d
e
d
f
r
om
f
ounda
ti
ona
l
to
pi
c
s
li
ke
to
pol
ogy,
th
r
ough
f
uz
z
y
s
ys
te
m
s
.
E
s
ta
bl
is
hi
ng
a
nd
de
f
in
in
g
th
e
s
e
c
or
e
c
onc
e
pt
s
in
e
vi
ta
bl
y
ge
ne
r
a
te
s
ne
w
que
s
ti
on
s
a
nd
pr
obl
e
m
s
,
s
uc
h
a
s
unde
r
s
ta
ndi
ng
th
e
r
e
la
ti
ons
hi
p
be
twe
e
n
di
f
f
e
r
e
nt
c
onc
e
pt
s
,
de
te
r
m
in
in
g
w
he
th
e
r
one
c
onc
e
pt
le
a
ds
to
a
not
he
r
,
a
nd
id
e
nt
if
yi
ng
th
e
c
ondi
ti
ons
r
e
qui
r
e
d
f
or
s
uc
h
tr
a
ns
it
io
ns
. A
ddi
ti
ona
ll
y, w
e
a
k t
he
s
e
c
ondi
ti
ons
c
a
n yi
e
ld
ne
w
c
onc
e
pt
s
t
ha
t
de
m
a
nd f
ur
th
e
r
e
xpl
or
a
ti
on.
T
he
c
onc
e
pt
of
f
uz
z
y
ne
ig
hbor
hood
hol
ds
a
s
ig
ni
f
ic
a
nt
pl
a
c
e
i
n
m
a
th
e
m
a
ti
c
s
,
a
nd
e
xt
e
nds
it
s
e
f
f
e
c
t
in
to
f
ie
ld
s
a
s
c
om
put
e
r
s
c
ie
nc
e
a
nd
e
ngi
ne
e
r
in
g
,
pa
r
ti
c
ul
a
r
l
y
in
th
e
c
ont
e
xt
o
f
da
ta
m
ode
li
ng
[
9]
–
[
11
]
.
T
he
r
e
f
or
e
,
it
is
e
s
s
e
nt
ia
l
to
e
xpl
or
e
th
e
to
pi
c
f
r
om
m
ul
ti
pl
e
a
n
gl
e
s
.
O
ne
s
u
c
h
a
s
p
e
c
t
in
vol
ve
s
e
xa
m
in
in
g
th
e
va
r
io
us
ty
pe
s
of
f
uz
z
y
ne
ig
hbor
hoods
,
a
m
ong
w
hi
c
h
th
e
f
uz
z
y
Q
-
ne
ig
hbor
hood
[
12]
,
[
13]
is
of
pa
r
ti
c
ul
a
r
in
te
r
e
s
t.
D
ur
in
g
our
s
tu
dy,
w
e
not
e
th
a
t
th
e
dua
l
of
th
e
f
uz
z
y
Q
-
ne
ig
hbor
hood
ha
s
r
e
c
e
iv
e
d
li
tt
le
to
no
a
tt
e
nt
io
n
in
pr
io
r
r
e
s
e
a
r
c
h.
Q
ue
s
ti
ons
n
a
tu
r
a
ll
y
a
r
is
e
:
w
ha
t
f
or
m
doe
s
th
is
dua
l
ta
k
e
?
,
i
s
it
pos
s
ib
le
to
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2252
-
8814
I
nt
J
A
dv A
ppl
S
c
i
,
V
ol
.
14
, N
o.
4
,
D
e
c
e
m
be
r
20
25
:
108
3
-
108
8
1084
c
ons
tr
uc
t
a
f
uz
z
y
to
pol
ogi
c
a
l
c
a
te
gor
y
ba
s
e
d
on
it
?
,
a
nd
th
e
s
e
a
r
e
s
om
e
of
th
e
c
e
nt
r
a
l
in
qui
r
ie
s
w
e
a
im
to
e
xpl
or
e
a
nd a
ddr
e
s
s
t
hr
oughout t
hi
s
s
tu
dy.
O
ur
s
tu
dy
pr
e
s
e
nt
s
a
n
a
ppr
oa
c
h
to
th
e
f
uz
z
y
Q
-
ne
ig
hbor
hood
of
a
f
uz
z
y
poi
nt
[
12]
,
s
pe
c
if
ic
a
ll
y
in
ve
s
ti
ga
ti
ng
it
s
r
e
la
ti
ons
hi
p
w
it
h
th
e
f
uz
z
y
ne
ig
hbor
hood
of
a
f
uz
z
y
poi
nt
[
14]
–
[
16]
.
P
r
e
vi
ous
s
tu
di
e
s
ha
ve
la
r
ge
ly
ove
r
lo
oke
d
th
is
s
pe
c
if
ic
r
e
la
ti
ons
hi
p
in
th
e
ir
a
na
ly
s
e
s
,
a
lt
hough
pr
io
r
r
e
s
e
a
r
c
h
ha
s
e
xa
m
in
e
d
f
uz
z
y
ne
ig
hbor
hood
s
ys
te
m
s
th
r
ough
c
ons
is
te
nt
f
unc
ti
ons
a
nd
qua
nt
a
le
-
ba
s
e
d
s
tr
uc
tu
r
e
s
a
s
[
17]
–
[
19]
.
I
n
c
ont
r
a
s
t,
th
is
pa
pe
r
br
in
gs
it
to
th
e
f
o
r
e
f
r
ont
,
o
f
f
e
r
in
g
a
c
le
a
r
e
xpl
a
na
ti
on
s
uppor
te
d
by
il
lu
s
tr
a
ti
ve
e
xa
m
pl
e
s
.
A
t
th
e
s
a
m
e
ti
m
e
,
w
e
in
tr
oduc
e
d
th
e
dua
l
of
th
e
f
uz
z
y
Q
-
ne
ig
hbor
hood
c
onc
e
pt
a
nd
c
la
r
if
ie
d
how
it
r
e
la
te
s
to
pr
e
vi
ous
ly
e
s
ta
bl
is
he
d
f
uz
z
y
n
e
ig
hbor
hood
s
tr
uc
tu
r
e
s
.
W
e
a
l
s
o
e
xa
m
in
e
w
he
th
e
r
r
e
la
xi
ng
c
e
r
ta
in
c
ondi
ti
ons
le
a
ds
to
ne
w
in
s
ig
ht
s
a
nd
e
xpl
or
e
th
e
c
onne
c
ti
ons
be
tw
e
e
n
th
e
s
e
s
ys
t
e
m
s
. T
hr
ough
di
s
c
u
s
s
io
n,
e
x
a
m
pl
e
s
, a
nd
th
e
or
e
ti
c
a
l
a
na
ly
s
is
.
E
xpl
or
in
g
th
e
c
a
te
gor
y
-
th
e
or
e
ti
c
a
pp
r
oa
c
h
to
f
uz
z
y
dua
l
Q
-
ne
ig
hbor
hood
c
a
n
le
a
d
to
th
e
de
ve
lo
pm
e
nt
of
a
ne
w
c
a
te
gor
ic
a
l
s
tr
uc
tu
r
e
th
a
t
di
f
f
e
r
s
f
unda
m
e
nt
a
ll
y
f
r
om
th
e
f
uz
z
y
f
il
te
r
f
r
a
m
e
w
or
k
ty
pi
c
a
ll
y i
nduc
e
d by f
uz
z
y ne
ig
hbor
hoods
a
nd Q
-
ne
ig
hbor
hood
.
2.
P
R
E
L
I
M
I
N
A
R
I
E
S
W
e
w
il
l
r
e
c
a
ll
s
om
e
c
onc
e
pt
s
th
a
t
w
e
ne
e
d
in
th
is
pa
pe
r
.
I
t
i
s
im
por
ta
nt
to
unde
r
s
ta
nd
th
e
pa
pe
r
pr
oc
e
dur
e
s
.
F
r
om
now
on
X
a
lwa
ys
d
e
not
e
s
a
none
m
pt
y
s
e
t
c
a
ll
e
d
th
e
uni
v
e
r
s
a
l
s
e
t,
it
s
e
le
m
e
nt
s
ha
ve
m
e
m
be
r
s
hi
p
1,
a
nd
A
is
a
f
uz
z
y
s
ub
s
e
t
of
X
,
w
hi
c
h
w
e
w
il
l
c
a
ll
la
t
e
r
f
uz
z
y
s
e
t.
O
n
th
e
c
ont
r
a
r
y,
th
e
e
m
pt
y
s
e
t
∅
wh
os
e
e
le
m
e
nt
s
ha
ve
m
e
m
be
r
s
hi
p 0. Al
s
o
,
don’
t
f
or
ge
t,
I
de
not
e
s
[
0
,
1
]
w
he
r
e
0
≤
μ
A
(
x
)
≤
1
, f
or
e
a
c
h
A
is
a
f
uz
z
y s
e
t
in
X
.
W
e
w
il
l
r
e
c
a
ll
s
om
e
c
onc
e
pt
s
th
a
t
w
e
ne
e
d
in
th
is
pa
pe
r
.
I
t
i
s
im
por
ta
nt
to
unde
r
s
ta
nd
th
e
pa
pe
r
pr
oc
e
dur
e
s
.
F
r
om
now
on
X
a
lwa
ys
d
e
not
e
s
a
none
m
pt
y
s
e
t
c
a
ll
e
d
th
e
uni
v
e
r
s
a
l
s
e
t,
it
s
e
le
m
e
nt
s
ha
ve
m
e
m
be
r
s
hi
p
1,
a
nd
A
is
a
f
uz
z
y
s
ub
s
e
t
of
X
,
w
hi
c
h
w
e
w
il
l
c
a
ll
la
t
e
r
f
uz
z
y
s
e
t.
O
n
th
e
c
ont
r
a
r
y,
th
e
e
m
pt
y
s
e
t
∅
wh
os
e
e
le
m
e
nt
s
ha
ve
m
e
m
be
r
s
hi
p 0. Al
s
o
,
don’
t
f
or
ge
t,
I
de
not
e
s
[
0
,
1
]
w
he
r
e
0
≤
μ
A
(
x
)
≤
1
, f
or
e
a
c
h
A
is
a
f
uz
z
y s
e
t
in
X
.
D
e
f
in
it
io
n
2.1:
a
f
uz
z
y
s
e
t
A
in
uni
ve
r
s
a
l
s
e
t
X
is
a
c
ol
le
c
ti
on
of
or
de
r
e
d
pa
ir
s
a
s
A
=
{
(
x
,
μ
A
(
x
)
)
:
x
∈
X
}
,
μ
A
(
x
)
∈
I
,
[
1]
,
[
2]
.
D
e
f
in
it
io
n
2.2:
a
f
uz
z
y
poi
nt
x
t
,
(
0
<
≤
1
)
,
in
X
is
a
f
uz
z
y
s
e
t
w
hi
c
h
ta
ke
s
th
e
va
lu
e
0
f
or
a
ll
y
∈
X
e
xc
e
pt
f
or
one
e
le
m
e
nt
,
s
a
y
x
∈
X
,
[
1]
,
[
3]
.
D
e
f
in
it
io
n
2.3:
l
e
t
x
t
be
f
uz
z
y
poi
nt
s
a
nd
A
,
B
a
r
e
f
uz
z
y
s
e
ts
in
X
[
1]
,
[
3]
th
e
n
w
e
s
a
y:
i
)
x
t
∈
A
if
f
t
≤
μ
A
(
x
)
.
O
th
e
r
th
a
n
th
a
t
x
t
∉
A
;
ii
)
B
⊆
A
if
f
μ
B
(
x
)
≤
μ
A
(
x
)
,
∀
x
∈
X
.
O
th
e
r
th
a
n
B
⊈
A
;
ii
i)
A
=
B
if
f
μ
A
(
x
)
=
μ
B
(
x
)
,
∀
x
∈
X
.
O
th
e
r
th
a
n
A
≠
B
;
iv
)
th
e
uni
on
of
f
uz
z
y
s
e
ts
A
i
⊆
X
ha
s
a
m
e
m
be
r
s
hi
p
f
un
c
ti
on
w
hi
c
h
de
f
in
e
d
f
or
e
a
c
h
x
∈
X
by:
μ
⋃
A
i
i
∈
I
(
x
)
=
m
a
x
{
μ
A
i
(
x
)
,
i
∈
I
}
;
v)
th
e
in
te
r
s
e
c
ti
on
of
f
uz
z
y
s
e
ts
A
i
⊆
X
,
i
=
1
,
2
,
,
…
,
n
ha
s
m
e
m
be
r
s
hi
p
f
unc
ti
on
w
hi
c
h
de
f
in
e
d
f
or
e
a
c
h
x
∈
X
by
by
μ
⋂
A
i
n
i
=
1
(
x
)
=
m
in
{
μ
A
i
(
x
)
,
i
=
1
,
2
,
,
…
,
n
}
;
a
nd
vi
)
th
e
c
om
pl
e
m
e
nt
of
a
f
uz
z
y
s
e
t
A
⊆
X
ha
s
a
m
e
m
be
r
s
hi
p
f
un
c
ti
on
w
hi
c
h
de
f
in
e
d
f
or
e
a
c
h
x
∈
X
by:
μ
A
c
(
x
)
=
1
−
μ
A
(
x
)
.
D
e
f
in
it
io
n
2.4:
th
e
s
uppor
t
of
s
ubs
e
t
A
(
s
up
A
)
is
th
e
c
r
is
p
s
e
t
of
a
ll
e
le
m
e
nt
s
in
X
th
a
t
ha
ve
a
non
-
z
e
r
o
m
e
m
be
r
s
hi
p,
i.
e
.,
s
up
A
=
{
x
∈
X
:
μ
A
(
x
)
>
0
}
,
[
1]
,
[
4]
.
T
he
or
e
m
2.1:
l
e
t
A
a
nd
B
be
two
f
uz
z
y
s
e
ts
in
th
e
uni
ve
r
s
a
l
s
e
t
X
[
4]
,
[
5]
,
th
e
n:
i)
A
⊆
B
if
f
f
o
r
e
a
c
h
x
t
∈
A
th
e
n
x
t
∈
B
;
ii
)
A
=
B
if
f
A
⊆
B
a
nd
B
⊆
A
;
ii
i)
x
t
∈
A
∪
B
if
f
x
t
∈
A
or
x
t
∈
B
;
a
nd
iv
)
x
t
∈
A
∩
B
if
f
x
t
∈
A
a
nd
x
t
∈
B
.
H
ow
e
ve
r
,
th
e
r
e
doe
s
n'
t
ne
e
d
to
e
xi
s
t
a
r
e
la
ti
on
be
twe
e
n
th
e
f
or
m
ul
a
s
x
t
∈
A
a
nd
x
t
∈
A
c
a
nd
w
e
ha
ve
x
t
∈
A
c
if
f
t
≤
1
−
μ
A
(
x
)
,
x
t
∉
A
c
if
f
t
>
1
−
μ
A
(
x
)
.
D
e
f
in
it
io
n
2.5:
l
e
t
X
be
a
uni
ve
r
s
a
l
s
e
t
[
6]
,
[
7]
,
th
e
n:
i)
a
f
uz
z
y
poi
nt
x
t
in
X
is
s
a
id
to
be
qu
a
s
i
-
c
oi
nc
id
e
nt
w
it
h
th
e
f
uz
z
y
s
e
t
A
in
X
,
de
not
e
d
by
x
t
qA
,
if
a
nd
onl
y
i
f
t
+
μ
A
(
x
)
>
1
,
a
nd
not
qua
s
i
-
c
oi
nc
id
e
nt
w
it
h t
he
f
uz
z
y s
e
t
A
, de
not
e
d by
x
t
↽
qA
, i
f
a
nd only i
f
t
+
μ
A
(
x
)
≤
1
;
a
nd i
i)
A
f
uz
z
y s
e
t
A
in
X
is
s
a
id
t
o be
qua
s
i
-
c
oi
nc
id
e
nt
w
it
h
th
e
f
uz
z
y
s
e
t
B
in
X
,
de
not
e
d
by
Aq
B
,
if
f
th
e
r
e
e
xi
s
t
s
x
in
X
s
uc
h
th
a
t
μ
A
(
x
)
+
μ
B
(
x
)
>
1
a
nd
not
qua
s
i
-
c
oi
nc
id
e
nt
w
it
h
th
e
f
uz
z
y
s
e
t
B
in
X
,
de
not
e
d
by
A
↽
qB
,
if
f
f
or
e
a
c
h
x
in
X
,
th
e
n
μ
A
(
x
)
+
μ
B
(
x
)
≤
1
.
T
he
or
e
m
2.2:
c
ons
id
e
r
X
a
s
a
uni
ve
r
s
a
l
s
e
t,
a
nd
le
t
A
a
nd
B
r
e
pr
e
s
e
nt
f
uz
z
y
s
ubs
e
ts
w
it
hi
n
X
[
8]
,
[
20]
, t
he
n:
i
)
A
↽
qB
if
f
A
⊆
B
c
;
ii
)
A
⊆
B
if
f
x
t
qB
,
∀
x
t
qA
;
ii
i)
A
↽
q
A
c
F
or
e
ve
r
y f
uz
z
y s
e
t
A
de
f
in
e
d w
it
hi
n
X
;
iv
)
A
∩
B
=
∅
th
e
n
A
↽
qB
;
a
nd v)
x
t
↽
qA
if
f
x
t
∈
A
c
.
R
e
m
a
r
k
2.1:
n
ot
e
th
a
t
if
Aq
B
th
e
n
B
q
A
a
nd
vi
c
e
ve
r
s
a
,
a
nd
s
o
A
↽
qB
.
N
ow
,
w
e
w
il
l
in
tr
oduc
e
th
e
c
onc
e
pt
s
of
f
uz
z
y
to
pol
ogi
c
a
l
s
pa
c
e
s
,
w
hi
c
h
w
e
w
il
l
ne
e
d
l
a
te
r
.
D
e
f
in
it
io
n
2.6:
a
f
uz
z
y
to
pol
ogy
T
,
is
de
f
in
e
d
a
s
a
c
ol
le
c
ti
on
of
f
uz
z
y
s
e
ts
on
a
uni
ve
r
s
a
l
s
e
t
X
th
a
t
s
a
ti
s
f
ie
s
th
e
f
ol
lo
w
in
g
c
ondi
ti
ons
[
3
]
:
i
)
∅
,
X
∈
T
;
ii
)
if
A
,
B
∈
T
th
e
n
A
∩
B
∈
T
;
a
nd
ii
i)
i
f
A
i
∈
T
f
or
e
ve
r
y
i
∈
I
,
th
e
n
∪
i
∈
I
A
i
∈
T
.
T
is
c
a
ll
e
d
a
f
uz
z
y
to
pol
ogy
f
o
r
X
,
a
nd
th
e
p
a
ir
(
X
,
T
)
I
t
is
c
a
ll
e
d
a
f
uz
z
y
to
pol
ogi
c
a
l
s
pa
c
e
.
E
ve
r
y
m
e
m
be
r
of
T
is
c
a
ll
e
d
a
f
uz
z
y
op
e
n
s
e
t.
A
f
uz
z
y s
e
t
is
c
lo
s
e
d i
f
f
i
ts
c
om
pl
e
m
e
nt
i
s
ope
n.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
A
dv A
ppl
S
c
i
I
S
S
N
:
2252
-
8814
N
e
w
appr
oa
c
h of
t
he
ne
ig
hbor
hood
s
tr
uc
tu
r
e
of
f
uz
z
y
poi
nt
s
(
A
m
e
r
H
imz
a A
lmy
al
y
)
1085
D
e
fi
n
it
io
n
2.7:
in
a
f
uz
z
y
to
pol
ogi
c
a
l
s
pa
c
e
(
X
,
T
)
,
a
f
uz
z
y
s
e
t
A
is
c
on
s
id
e
r
e
d
a
f
uz
z
y
n
e
ig
hbor
hood
of
a
f
uz
z
y
poi
nt
x
t
if
th
e
r
e
e
xi
s
ts
a
f
uz
z
y
ope
n
s
e
t
U
s
uc
h
th
a
t
x
t
∈
U
⊆
A
.
T
he
c
ol
le
c
ti
on
of
a
ll
f
uz
z
y
ne
ig
hbor
hoods
of
x
t
is
r
e
f
e
r
r
e
d
to
a
s
th
e
f
uz
z
y
ne
ig
hbor
hood
s
ys
te
m
a
t
x
t
a
nd
is
de
not
e
d
by
N
x
t
,
[
3]
.
E
xa
m
pl
e
2.1:
le
t
X
=
{
x
,
y
}
a
nd
T
=
{
∅
,
X
,
{
x
0
.
5
}
,
{
y
0
.
6
}
,
{
x
0
.
5
,
y
0
.
6
}
}
,
T
is
a
f
uz
z
y
to
pol
ogy
on
X
.
L
e
t
A
=
{
x
0
.
6
,
y
0
.
5
}
th
e
n
A
is
a
f
uz
z
y
ne
ig
hbor
hood
on
x
t
,
∀
t
≤
0
.
5
,
but
A
is
a
n
i
m
pos
s
ib
le
f
uz
z
y
ne
ig
hbor
hood on
y
t
,
0
<
≤
1
.
T
he
or
e
m
2.3:
a
f
uz
z
y s
e
t
A
is
c
ons
id
e
r
e
d f
uz
z
y
ope
n i
f
f
f
o
r
e
ve
r
y f
uz
z
y poin
t
x
t
c
ont
a
in
e
d
in
A
,
A
is
a
f
uz
z
y
ne
ig
hbor
hood
of
x
t
,
[
3]
.
E
xa
m
pl
e
2.2:
it
'
s
c
le
a
r
th
a
t
A
in
e
xa
m
pl
e
(
2.1)
is
no
t
f
uz
z
y ope
n.
D
e
f
in
it
io
n 2.8:
le
t
A
be
a
f
uz
z
y s
e
t
in
a
f
uz
z
y t
opol
og
ic
a
l
s
pa
c
e
(
X
,
T
)
. T
he
i
nt
e
r
io
r
A
°
a
nd c
lo
s
ur
e
A
̅
a
r
e
de
f
in
e
d a
s
(
1)
a
nd (
2)
[
21
]
,
[
22]
.
°
=
⋃
{
:
⊆
}
(
1)
̅
=
⋂
{
:
⊆
}
(
2)
E
xa
m
pl
e
2.3:
not
e
in
e
xa
m
pl
e
(
2.1
)
,
A
°
=
{
x
0
.
5
}
a
nd
A
̅
=
X
.
D
e
f
in
it
io
n
2.9:
a
f
uz
z
y
f
il
te
r
ℱ
on
a
uni
ve
r
s
a
l
s
e
t
X
is
a
c
ol
le
c
ti
on
of
f
uz
z
y
s
e
t
s
in
X
th
a
t
s
a
ti
s
f
y
th
e
s
e
c
ondi
ti
ons
[
23
]
,
[
24]
:
i)
If
F
1
a
nd
F
2
a
r
e
e
le
m
e
nt
s
of
ℱ
,
th
e
n
F
1
⋂
F
2
m
us
t
a
ls
o
be
lo
ng
to
ℱ
;
ii
)
If
F
is
a
n
e
le
m
e
nt
of
ℱ
a
nd
F
⊆
G
,
th
e
n
G
m
us
t
a
ls
o
be
lo
ng t
o
ℱ
;
a
nd i
ii
)
∅
∉
ℱ
.
R
e
m
a
r
k
2.2
:
f
r
om
3
in
th
e
de
f
in
it
io
n
a
bove
,
th
e
n
th
e
f
il
te
r
is
i
m
pos
s
ib
le
i
n
to
pol
ogy
a
nd
vi
c
e
ve
r
s
a
.
E
xa
m
pl
e
2.4:
th
e
f
uz
z
y
ne
ig
hbor
hood
s
ys
te
m
N
x
t
of
a
ny
f
uz
z
y
p
oi
nt
x
t
in
a
uni
ve
r
s
a
l
s
e
t
X
is
f
uz
z
y
f
il
te
r
.
P
r
oof
:
i)
le
t
G
x
t
&
V
x
t
∈
N
x
t
⟹
∃
B
&
.
x
t
∈
B
⊆
G
x
t
&
x
t
∈
C
⊆
V
x
t
⟹
x
t
∈
B
∩
C
⊆
G
x
t
∩
V
x
t
⟹
G
x
t
∩
V
x
t
∈
N
x
t
;
ii
)
If
G
x
t
∈
N
x
t
⟹
∃
B
in
T
s
.
t
x
t
∈
B
⊆
G
x
t
a
n
d
if
G
x
t
⊆
M
⟹
M
∈
N
x
t
s
in
c
e
x
t
∈
B
⊆
M
;
ii
i)
∅
∉
N
x
t
s
in
c
e
∅
it
ha
s
no
e
le
m
e
nt
w
it
h
t
>
0
;
iv
)
D
e
f
in
it
io
n
2.10:
a
f
uz
z
y
S
upr
a
to
pol
ogy
δ
⊆
I
X
is
de
f
in
e
d
a
s
f
ol
lo
w
s
[
25]
:
–
∅
,
X
∈
δ
.
–
If
B
j
∈
δ
,
∀
j
∈
J
, t
he
n
⋃
B
j
∈
δ
j
∈
J
.
R
e
m
a
r
k
2.3:
f
r
om
th
e
de
f
in
it
io
n
a
bove
,
e
ve
r
y
f
uz
z
y
to
pol
og
y
is
a
f
uz
z
y
S
upr
a
to
pol
ogy
,
but
th
e
c
onve
r
s
e
is
not
tr
ue
in
ge
ne
r
a
l
,
a
s
f
ol
lo
w
s
:
e
xa
m
pl
e
2.5:
in
e
xa
m
pl
e
2.1,
le
t
δ
=
{
∅
,
X
,
{
x
0
.
5
,
y
0
.
6
}
,
{
x
0
.
6
,
y
0
.
5
}
,
{
x
0
.
6
,
y
0
.
6
}
}
is
a
f
uz
z
y
S
upr
a
to
pol
ogy
on
X
,
but
it
is
not
a
f
uz
z
y
to
po
lo
gy.
R
e
m
a
r
k
2.4:
a
s
poi
nt
e
d
out
in
de
f
in
it
io
n
2.9
(
3)
,
th
a
t
f
uz
z
y
S
upr
a
to
pol
ogy
is
im
pos
s
ib
le
is
f
uz
z
y
to
pol
ogy
,
a
nd s
o
th
e
c
onve
r
s
e
i
s
t
r
ue
.
3.
Q
-
N
E
I
G
H
B
O
R
H
O
O
D
S
O
F
A
F
U
Z
Z
Y
P
O
I
N
T
I
n
th
is
s
e
c
ti
on,
w
e
f
ir
s
t
h
ig
hl
ig
ht
s
om
e
of
th
e
c
la
im
s
m
a
de
a
bout
th
e
f
uz
z
y
Q
-
ne
ig
hbor
hood
of
a
f
uz
z
y
poi
nt
.
W
e
th
e
n
e
xpl
or
e
it
s
r
e
la
ti
ons
hi
p
w
it
h
th
e
f
uz
z
y
ne
ig
hbor
hood,
w
hi
c
h
is
a
pr
e
li
m
in
a
r
y
s
te
p
to
w
a
r
ds
c
ons
tr
uc
ti
ng
a
ne
w
ne
ig
hbor
hood
in
th
e
ne
xt
c
ha
pt
e
r
.
T
hi
s
w
il
l
e
nc
our
a
ge
us
to
in
f
e
r
ne
w
r
e
la
ti
ons
hi
ps
a
nd t
he
or
e
m
s
, w
hi
c
h
w
e
w
il
l
s
uppor
t
w
it
h e
xa
m
pl
e
s
.
D
e
f
in
it
io
n
3.1
[
12
]
:
a
f
uz
z
y
s
e
t
A
in
(
X
,
T
)
is
r
e
f
e
r
r
e
d
to
a
s
a
f
uz
z
y
qua
s
i
-
ne
ig
hbor
hood
(
or
f
uz
z
y
Q
-
ne
ig
hbor
hood)
of
x
t
if
f
th
e
r
e
e
xi
s
ts
a
B
∈
T
s
uc
h
th
a
t
x
t
qB
⊆
A
.
T
he
c
ol
le
c
ti
on
of
a
ll
f
uz
z
y
Q
-
ne
ig
hbor
hoods
of
x
t
is
c
a
ll
e
d
th
e
s
ys
te
m
of
f
uz
z
y
Q
-
ne
ig
hbor
hoods
of
x
t
a
nd
de
not
e
d
by
QN
x
t
.
N
ot
e
3.1:
i)
a
f
uz
z
y
Q
-
ne
ig
hbor
hood
s
of
a
f
uz
z
y
poi
nt
x
t
ge
ne
r
a
l
m
a
y
be
c
ont
a
in
in
g
th
e
poi
nt
it
s
e
lf
x
t
or
no
;
ii
)
a
f
uz
z
y
s
e
t
A
th
a
t
qua
s
i
-
c
oi
nc
id
e
s
w
it
h
a
f
uz
z
y
poi
nt
x
t
ge
ne
r
a
ll
y
not
ne
c
e
s
s
a
r
y
,
it
is
f
uz
z
y
Q
-
ne
ig
hbor
hood
of
x
t
;
ii
i)
in
th
e
f
ol
lo
w
in
g
,
w
e
w
il
l
gi
ve
e
xa
m
pl
e
s
f
or
to
w
not
e
s
a
bo
ve
.
E
xa
m
pl
e
s
3.1
:
in
e
xa
m
pl
e
2.1
:
i
)
w
e
not
e
th
a
t
th
e
f
uz
z
y
s
e
t
A
is
a
f
uz
z
y
Q
-
ne
ig
hbor
hood
of
th
e
f
uz
z
y
p
oi
nt
x
0
.
7
s
in
c
e
x
0
.
7
q
{
x
0
.
5
}
⊆
A
but
x
0
.
7
∉
A
,
w
hi
le
A
is
th
e
f
uz
z
y
Q
-
ne
ig
hbor
hood
of
x
0
.
6
s
in
c
e
x
0
.
6
q
{
x
0
.
5
}
⊆
A
a
nd
x
0
.
6
∈
A
;
a
nd
ii
)
We
not
e
A
is
qua
s
i
-
c
oi
nc
id
e
nt
w
it
h
y
0
.
6
but
A
is
not
a
f
uz
z
y Q
-
ne
ig
hbor
hood of
a
f
uz
z
y point
y
0
.
6
s
in
c
e
∄
B
∈
T
s
.
t
y
0
.
6
qB
⊆
A
.
N
ot
e
3.2:
w
e
a
ls
o
not
e
th
a
t
th
e
c
onc
e
pt
s
of
f
uz
z
y
ne
ig
hbor
hood
a
nd
f
uz
z
y
Q
-
ne
ig
hbor
hood
a
r
e
s
e
pa
r
a
te
a
s
f
ol
lo
w
s
:
e
xa
m
pl
e
3.2:
in
e
xa
m
pl
e
2.1,
A
is
th
e
f
uz
z
y Q
-
ne
ig
hbor
hood
of
th
e
f
uz
z
y
poi
nt
x
0
.
6
s
in
c
e
x
0
.
6
q
{
x
0
.
5
}
⊆
A
but
is
not
a
f
uz
z
y
n
e
ig
hbor
hood
s
in
c
e
∄
B
∈
T
s
.
t
x
0
.
6
∈
B
⊆
A
a
nd
s
o
A
is
a
f
uz
z
y
ne
ig
hbor
hood
of
th
e
f
uz
z
y
poi
nt
x
0
.
4
s
in
c
e
x
0
.
4
⊆
{
x
0
.
5
}
⊆
A
,
but
it
is
not
f
uz
z
y
Q
-
ne
ig
hbor
hood
s
in
c
e
∄
B
∈
T
s
.
t
x
0
.
4
qB
⊆
A
.
R
e
m
a
r
k 3.1:
w
e
obs
e
r
ve
t
ha
t
if
a
f
uz
z
y
s
e
t
is
not
qua
s
i
-
c
oi
nc
id
e
nt
w
it
h a
f
uz
z
y point
, t
he
n i
t
is
im
pos
s
ib
le
f
or
it
to
be
th
e
f
uz
z
y
Q
-
ne
ig
hbor
hood
of
th
a
t
poi
nt
.
P
r
opos
it
io
n
3.1:
F
o
r
a
ny
x
t
∈
X
,
w
he
r
e
0
<
≤
1
, t
he
n
:
i)
QN
x
t
is
not
f
uz
z
y t
opol
ogy or
f
uz
z
y
S
upr
a
to
pol
ogy
;
a
nd i
i)
QN
x
t
is
a
f
uz
z
y f
il
te
r
on
X
.
P
r
oof
:
f
r
om
r
e
m
a
r
k
3.1
∅
∉
QN
x
t
.
L
e
t
G
,
V
∈
QN
x
t
,
th
e
n
∃
B
,
C
in
T
s
.t
.
x
t
qB
⊆
G
a
nd
x
t
qC
⊆
V
⇒
x
t
qB
⋂
C
⊆
G
⋂
V
⇒
G
⋂
V
∈
QN
x
t
.
L
e
t
G
∈
QN
x
t
⇒
∃
B
∈
T
s
.
t
.
x
t
qB
⊆
G
a
nd
if
G
⊆
V
⇒
x
t
qB
⊆
V
⇒
V
∈
QN
x
t
.
F
r
om
pr
oof
1,
∅
∉
QN
x
t
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2252
-
8814
I
nt
J
A
dv A
ppl
S
c
i
,
V
ol
.
14
, N
o.
4
,
D
e
c
e
m
be
r
20
25
:
108
3
-
108
8
1086
F
r
om
not
e
3.2
a
nd
e
xa
m
pl
e
3.2,
w
e
know
th
e
c
onc
e
pt
s
of
f
u
z
z
y
ne
ig
hbor
hood
s
ys
te
m
a
nd
f
uz
z
y
Q
-
ne
ig
hbor
hood
s
a
r
e
s
e
pa
r
a
te
,
but
a
que
s
ti
on
m
ig
ht
a
r
is
e
in
our
m
in
ds
:
w
ha
t
is
th
e
na
tu
r
e
of
th
e
r
e
la
ti
on
be
twe
e
n
th
e
e
le
m
e
nt
s
in
th
e
f
uz
z
y
n
e
ig
hbor
hood
a
nd
f
uz
z
y
Q
-
n
e
ig
hbor
hood
s
ys
te
m
a
t
th
e
s
a
m
e
f
uz
z
y
poi
nt
?
.
T
he
or
e
m
3.1:
le
t
x
t
∈
X
be
a
f
uz
z
y
poi
nt
a
nd
le
t
N
x
t
a
nd
QN
x
t
a
r
e
f
u
z
z
y
ne
ig
hbor
hood
s
a
nd
f
uz
z
y
Q
-
ne
ig
hbor
hood
s
ys
te
m
a
t
x
t
,
r
e
s
pe
c
ti
ve
ly
,
th
e
n
e
a
c
h
e
le
m
e
nt
A
∈
N
x
t
is
qu
a
s
i
-
c
oi
nc
id
e
nt
w
it
h
e
a
c
h
B
∈
QN
x
t
a
nd
s
o
c
onve
r
s
e
.
P
r
oof
:
le
t
A
∈
N
x
t
⇒
∃
U
is
f
uz
z
y
ope
n
s
.t
.
x
t
∈
U
⊆
A
.
L
e
t
B
∈
QN
x
t
⇒
∃
V
is
f
uz
z
y
ope
n s
.t
.
x
t
qV
⊆
B
a
nd s
a
m
e
t
im
e
.
x
t
qB
⇒
t
+
μ
B
(
x
)
>
1
(
3
)
B
ut
,
x
t
∈
A
⇒
t
≤
μ
A
(
x
)
th
e
n
f
r
om
(
1)
μ
A
(
x
)
+
μ
B
(
x
)
>
1
⇒
a
nd
s
o
by
th
e
or
e
m
2.1
B
q
A
.
T
he
or
e
m
3.2:
f
or
a
ny
x
t
∈
X
t
he
n
,
N
x
t
⊆
QN
x
t
or
QN
x
t
⊆
N
x
t
.
P
r
oof
:
le
t
x
t
∈
X
.
T
he
n
t
≤
0
.
5
or
t
>
0
.
5
.
I
f
t
≤
0
.
5
,
le
t
B
∈
QN
x
t
th
e
n
∃
V
is
ope
n
s
.t
.
x
t
qV
⊆
B
⇒
μ
V
(
x
)
>
0
.
5
⇒
x
t
∈
V
⊆
B
⇒
B
∈
N
x
t
⇒
QN
x
t
⊆
N
x
t
.
N
ow
,
if
t
>
0
.
5
,
le
t
A
∈
N
x
t
⇒
∃
U
is
a
f
uz
z
y
ope
n
s
e
t
s
.t
.
x
t
∈
U
⊆
A
a
nd
s
in
c
e
t
>
0
.
5
⇒
μ
U
(
x
)
+
t
>
1
⇒
x
t
qU
⊆
A
⇒
A
∈
QN
x
t
⇒
N
x
t
⊆
QN
x
t
.
T
he
or
e
m
3.3:
f
or
a
ny
x
t
∈
X
.
A
ny
s
ys
te
m
Γ
w
hi
c
h
is
a
f
uz
z
y
ne
ig
hbor
hood
a
nd
a
Q
-
ne
ig
hbor
hood
s
y
s
te
m
on
x
t
th
e
n
e
v
e
r
y
e
l
e
m
e
nt
ha
s
m
e
m
be
r
s
hi
p
a
t
x
la
r
ge
r
th
a
n
0
.
5
.
P
r
oof
:
le
t
x
t
∈
X
,
t
≤
0
.
5
or
t
>
0
.
5
. I
f
t
≤
0
.
5
a
nd
ς
∈
Γ
⇒
ς
∈
QN
x
t
a
nd by the
or
e
m
3.2
⇒
μ
ς
(
x
)
>
0
.
5
, a
nd
if
t
>
0
.
5
⇒
∈
N
x
t
⇒
μ
ς
(
x
)
>
0
.
5
.
4.
DQ
-
N
E
I
G
H
B
O
R
H
O
O
D
O
F
A
F
U
Z
Z
Y
P
O
I
N
T
I
n
th
is
s
e
c
ti
on,
w
e
w
il
l
pr
e
s
e
nt
a
nd
pr
ove
th
a
t
th
e
dua
l
i
s
n’
t
e
qui
va
le
nt
to
f
uz
z
y
Q
-
ne
ig
hbor
hood
.
D
e
f
in
it
io
n
4.1:
a
f
uz
z
y
s
e
t
A
in
f
uz
z
y
to
pol
ogi
c
a
l
s
pa
c
e
(
X
,
T
)
is
c
a
ll
e
d
a
f
uz
z
y
dua
l
qua
s
i
-
ne
ig
hbor
hood
(
in
br
ie
f
,
f
uz
z
y
D
Q
-
ne
ig
hbor
hood)
of
a
f
uz
z
y
poi
nt
x
t
if
f
x
t
qA
a
nd
th
e
r
e
e
xi
s
t
s
V
∈
T
s
.t
.
x
t
qV
⊆
A
c
.
T
he
f
a
m
il
y
of
a
ll
th
e
f
uz
z
y
D
Q
-
ne
ig
hbor
hoods
a
t
a
f
uz
z
y
poi
nt
x
t
is
c
a
ll
e
d
th
e
f
uz
z
y
D
Q
-
ne
ig
hbor
hood
s
y
s
te
m
of
x
t
,
w
hi
c
h
is
de
not
e
d
by
N
D
x
t
.
R
e
m
a
r
k
4.1:
f
r
om
de
f
in
it
io
n
2.1,
w
e
not
e
th
e
f
ol
lo
w
in
g:
i)
i
f
A
is
a
f
uz
z
y
DQ
-
ne
ig
hbor
hood
of
x
t
th
e
n
x
t
q
A
c
;
ii
)
A
is
a
f
uz
z
y
D
Q
-
ne
ig
hbor
ho
od
of
x
t
if
f
x
t
qA
a
nd
A
c
is
th
e
f
uz
z
y
Q
-
ne
ig
hbor
hood
of
x
t
;
a
nd
ii
i)
If
A
is
a
f
uz
z
y
Q
-
ne
ig
hbor
hood a
nd D
Q
-
ne
ig
hbor
hood
of
x
t
th
e
n
s
o
A
c
.
P
r
oof
:
th
e
pr
oof
,
be
in
g
s
tr
a
ig
ht
f
o
r
w
a
r
d,
is
om
it
te
d.
T
he
f
ol
lo
w
in
g
e
xa
m
pl
e
e
xpl
a
in
s
th
a
t
th
e
two
c
onc
e
pt
s
,
f
uz
z
y
DQ
-
ne
ig
hbor
hood
a
nd
-
ne
ig
hbor
hood
s
a
r
e
not
e
qui
va
le
nt
.
E
xa
m
pl
e
4.1
:
le
t
X
=
{
a
,
b
}
a
nd
T
=
{
∅
,
X
,
{
a
0
.
4
,
b
0
.
4
}
}
.
L
e
t
A
1
=
{
a
0
.
3
,
b
0
.
3
}
,
A
2
=
{
a
0
.
7
,
b
0
.
7
}
a
nd
A
3
=
{
a
0
.
5
,
b
0
.
5
}
th
e
n
A
1
is
th
e
f
uz
z
y
D
Q
-
ne
ig
hbor
hood
of
a
0
.
9
but
it
is
not
f
uz
z
y Q
-
ne
ig
hbor
hood
of
a
0
.
9
.
A
2
is
th
e
f
uz
z
y
Q
-
ne
ig
hbor
hood
of
a
0
.
9
but
it
is
not
f
uz
z
y
D
Q
-
ne
ig
hbor
hood
of
a
0
.
9
.
A
3
is
a
f
uz
z
y
DQ
-
ne
ig
hbor
hood
a
nd
Q
-
ne
ig
hbor
hood
o
f
a
0
.
9
.
N
ot
e
4.1:
a
f
uz
z
y
D
Q
-
ne
ig
hbor
hood
s
e
t,
a
lo
ng
w
it
h
th
e
c
om
pl
e
m
e
nt
of
a
f
uz
z
y
poi
nt
,
doe
s
not
in
c
lu
de
th
e
f
uz
z
y
poi
nt
it
s
e
lf
.
T
he
n
e
ig
hbor
hood
s
tr
uc
tu
r
e
of
a
poi
nt
th
a
t
doe
s
not
c
ont
a
in
th
e
poi
nt
it
s
e
lf
ha
s
be
e
n
e
xpl
or
e
d
in
[
12]
.
T
he
or
e
m
4.1:
a
f
uz
z
y
D
Q
-
ne
ig
hbor
hood
s
ys
te
m
N
D
x
t
of
a
f
uz
z
y
poi
nt
x
t
in
a
f
uz
z
y
to
pol
ogi
c
a
l
s
p
a
c
e
(
X
,
T
)
s
a
ti
s
f
ie
s
th
e
f
ol
lo
w
in
g:
i)
∅
,
X
∉
N
D
x
t
;
ii
)
If
A
j
∈
N
D
x
t
,
j
∈
J
th
e
n
⋂
A
j
∈
N
D
x
t
j
∈
J
f
or
a
ny
in
de
x
J
;
a
nd
ii
i)
If
A
l
∈
N
D
x
t
,
l
∈
L
th
e
n
⋃
A
l
∈
N
D
x
t
l
∈
L
f
o
r
a
ny
f
in
it
e
in
de
x
L
.
P
r
oof
:
le
t
(
X
,
T
)
be
a
f
uz
z
y
to
pol
ogi
c
a
l
s
pa
c
e
a
nd
x
t
∈
X
f
ix
e
d
f
uz
z
y
poi
nt
.
∅
↽
q
x
t
,
∀
x
t
∈
X
⇒
∅
∉
N
D
x
t
.
N
ow
if
X
∈
N
D
x
t
⇒
∅q
x
t
a
nd
th
is
c
ont
r
a
di
c
ti
on,
th
e
r
e
f
or
e
,
X
∉
N
D
x
t
.
F
or
a
ny
in
de
x
J
,
le
t
A
j
∈
N
D
x
t
,
j
∈
J
th
e
n
∃
U
j
∈
T
s
uc
h
th
a
t
x
t
q
U
j
⊆
A
j
c
⇒
x
t
q
⋃
U
j
⊆
⋃
A
j
c
j
∈
J
j
∈
J
⇒
⋃
A
j
c
j
∈
J
is
th
e
Q
-
ne
ig
hbor
hood
of
x
t
,
but
A
j
q
x
t
,
∀
j
∈
J
⇒
⋂
A
j
q
x
t
j
∈
J
a
nd
by
R
e
m
a
r
k
4.1,2
⇒
⋂
A
j
∈
N
D
x
t
j
∈
J
.
F
or
a
ny
f
in
it
e
in
de
x
L
,
le
t
A
l
∈
N
D
x
t
,
l
∈
L
th
e
n
∃
V
l
∈
T
s
uc
h
th
a
t
x
t
q
V
l
⊆
A
l
⇒
x
t
q
⋂
V
l
⊆
⋂
A
l
c
l
∈
L
l
∈
L
⇒
⋂
A
l
c
l
∈
L
is
th
e
Q
-
ne
ig
hbor
hood of
x
t
, but
A
l
q
x
t
,
∀
l
∈
L
⇒
⋃
A
l
q
x
t
l
∈
L
a
nd by R
e
m
a
r
k 4.1,2
⇒
⋃
A
l
∈
N
D
x
t
l
∈
L
.
I
n
f
a
c
t,
f
r
om
th
e
or
e
m
4.1,
N
D
x
t
it
is
n'
t
f
uz
z
y
to
pol
ogy
or
f
uz
z
y
S
u
pr
a
to
pol
ogy.
F
ur
th
e
r
m
or
e
,
it
is
not
a
f
uz
z
y
f
il
te
r
,
a
nd
th
is
is
th
e
oppo
s
it
e
of
N
x
t
w
hi
c
h
pr
ove
s
it
e
a
r
li
e
r
.
T
he
f
ol
lo
w
in
g
e
x
a
m
pl
e
e
xpl
a
in
s
th
a
t
:
e
xa
m
pl
e
4.2:
in
e
xa
m
pl
e
2.1,
A
1
is
th
e
f
uz
z
y
D
Q
-
ne
ig
hbor
hood
of
a
0
.
9
,
i.
e
.,
A
1
∈
N
D
a
0
.
9
a
nd
A
1
⊆
A
2
but
A
2
∉
N
D
a
0
.
9
.
P
r
opos
it
io
n
4.1:
in
a
f
uz
z
y
D
Q
-
ne
ig
hbor
hood
s
ys
te
m
N
D
x
t
of
a
f
uz
z
y
poi
nt
x
t
in
a
f
uz
z
y
to
pol
ogi
c
a
l
s
pa
c
e
(
X
,
T
)
,
le
t
A
∈
ND
x
t
a
nd
B
⊆
A
,
if
x
t
qB
th
e
n
B
∈
N
D
x
t
.
P
r
oof
:
S
in
c
e
B
⊆
A
,
th
e
n
A
c
⊆
B
c
.
x
t
qU
⊆
A
c
⊆
B
c
, f
or
s
om
e
U
∈
T
⇒
B
∈
N
D
x
t
.
5.
R
E
S
U
L
T
S
A
N
D
D
I
S
C
U
S
S
I
O
N
T
he
r
e
s
ul
t
of
t
hi
s
s
tu
dy s
how
s
t
ha
t
th
e
f
uz
z
y Q
-
ne
ig
hbor
hood
s
ys
te
m
i
s
a
ls
o a
f
uz
z
y f
il
te
r
. I
t
is
w
or
th
obs
e
r
vi
ng
he
r
e
th
a
t
th
e
c
onc
e
pt
s
of
f
uz
z
y
Q
-
ne
ig
hbor
hood
a
nd
f
uz
z
y
ne
ig
hbor
hood
s
ys
te
m
s
,
a
lt
hough
unl
ik
e
e
a
c
h ot
he
r
, a
r
e
s
tr
uc
tu
r
a
ll
y r
e
la
te
d
;
th
e
f
or
m
e
r
is
l
im
it
e
d i
n
t
he
l
a
tt
e
r
a
nd vic
e
ve
r
s
a
. T
hi
s
i
nt
e
r
r
e
la
ti
ons
hi
p ha
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
A
dv A
ppl
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c
i
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:
2252
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8814
N
e
w
appr
oa
c
h of
t
he
ne
ig
hbor
hood
s
tr
uc
tu
r
e
of
f
uz
z
y
poi
nt
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(
A
m
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r
H
imz
a A
lmy
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)
1087
be
e
n r
e
la
ti
ve
ly
ne
gl
e
c
te
d i
n l
it
e
r
a
tu
r
e
[
10]
–
[
13]
, a
lt
hough it
s
or
i
gi
na
l
in
te
r
pr
e
ta
ti
on oc
c
ur
s
i
n t
he
m
ode
ll
in
g of
th
e
be
ha
vi
or
of
f
uz
z
y t
opol
ogi
c
a
l
obj
e
c
ts
.
T
he
m
a
in
c
ont
r
ib
ut
io
n
of
th
is
pa
pe
r
is
th
e
de
f
in
it
io
n
a
nd
s
tu
dy
of
th
e
dua
li
ty
of
th
e
f
uz
z
y
Q
-
ne
ig
hbor
hood
s
ys
te
m
in
r
e
la
ti
on
to
a
f
ix
e
d
f
uz
z
y
poi
nt
.
W
e
ha
ve
s
how
n
th
a
t
th
is
two
-
c
la
s
s
s
y
s
te
m
is
ge
ne
r
a
ll
y
not
a
poi
nt
-
non
-
in
c
lu
s
io
n
or
f
uz
z
if
yi
ng
s
ys
te
m
.
T
hi
s
de
vi
a
ti
on
f
r
om
tr
a
di
ti
ona
l
s
tr
uc
tu
r
e
s
in
c
lu
de
s
th
e
m
e
th
odol
ogy of
ne
w
t
he
or
e
ti
c
a
l
unde
r
s
ta
ndi
ngs
, e
s
pe
c
ia
ll
y w
he
n i
t
c
om
e
s
t
o unde
r
s
ta
ndi
ng
ne
ig
hbor
hoo
d
s
ys
te
m
s
th
a
t
do
not
de
pe
nd
on
a
s
pe
c
if
ic
r
e
f
e
r
e
nc
e
poi
nt
.
B
y
di
s
ti
ngui
s
hi
ng
be
twe
e
n
or
ig
in
a
l
a
nd
dua
l
s
ys
te
m
s
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w
e
c
a
n
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la
s
s
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f
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z
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s
pa
c
e
s
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in
nova
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ve
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ys
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ghl
ig
ht
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g
th
e
va
r
ia
ti
on
of
be
ha
vi
or
s
th
a
t
c
a
n
a
r
is
e
in
f
uz
z
y
f
r
a
m
e
w
or
ks
.
F
ur
th
e
r
m
or
e
,
a
d
e
ta
il
e
d
s
tu
dy
of
t
he
c
onne
c
ti
on
s
be
tw
e
e
n
f
uz
z
y
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ig
hbor
hood
s
ys
te
m
s
,
f
uz
z
y
Q
-
ne
ig
hbor
hood
s
ys
te
m
s
,
a
nd
th
e
ir
double
s
r
e
ve
a
le
d
a
m
or
e
c
om
pl
e
x
s
tr
uc
tu
r
e
.
T
hi
s
r
e
s
e
a
r
c
h
hi
ghl
ig
ht
s
th
e
pr
om
is
e
of
a
dopt
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g
c
a
te
gor
ic
a
l
m
e
th
ods
,
s
uc
h
a
s
D
Q
ne
ig
hbor
hood
s
ys
te
m
s
,
to
c
r
e
a
te
s
pe
c
ia
l
m
a
th
e
m
a
ti
c
a
l
s
pa
c
e
s
.
T
he
s
e
s
ys
te
m
s
c
oul
d
pr
ovi
de
a
r
ic
he
r
unde
r
s
ta
ndi
ng
of
c
onc
e
pt
s
s
u
c
h
a
s
pr
oxi
m
it
y,
c
ont
in
ui
ty
, a
nd s
e
pa
r
a
ti
on i
n f
uz
z
y e
nvi
r
onm
e
nt
s
.
T
he
s
tu
dy r
a
is
e
s
s
om
e
i
nt
e
r
e
s
ti
ng que
s
ti
ons
a
bout
ne
ig
hbor
hoo
d
s
t
ha
t
do not f
it
th
e
us
ua
l
f
uz
z
y point
m
ode
l.
T
he
s
e
ki
nds
of
s
ys
t
e
m
s
a
r
e
c
ount
e
r
pr
oduc
ti
ve
to
tr
a
di
ti
ona
l
id
e
a
s
a
nd
hi
ghl
ig
ht
th
e
ne
e
d
f
or
m
or
e
r
e
s
e
a
r
c
h
in
to
pr
a
c
ti
c
a
l,
r
e
a
l
-
w
or
ld
m
ode
ls
of
c
onf
us
io
n.
I
n
th
e
f
ut
ur
e
,
f
ut
ur
e
r
e
s
e
a
r
c
h
c
oul
d
e
xpl
or
e
th
e
s
e
s
tr
uc
tu
r
e
s
w
it
hi
n
c
a
te
gor
ic
a
l,
s
p
a
ti
a
l,
or
a
lg
e
br
a
ic
f
r
a
m
e
w
or
ks
,
pa
vi
ng
th
e
w
a
y
f
or
br
oa
de
r
a
nd
m
or
e
a
ppl
ic
a
bl
e
t
he
or
ie
s
i
n f
uz
z
y t
opol
ogy a
nd r
e
la
te
d
a
r
e
a
s
.
6.
C
O
N
C
L
U
S
I
O
N
T
he
f
uz
z
y
Q
-
ne
ig
hbor
hood
s
y
s
te
m
is
a
f
uz
z
y
f
il
te
r
.
F
or
a
ny
gi
ve
n
f
uz
z
y
poi
nt
,
it
c
ont
a
in
s
e
it
he
r
th
e
f
uz
z
y
ne
ig
hbor
hood
s
ys
te
m
or
c
ont
a
in
s
w
it
hi
n i
t
a
n i
m
por
ta
nt
r
e
la
ti
ons
hi
p t
ha
t
ha
s
r
e
m
a
in
e
d l
a
r
ge
ly
unknown
in
pr
e
vi
ous
s
tu
di
e
s
.
I
n
th
is
pa
pe
r
,
w
e
a
ddr
e
s
s
th
is
g
a
p
by
pr
ov
id
in
g
a
c
le
a
r
a
na
ly
s
is
s
uppor
te
d
by
il
lu
s
tr
a
te
d
e
xa
m
pl
e
s
.
F
ur
th
e
r
m
or
e
,
w
e
in
tr
oduc
e
d
th
e
dua
l
of
th
e
f
uz
z
y
Q
-
ne
ig
hbor
hood
s
ys
te
m
,
de
m
ons
tr
a
ti
ng
th
a
t
it
ne
it
he
r
f
or
m
s
a
f
uz
z
y
f
il
te
r
nor
c
oi
nc
id
e
s
w
it
h
th
e
or
ig
in
a
l
f
uz
z
y
Q
-
ne
ig
hbor
hood
s
ys
te
m
.
A
s
f
a
m
il
ie
s
of
f
uz
z
y
s
e
ts
,
th
e
s
e
s
y
s
te
m
s
pr
ovi
de
a
m
e
a
n
s
of
c
ha
r
a
c
te
r
iz
in
g
s
p
e
c
if
ic
c
la
s
s
e
s
of
f
uz
z
y
s
p
a
c
e
s
,
th
us
f
a
c
il
it
a
ti
ng
br
oa
de
r
ge
ne
r
a
li
z
a
ti
ons
a
nd de
e
p
e
r
s
tr
uc
tu
r
a
l
in
s
ig
ht
s
.
F
U
N
D
I
N
G
I
N
F
O
R
M
A
T
I
O
N
A
ut
hor
s
s
ta
te
no f
undi
ng i
nvol
ve
d.
A
U
T
H
O
R
C
O
N
T
R
I
B
U
T
I
O
N
S
S
T
A
T
E
M
E
N
T
T
hi
s
jo
ur
na
l
us
e
s
th
e
C
ont
r
ib
ut
or
R
ol
e
s
T
a
xonomy
(
C
R
e
di
T
)
to
r
e
c
ogni
z
e
in
di
vi
dua
l
a
ut
hor
c
ont
r
ib
ut
io
ns
, r
e
duc
e
a
ut
hor
s
hi
p di
s
put
e
s
,
a
nd f
a
c
il
it
a
te
c
ol
la
bo
r
a
ti
on.
N
am
e
o
f
A
u
t
h
or
C
M
So
Va
Fo
I
R
D
O
E
Vi
Su
P
Fu
A
m
e
r
H
im
z
a
A
lm
ya
ly
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
J
w
ngs
a
r
M
o
s
ha
ha
r
y
✓
✓
✓
✓
C
:
C
onc
e
pt
ua
l
i
z
a
t
i
on
M
:
M
e
t
hodol
ogy
So
:
So
f
t
w
a
r
e
Va
:
Va
l
i
da
t
i
on
Fo
:
Fo
r
m
a
l
a
na
l
ys
i
s
I
:
I
nve
s
t
i
ga
t
i
on
R
:
R
e
s
our
c
e
s
D
:
D
a
t
a
C
ur
a
t
i
on
O
:
W
r
i
t
i
ng
-
O
r
i
gi
na
l
D
r
a
f
t
E
:
W
r
i
t
i
ng
-
R
e
vi
e
w
&
E
di
t
i
ng
Vi
:
Vi
s
ua
l
i
z
a
t
i
on
Su
:
Su
pe
r
vi
s
i
on
P
:
P
r
oj
e
c
t
a
dm
i
ni
s
t
r
a
t
i
on
Fu
:
Fu
ndi
ng a
c
qui
s
i
t
i
on
C
O
N
F
L
I
C
T
O
F
I
N
T
E
R
E
S
T
S
T
A
T
E
M
E
N
T
A
ut
hor
s
s
ta
te
no c
onf
li
c
t
of
i
nt
e
r
e
s
t.
D
A
T
A
A
V
A
I
L
A
B
I
L
I
T
Y
D
a
ta
a
va
il
a
bi
li
ty
is
not
a
ppl
ic
a
bl
e
to
th
is
pa
pe
r
a
s
no
ne
w
d
a
ta
w
e
r
e
c
r
e
a
te
d
or
a
na
ly
z
e
d
in
th
is
s
tu
dy
.
R
E
F
E
R
E
N
C
E
S
[
1]
L
. A
. Z
a
de
h, “
F
uz
z
y s
e
t
s
,”
I
nf
or
m
at
i
on and C
ont
r
ol
, vol
. 8, no. 3, pp. 338
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353,
1965, doi
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S
0019
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9958(
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A
. K
a
uf
m
a
n a
nd M
. M
. G
upt
a
,
I
nt
r
odu
c
t
i
on t
o f
uz
z
y
a
r
i
t
hm
e
t
i
c
. V
a
n N
o
s
t
r
a
nd
R
e
i
nhol
d C
om
pa
ny N
e
w
Y
or
k, 1991.
[
3]
C
.
L
.
C
ha
ng,
“
F
uz
z
y
t
opol
ogi
c
a
l
s
pa
c
e
s
,”
J
our
nal
of
M
at
he
m
at
i
c
al
A
nal
y
s
i
s
and
A
ppl
i
c
at
i
ons
,
vol
.
24,
pp.
182
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192,
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doi
:
10.1016/
0022
-
247X
(
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[
4]
T
. J
. R
os
s
,
F
uz
z
y
L
ogi
c
w
i
t
h E
ngi
ne
e
r
i
ng A
ppl
i
c
at
i
ons
. H
oboke
n, N
J
, U
S
A
:
J
ohn W
i
l
e
y &
S
ons
, 2005.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2252
-
8814
I
nt
J
A
dv A
ppl
S
c
i
,
V
ol
.
14
, N
o.
4
,
D
e
c
e
m
be
r
20
25
:
108
3
-
108
8
1088
[
5]
R
. L
ow
e
n, “
F
uz
z
y
t
opol
ogi
c
a
l
s
p
a
c
e
s
a
nd f
uz
z
y c
om
pa
c
t
ne
s
s
,
”
J
ou
r
nal
of
M
at
he
m
at
i
c
al
A
nal
y
s
i
s
and
A
ppl
i
c
at
i
ons
, vol
. 56, no.
3,
pp. 621
–
633, 1976, doi
:
10.1016/
0022
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247X
(
76)
90029
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[
6]
A
.
P
.
Š
os
t
a
k,
“
O
n
a
f
uz
z
y
t
opol
ogi
c
a
l
s
t
r
uc
t
ur
e
,”
i
n
P
r
oc
e
e
di
ngs
of
t
he
13t
h
W
i
nt
e
r
Sc
hool
on
A
bs
t
r
a
c
t
A
nal
y
s
i
s
,
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[
7]
A
.
A
.
R
a
m
a
da
n,
“
S
m
oot
h
t
opol
ogi
c
a
l
s
pa
c
e
s
,”
F
uz
z
y
Se
t
s
and
Sy
s
t
e
m
s
,
vol
.
48,
no.
3,
pp.
371
–
375,
1992,
doi
:
10.1016/
0165
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0114(
92)
90352
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5.
[
8]
A
.
V
a
di
ve
l
a
nd
C
.
J
.
S
unda
r
,
“
N
e
ut
r
os
ophi
c
δ
-
ope
n
m
a
ps
a
nd
n
e
ut
r
os
ophi
c
δ
-
c
l
os
e
d
m
a
p
s
,”
A
m
e
r
i
c
an
J
our
nal
of
M
at
he
m
at
i
c
al
and C
om
put
e
r
M
ode
l
i
ng
, vol
. 13, no. 2, pp. 66
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74, 2021.
[
9]
R
.
A
.
-
G
da
i
r
i
,
A
.
A
.
N
a
s
e
f
,
M
.
A
.
E
.
-
G
a
ya
r
,
a
nd
M
.
K
.
E
.
-
B
a
bl
y,
“
O
n
f
uz
z
y
poi
nt
a
ppl
i
c
a
t
i
ons
of
f
uz
z
y
t
opol
ogi
c
a
l
s
pa
c
e
s
,
”
I
nt
e
r
nat
i
onal
J
our
nal
of
F
uz
z
y
L
ogi
c
and
I
nt
e
l
l
i
ge
nt
Sy
s
t
e
m
s
,
vol
.
23,
no.
2,
pp.
162
–
172,
2023,
doi
:
10.5391/
I
J
F
I
S
.2023.23.2.162.
[
10]
B
.
P
.
V
a
r
ol
a
nd
H
.
M
a
l
koç
,
“
B
i
pol
a
r
f
uz
z
y
S
upr
a
t
opol
ogy
vi
a
(
Q
-
)
ne
i
ghbor
hood
a
nd
i
t
s
a
ppl
i
c
a
t
i
on
i
n
d
a
t
a
m
i
ni
ng
pr
oc
e
s
s
,
”
Sy
m
m
e
t
r
y
, vol
. 16, no. 2, 2024, doi
:
10.3390/
s
ym
16020216.
[
11]
A
.
T
.
a
nd
P
.
M
a
da
n,
“
D
e
c
i
s
i
on
m
a
ki
ng
ba
s
e
d
on
r
ough
a
ppr
oxi
m
a
t
i
ons
ge
ne
r
a
l
i
z
e
d
by
j
-
f
uz
z
y
ne
i
ghbor
hoods
,”
J
our
nal
of
I
nt
e
l
l
i
ge
nt
and F
uz
z
y
Sy
s
t
e
m
s
, vol
. 48, no. 3, pp. 279
–
289, 2024, doi
:
10.3233/
J
I
F
S
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239107.
[
12]
P
. P
.
-
M
i
ng a
nd L
. Y
.
-
M
i
ng, “
F
uz
z
y t
opol
ogy. I
. N
e
i
ghbor
hood s
t
r
uc
t
ur
e
of
a
f
uz
z
y poi
nt
a
nd M
oor
e
-
S
m
i
t
h c
onve
r
ge
nc
e
,”
J
our
na
l
of
M
at
he
m
at
i
c
al
A
nal
y
s
i
s
and A
ppl
i
c
at
i
ons
, vol
. 76, no. 2, pp. 571
–
599, 1980.
[
13]
R
. G
a
o
a
nd J
. W
u, “
F
i
l
t
e
r
w
i
t
h i
t
s
a
ppl
i
c
a
t
i
ons
i
n f
uz
z
y
s
of
t
t
opol
ogi
c
a
l
s
p
a
c
e
s
,
”
A
I
M
S M
at
he
m
at
i
c
s
, vol
. 6, no. 3,
pp. 2359
–
2368,
2021, doi
:
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m
a
t
h.2021143.
[
14]
L
. R
e
z
ni
k,
F
uz
z
y
C
ont
r
ol
l
e
r
s
H
andbook
. M
e
l
bour
ne
, A
u
s
t
r
a
l
i
a
:
E
l
s
e
vi
e
r
, 1998.
[
15]
I
.
Z
a
ha
n
a
nd
R
.
N
a
s
r
i
n,
“
A
n
i
nt
r
oduc
t
i
on
t
o
f
uz
z
y
t
opol
ogi
c
a
l
s
pa
c
e
s
,”
A
dv
anc
e
s
i
n
P
ur
e
M
at
he
m
at
i
c
s
,
vol
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5,
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a
pm
.2021.115034.
[
16]
I
.
J
a
w
a
r
ne
h,
A
.
U
.
A
l
kour
i
,
M
.
H
a
z
a
i
m
e
h,
a
nd
Z
.
A
l
t
a
w
a
l
l
be
h,
“
O
n
t
he
f
uz
z
y
t
opol
ogi
c
a
l
s
pa
c
e
s
on
a
f
uz
z
y
s
pa
c
e
(
X
,
I
)
,”
A
I
P
C
onf
e
r
e
nc
e
P
r
oc
e
e
di
ngs
, vol
. 3094, no. 1, 2024, doi
:
10.1063/
5.0210144.
[
17]
B
.
Q
i
n,
K
.
H
u,
a
nd
Q
.
X
ue
,
“
T
he
c
ons
i
s
t
e
nt
f
unc
t
i
ons
a
nd
r
e
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f
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“
Q
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m
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F
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m
e
nt
a
l
s
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a
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c
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t
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on
t
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L
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s
i
-
t
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e
s
,
L
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s
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-
uni
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t
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m
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m
s
-
ba
s
e
d
pe
s
s
i
m
i
s
t
i
c
r
ough
s
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t
s
a
nd
t
he
i
r
a
ppl
i
c
a
t
i
on
s
i
n
i
nc
om
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t
e
i
nf
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m
a
t
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on
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e
m
s
,”
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c
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l
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r
a
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t
e
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on
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s
f
r
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R
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gul
a
r
i
t
y
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pa
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A
m
e
r
i
c
an
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ade
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3.
B
I
O
G
R
A
P
H
I
E
S
O
F
A
U
T
H
O
R
S
Amer
Himza
Almyaly
is
an
Associate
Professor
at
the
Colle
ge
of
Science,
Al
-
Muthanna
University,
Iraq.
He
holds
a
Ph.D.
in
Mathematical
Sciences
and
a
Higher
Diploma
in
Computer
Science.
His
research
interests
include
topolog
y,
fuzzy
topology,
fuzzy
sets,
and
functional
analysis
.
He
served
as
the
Head
of
the
Departm
ent
of
Mathematics
and
Computer
Applicat
ions
at
the
College
of
Science
from
2019
to
2020,
and
he
is
also
the
Head
of
the
Mathematics
Department
at
the
College
of
Education
for
Pure
Sciences
from
2023
to
2025, Al
-
Muthanna Univer
sity. He can be
contacted via
email at: a
merhimzi@mu.edu.iq
.
Jwngsar
Moshahary
is
an
Assistant
Professor
at
SCS
College
o
f
Agriculture,
Assam
Agricultural
University.
He
holds
a
Ph.D.
in
Mathematical
Sc
iences,
and
his
research
interests
include
topology,
fuzzy
topology,
fuzzy
s
ets,
and
their
ap
plications
.
He
has
been
actively
engaged
in
teaching
and
resea
rch
since
2019.
He
can
be
contacted
via
email
at
:
jwngsar.mosha
hary@aau.ac.in
.
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