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7
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.
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P
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[
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Evaluation Warning : The document was created with Spire.PDF for Python.
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I
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b.
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1
1
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.
d.
P
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4
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P
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5
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h
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all
∈
[
0
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1
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[
1
9
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.
2.
RE
SU
L
T
S
a.
T
h
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r
em
2
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1
.
L
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s
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b.
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th
e
f
o
llo
w
in
g
q
u
e
s
tio
n
i
s
i
m
p
o
r
ta
n
t:
c.
Q
u
est
io
n
2
.
3
.
L
et
(
,
ℑ
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b
e
a
C
DH
f
u
zz
y
to
p
o
lo
g
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s
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ac
e.
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s
it tr
u
e
th
at
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=
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f
o
r
all
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.
T
h
e
f
o
llo
w
i
n
g
e
x
a
m
p
le
g
i
v
es
a
n
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e
a
n
s
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er
o
f
Q
u
est
io
n
2
.
3
:
d.
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x
a
m
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le
2
.
4
.
Fo
r
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i
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0
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<
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let
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{
,
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d
d
ef
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n
e
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{
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1
,
2
,
4
,
2
∪
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}
.
T
h
en
(
,
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is
C
DH
a
n
d
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=
{
}
b
u
t
ℑ
₀
=
f
o
r
all
∈
.
T
h
e
f
o
llo
w
i
n
g
t
w
o
le
m
m
as
w
i
ll b
e
u
s
ed
in
t
h
e
f
o
llo
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n
g
m
ai
n
r
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e.
L
e
m
m
a
2
.
5
.
L
et
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,
ℑ
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e
a
f
u
zz
y
t
o
p
o
lo
g
ical
s
p
ac
e
an
d
let
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b
e
a
f
u
zz
y
h
o
m
o
g
e
n
eo
u
s
co
m
p
o
n
en
t
o
f
(
,
ℑ
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it
h
ℑ
∉
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₀
.
T
h
en
−
ℑ
is
d
en
s
e
in
(
,
ℑ
₀
)
.
P
r
o
o
f
.
A
s
s
u
m
e
o
n
th
e
co
n
tr
ar
y
t
h
at
−
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n
o
t
d
en
s
e
in
(
,
ℑ
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.
T
h
en
th
er
e
ex
is
ts
a
n
o
n
-
e
m
p
t
y
s
et
∈
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s
u
c
h
th
a
t
∩
(
−
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=
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.
T
h
u
s
⊆
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d
b
y
P
r
o
p
o
s
it
io
n
1
.
1
w
e
h
av
e
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∈
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₀
,
a
co
n
tr
ad
ictio
n
.
f.
L
e
m
m
a
2
.
6
.
L
et
(
,
ℑ
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b
e
a
C
D
H
f
u
zz
y
to
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o
lo
g
ical
s
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ac
e.
Su
p
p
o
s
e
t
h
at
t
h
er
e
e
x
is
t
s
a
f
u
zz
y
h
o
m
o
g
en
eo
u
s
co
m
p
o
n
e
n
t
ℑ
o
f
(
,
ℑ
)
w
it
h
ℑ
∉
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₀
.
L
et
S b
e
a
co
u
n
tab
le
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e
n
s
e
s
u
b
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o
f
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,
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(
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e
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ar
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e
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g
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n
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∩
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(
−
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iii.
≠
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en
s
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,
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r
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ce
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−
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h
u
s
∩
⊆
∅
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(
ℑ
∩
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⊆
ℑ
.
ii)
L
et
∈
(
−
(
)
)
∩
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ce
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,
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n
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.
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ce
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th
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iii)
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p
p
o
s
e
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e
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n
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ar
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−
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−
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{
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th
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en
s
e
in
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,
ℑ
₀
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d
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y
P
r
o
p
o
s
itio
n
1
.
2
(
i)
,
ℚ
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an
d
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ar
e
b
o
th
d
en
s
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I
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H,
th
en
t
h
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a
f
u
zz
y
h
o
m
eo
m
o
r
p
h
is
m
ℎ
:
(
,
ℑ
)
→
(
,
ℑ
)
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Ho
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(
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(
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e
n
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(
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y
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p
o
s
itio
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d
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o
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d
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n
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n
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I
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h
at
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s
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th
at
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en
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en
s
e
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d
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e
h
a
v
e
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(
∩
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in
ce
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h
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s
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h
er
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o
r
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∈
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.
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e
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w
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n
g
i
s
t
h
e
m
ai
n
r
es
u
lt o
f
th
i
s
p
ap
er
:
g.
T
h
eo
r
em
2
.
7
.
I
f
(
,
ℑ
)
is
a
C
DH
f
u
z
z
y
to
p
o
lo
g
ical
s
p
ac
e,
th
e
n
e
v
e
r
y
f
u
zz
y
h
o
m
o
g
e
n
eo
u
s
co
m
p
o
n
en
t
ℑ
o
f
(
,
ℑ
)
is
o
p
en
in
(
X,
ℑ₀
)
.
P
r
o
o
f
.
Su
p
p
o
s
e
o
n
th
e
co
n
tr
ar
y
t
h
at
f
o
r
s
o
m
e
∈
,
ℑ
∉
ℑ
₀
.
Sin
ce
(
,
ℑ
)
is
C
DH,
th
en
b
y
P
r
o
p
o
s
itio
n
1
.
4
,
(
,
ℑ
₀
)
is
C
DH.
C
h
o
o
s
e
a
co
u
n
tab
le
d
en
s
e
s
et
o
f
(
,
ℑ
₀
)
.
L
et
=
−
(
∩
(
−
ℑ
)
)
a
nd
=
(
(
∩
)
∪
(
∩
(
−
ℑ
)
)
)
−
(
)
.
B
y
L
e
m
m
a
2
.
5
,
ℑ
is
d
en
s
e
in
(
,
ℑ
₀
)
.
S
in
ce
∈
ℑ
₀
an
d
b
y
L
e
m
m
a
2
.
6
(
i
ii)
≠
∅
,
th
en
∩
(
−
ℑ
)
≠
∅
.
C
h
o
o
s
e
∈
∩
(
−
ℑ
)
an
d
let
=
∪
{
}
.
Sin
ce
⊆
,
th
en
an
d
ar
e
b
o
th
co
u
n
tab
le.
A
ls
o
,
b
y
L
e
m
m
a
2
.
6
(
iv
)
an
d
ar
e
d
en
s
e
in
(
,
ℑ
₀
)
.
T
h
en
b
y
P
r
o
p
o
s
itio
n
1
.
2
(
i)
,
ℚ
(
)
an
d
ℚ
(
)
ar
e
t
w
o
co
u
n
tab
le
d
e
n
s
e
o
f
th
e
C
DH
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
(
,
ℑ
)
.
T
h
u
s
th
er
e
i
s
a
f
u
zz
y
h
o
m
eo
m
o
r
p
h
is
m
ℎ
:
(
,
ℑ
)
→
(
,
ℑ
)
s
u
c
h
th
a
t
ℎ
(
(
ℚ
(
)
)
)
=
(
ℚ
(
)
)
.
So
ℎ
(
)
=
.
Sin
ce
∈
−
ℑ
,
th
en
b
y
P
r
o
p
o
s
itio
n
1
.
3
,
ℎ
(
)
∈
−
ℑ
.
Sin
ce
ℎ
(
)
∈
⊆
,
th
en
ℎ
(
)
∈
∩
(
−
ℑ
)
⊆
(
∩
(
−
ℑ
)
)
an
d
s
o
ℎ
(
)
∈
−
=
(
−
)
.
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ce
ℎ
(
)
∈
,
th
en
ℎ
(
)
∈
−
(
)
.
T
h
er
ef
o
r
e,
ℎ
(
)
∈
−
(
)
.
Set
=
ℎ⁻
¹
(
−
(
)
)
.
B
y
P
r
o
p
o
s
itio
n
1
.
5
,
ℎ
:
(
,
ℑ
₀
)
→
(
,
ℑ
₀
)
is
co
n
tin
u
o
u
s
at
an
d
s
o
th
er
e
ex
is
t
s
∈
ℑ
₀
s
u
ch
t
h
at
∈
an
d
ℎ
(
)
⊆
−
(
)
.
Sin
c
e
∈
∩
∈
ℑ
₀
an
d
is
d
e
n
s
e
i
n
(
,
ℑ
₀
)
,
th
e
n
t
h
er
e
e
x
is
t
s
∈
∩
∩
⊆
∩
.
B
y
L
e
m
m
a
2
.
6
(
i)
,
w
e
h
av
e
∈
ℑ
an
d
b
y
P
r
o
p
o
s
itio
n
1
.
3
ℎ
(
)
∈
ℑ
.
Sin
ce
∈
,
ℎ
(
)
∈
ℎ
(
)
⊆
−
(
)
.
A
ls
o
,
s
in
ce
∈
⊆
,
ℎ
(
)
∈
ℎ
(
)
=
.
T
h
er
ef
o
r
e,
ℎ
(
)
∈
(
−
(
)
)
∩
an
d
b
y
L
e
m
m
a
2
.
6
(
ii),
ℎ
(
)
∈
−
ℑ
,
a
co
n
tr
ad
ictio
n
.
h.
C
o
r
o
llar
y
2
.
8
.
I
f
(
,
ℑ
)
is
a
C
D
H
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e,
th
e
n
ev
er
y
f
u
zz
y
h
o
m
o
g
en
eo
u
s
co
m
p
o
n
en
t
ℑ
o
f
(
,
ℑ
)
is
clo
p
en
in
(
X,
ℑ₀
)
.
R
ec
all
th
at
a
f
u
zz
y
to
p
o
lo
g
ica
l
s
p
ac
e
(
,
ℑ
)
is
s
aid
to
b
e
h
o
m
o
g
en
eo
u
s
[
1
6
]
if
f
o
r
an
y
t
w
o
p
o
in
ts
₁
,
₂
in
,
th
er
e
ex
i
s
ts
a
f
u
zz
y
h
o
m
eo
m
o
r
p
h
is
m
ℎ
:
(
,
ℑ
)
→
(
,
ℑ
)
s
u
c
h
th
at
ℎ
(
₁
)
=
₂
.
A
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
(
,
ℑ
)
is
h
o
m
o
g
en
eo
u
s
i
f
f
ℑ
=
f
o
r
all
∈
.
i.
C
o
r
o
llar
y
2
.
9
.
I
f
(
,
ℑ
)
is
a
C
D
H
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
an
d
(
,
ℑ
₀
)
is
co
n
n
ec
ted
,
th
e
n
(
,
ℑ
)
is
h
o
m
o
g
en
eo
u
s
.
P
r
o
o
f
.
L
et
∈
.
A
cc
o
r
d
in
g
to
C
o
r
o
llar
y
2
.
8
,
ℑ
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Evaluation Warning : The document was created with Spire.PDF for Python.
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RE
F
E
R
E
NC
E
S
[1
]
S
.
A
.
G
h
o
u
r
a
n
d
A
.
F
o
ra
,
“
On
C
DH
f
u
z
z
y
sp
a
c
e
s,
”
J
o
u
rn
a
l
o
f
In
t
e
ll
ig
e
n
t
&
Fu
zz
y
S
y
ste
ms
,
v
o
l.
3
0
,
p
p
.
9
3
5
-
9
4
1
,
2
0
1
6
.
[2
]
L
.
A
.
Zad
e
h
,
“
F
u
z
z
y
S
e
ts,
”
In
fo
rm
a
n
d
c
o
n
tro
l
,
v
o
l.
8
,
p
p
.
3
3
8
-
3
5
3
,
1
9
6
5
.
[3
]
C.
L
.
Ch
a
n
g
,
“
F
u
z
z
y
T
o
p
o
lo
g
ica
l
S
p
a
c
e
s,
”
J
o
u
rn
a
l
o
f
M
a
th
e
ma
ti
c
a
l
A
n
a
lys
is
a
n
d
A
p
p
li
c
a
ti
o
n
s
,
v
o
l.
2
4
,
p
p
.
1
8
2
-
1
9
0
,
1
9
6
8
.
[4
]
C.
K.
W
o
n
g
,
“
Co
v
e
rin
g
p
ro
p
e
rti
e
s
o
f
f
u
z
z
y
to
p
o
lo
g
ica
l
sp
a
c
e
s,
”
J
o
u
rn
a
l
o
f
M
a
t
h
e
ma
ti
c
a
l
An
a
lys
is
a
n
d
Its
Ap
p
li
c
a
ti
o
n
s
,
v
o
l
.
4
3
,
p
p
.
6
9
7
-
7
0
4
,
1
9
7
3
.
[5
]
C.
K.
W
o
n
g
,
“
F
u
z
z
y
p
o
in
ts
a
n
d
lo
c
a
l
p
ro
p
e
rti
e
s
o
f
f
u
z
z
y
to
p
o
l
o
g
y
,
”
J
o
u
rn
a
l
o
f
M
a
th
e
ma
t
ica
l
A
n
a
lys
is
a
n
d
Its
Ap
p
li
c
a
ti
o
n
s
,
v
o
l
.
4
6
,
p
p
.
3
1
6
-
3
2
8
,
1
9
7
4
.
[6
]
R.
L
o
we
n
,
“
A
c
o
m
p
a
riso
n
o
f
d
iffere
n
t
c
o
m
p
a
c
tn
e
ss
n
o
ti
o
n
s
i
n
f
u
z
z
y
to
p
o
lo
g
ica
l
sp
a
c
e
s,
”
J
o
u
rn
a
l
o
f
M
a
th
e
ma
ti
c
a
l
A
n
a
lys
is
a
n
d
Its
A
p
p
l
ica
ti
o
n
s
,
v
o
l
.
6
4
,
p
p
.
4
4
6
-
4
5
4
,
1
9
7
8
.
[7
]
M
.
H.
G
h
a
n
im
,
e
t
a
l.
,
“
S
e
p
a
ra
ti
o
n
a
x
io
m
s,
su
b
sp
a
c
e
s
a
n
d
su
m
s
in
f
u
z
z
y
to
p
o
lo
g
y
,
”
J
o
u
rn
a
l
o
f
M
a
th
e
ma
ti
c
a
l
An
a
lys
is
a
n
d
Its
A
p
p
l
ica
ti
o
n
s
,
v
o
l
.
1
0
2
,
p
p
.
1
8
9
-
2
0
2
,
1
9
8
4
.
[8
]
A
.
A
.
F
o
ra
,
“
S
e
p
a
ra
ti
o
n
a
x
io
m
s
fo
r
f
u
z
z
y
sp
a
c
e
s,
”
Fu
zz
y
se
ts a
n
d
sy
ste
ms
,
v
o
l.
3
3
,
p
p
.
5
9
-
7
5
,
1
9
8
9
.
[9
]
P
.
A
.
S
a
h
a
,
“
Co
u
p
led
c
o
i
n
c
id
e
n
c
e
p
o
i
n
t
t
h
e
o
re
m
in
a
G
-
c
o
m
p
lete
f
u
z
z
y
m
e
tri
c
sp
a
c
e
,
”
J
o
u
rn
a
l
o
f
P
h
y
sic
a
l
S
c
ien
c
e
s
,
v
o
l.
1
9
p
p
.
2
3
-
2
8
,
2
0
1
4
.
[1
0
]
S
.
A
.
G
h
o
u
r,
“
Ho
m
o
g
e
n
e
it
y
in
f
u
z
z
y
sp
a
c
e
s
a
n
d
th
e
i
r
in
d
u
c
e
d
sp
a
c
e
s,
”
Qu
e
stio
n
s
a
n
d
An
sw
e
rs
in
Ge
n
e
ra
l
T
o
p
o
lo
g
y,
v
o
l
.
2
1
,
p
p
.
1
8
5
-
1
9
5
,
2
0
0
3
.
[1
1
]
S. A
.
G
h
o
u
r,
“
S
L
H f
u
z
z
y
sp
a
c
e
s,
”
Af
ric
a
n
Di
a
sp
o
ra
J
o
u
rn
a
l
o
f
M
a
th
e
ma
t
ics
,
v
o
l.
2
,
p
p
.
6
1
-
6
7
,
2
0
0
4
.
[1
2
]
S
.
A
.
G
h
o
u
r
a
n
d
A
.
F
o
ra
,
“
M
in
i
m
a
li
t
y
a
n
d
Ho
m
o
g
e
n
e
it
y
in
F
u
z
z
y
S
p
a
c
e
s,
”
J
o
u
rn
a
l
o
f
Fu
zz
y
M
a
t
h
e
ma
ti
c
s,
v
o
l.
1
2
,
p
p
.
7
2
5
--
7
3
7
,
2
0
0
4
.
[1
3
]
S
.
A
.
G
h
o
u
r,
“
L
o
c
a
l
h
o
m
o
g
e
n
e
it
y
in
f
u
z
z
y
to
p
o
l
o
g
ica
l
sp
a
c
e
s,
”
In
ter
n
a
ti
o
n
a
l
J
o
u
r
n
a
l
o
f
M
a
th
e
ma
t
ics
a
n
d
M
a
th
e
ma
ti
c
a
l
S
c
ien
c
e
s
,
A
rt.
ID 8
1
4
9
7
,
v
o
l.
14
,
2
0
0
6
.
[1
4
]
S
.
A
.
G
h
o
u
r,
“
S
o
m
e
Ge
n
e
ra
li
z
a
ti
o
n
s
o
f
M
in
im
a
l
F
u
z
z
y
Op
e
n
S
e
ts
,
”
Acta
M
a
th
e
ma
ti
c
a
Un
ive
rs
it
a
ti
s
Co
me
n
ia
n
a
e
,
v
o
l.
7
5
,
p
p
.
1
0
7
-
1
1
7
,
2
0
0
6
.
[1
5
]
S
.
A
.
G
h
o
u
r
a
n
d
K.
A
.
Zo
u
b
i,
“
On
so
m
e
o
rd
in
a
ry
a
n
d
f
u
z
z
y
h
o
m
o
g
e
n
e
it
y
t
y
p
e
s,
”
Acta
M
a
th
e
ma
ti
c
a
Un
ive
rs
it
a
ti
s
Co
me
n
ia
n
a
e
,
v
o
l.
7
7
,
p
p
.
1
9
9
-
2
0
8
,
2
0
0
8
.
[1
6
]
A
.
F
o
ra
an
d
S
.
A
.
G
h
o
u
r,
“
Ho
m
o
g
e
n
e
it
y
in
F
u
z
z
y
S
p
a
c
e
s,
”
Qu
e
s
ti
o
n
s
a
n
d
A
n
swe
rs
in
Ge
n
e
ra
l
T
o
p
o
l
o
g
y
,
v
o
l.
1
9
,
pp.
1
5
9
-
1
6
4
,
2
0
0
1
.
[1
7
]
S
.
A
.
G
h
o
u
r
a
n
d
A
.
Az
a
iz
e
h
,
“
F
u
z
z
y
Ho
m
o
g
e
n
e
o
u
s
Bit
o
p
o
lo
g
ica
l
S
p
a
c
e
s,
”
In
ter
n
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
El
e
c
trica
l
a
n
d
Co
mp
u
ter
E
n
g
in
e
e
rin
g
,
v
o
l.
8
,
p
p
.
2
0
8
8
-
8
7
0
8
,
2
0
1
8
.
[1
8
]
R.
Be
n
n
e
tt
,
“
Co
u
n
tab
le
d
e
n
se
h
o
m
o
g
e
n
e
o
u
s sp
a
c
e
s,
”
Fu
n
d
a
me
n
t
a
M
a
th
e
ma
ti
c
a
e
,
v
o
l.
7
4
,
p
p
.
1
8
9
-
1
9
4
,
1
9
7
2
.
[1
9
]
G
.
J.
W
a
n
g
,
“
T
h
e
o
r
y
o
f
L
-
f
u
z
z
y
t
o
p
o
lo
g
ica
l
sp
a
c
e
,
”
S
h
a
n
x
i
No
rm
a
l
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