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a
m
i
n
i
m
al
o
p
en
s
et
in
[
1
1
]
if
an
y
o
p
en
s
et
in
w
h
ic
h
is
co
n
tain
ed
in
is
∅
o
r
.
T
h
e
au
t
h
o
r
s
i
n
[
1
2
]
ex
te
n
d
ed
t
h
e
co
n
ce
p
t
m
in
i
m
al
o
p
en
s
et
t
o
in
cl
u
d
e
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
es
as
f
o
llo
w
s
:
A
f
u
zz
y
o
p
en
s
e
t
o
f
a
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
(
,
ℑ
)
is
ca
lled
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
i
n
if
is
n
o
n
z
er
o
an
d
th
er
e
is
n
o
n
o
n
ze
r
o
f
u
zz
y
o
p
en
s
et
s
u
c
h
th
at
<
,
an
d
th
en
A
l
G
h
o
u
r
co
n
ti
n
u
ed
th
e
s
t
u
d
y
o
f
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
ets
in
[
1
3
,
1
4
]
.
R
ec
en
tl
y
,
th
e
a
u
t
h
o
r
s
i
n
[
1
5
]
in
tr
o
d
u
ce
d
an
d
in
v
esti
g
ated
t
wo
t
y
p
es
o
f
m
i
n
i
m
al
o
p
en
s
ets
i
n
b
ito
p
o
lo
g
ical
s
p
a
ce
s
an
d
u
s
i
n
g
t
h
e
m
th
e
y
o
b
ta
in
ed
s
o
m
e
h
o
m
o
g
e
n
eit
y
r
es
u
l
ts
i
n
b
ito
p
o
lo
g
ical
s
p
ac
es.
As
d
ef
i
n
ed
in
[
1
6
]
,
f
o
r
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
(
,
ℑ
)
,
th
e
ass
o
ciate
d
to
p
o
lo
g
ical
s
p
ac
e
{
⁻
¹
(
,
1
]
:
∈
ℑ
}
is
ca
lled
th
e
-
c
u
t
(
lev
el)
to
p
o
lo
g
ical
s
p
ac
e
an
d
d
en
o
ted
b
y
ℑ
.
C
u
t
to
p
o
lo
g
ical
s
p
ac
e
s
h
av
e
b
ee
n
s
t
u
d
ied
in
d
ee
p
b
y
a
n
u
m
b
er
o
f
au
t
h
o
r
s
.
So
m
e
au
th
o
r
s
w
er
e
u
s
ed
cu
t
to
p
o
lo
g
ical
s
p
ac
es
f
o
r
s
o
lv
in
g
s
o
m
e
p
r
o
b
lem
s
o
f
f
u
zz
y
to
p
o
lo
g
y
b
y
r
ed
u
cin
g
th
e
m
to
s
tan
d
ar
d
p
r
o
b
lem
s
o
f
g
e
n
er
al
to
p
o
lo
g
y
(
s
ee
[
1
7
-
2
6
]
)
.
A
l
s
o
cu
t to
p
o
lo
g
ical
s
p
ac
es
h
a
v
e
s
h
o
w
n
to
b
e
u
s
e
f
u
l in
f
u
zz
y
au
to
m
ata
t
h
eo
r
y
i
n
[
2
7
-
2
9
]
.
I
n
t
h
is
p
ap
er
t
h
e
w
e
co
n
ti
n
u
e
th
e
s
tu
d
y
o
f
t
h
e
co
n
ce
p
t
s
o
f
m
i
n
i
m
alit
y
a
n
d
h
o
m
o
g
e
n
eit
y
i
n
t
h
e
f
u
zz
y
co
n
tex
t.
C
o
n
cr
etel
y
,
in
Sectio
n
2
w
e
in
tr
o
d
u
ce
t
w
o
n
e
w
n
o
tio
n
s
o
f
m
i
n
i
m
alit
y
i
n
f
u
zz
y
b
i
to
p
o
lo
g
ical
s
p
ac
es
w
h
ic
h
ar
e
ca
lled
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
a
n
d
p
air
w
is
e
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et.
Se
v
er
al
r
elatio
n
s
h
ip
s
b
et
w
ee
n
s
u
c
h
n
o
tio
n
s
a
n
d
a
k
n
o
w
n
o
n
e
ar
e
g
i
v
e
n
i
n
T
h
eo
r
e
m
2
.
2
,
T
h
eo
r
em
2
.
5
,
T
h
eo
r
em
2
.
8
an
d
T
h
eo
r
em
2
.
1
0
.
Mo
r
eo
v
er
,
in
t
h
e
s
a
m
e
s
ec
tio
n
,
w
e
p
r
o
v
id
e
r
es
u
lt
s
ab
o
u
t
t
h
e
tr
an
s
f
o
r
m
atio
n
o
f
m
i
n
i
m
al,
a
n
d
p
air
w
is
e
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
ets
o
f
a
f
u
zz
y
b
it
o
p
o
lo
g
ical
s
p
ac
e,
v
ia
f
u
zz
y
co
n
ti
n
u
o
u
s
a
n
d
f
u
zz
y
o
p
en
m
ap
p
in
g
s
,
an
d
p
air
w
is
e
co
n
tin
u
o
u
s
a
n
d
p
air
w
i
s
e
o
p
en
m
ap
p
in
g
s
,
r
esp
ec
ti
v
el
y
.
Se
ctio
n
3
is
d
e
v
o
ted
t
o
p
r
esen
t
t
w
o
n
e
w
n
o
tio
n
s
o
f
h
o
m
o
g
en
eit
y
in
t
h
e
f
u
zz
y
f
r
a
m
e
w
o
r
k
.
I
n
f
ac
t,
w
e
i
n
tr
o
d
u
ce
t
h
e
n
o
tio
n
s
o
f
h
o
m
o
g
e
n
eo
u
s
a
n
d
p
air
w
is
e
h
o
m
o
g
en
eo
u
s
f
u
zz
y
b
ito
p
o
lo
g
ical
s
p
ac
e
s
.
Sev
er
al
r
elat
io
n
s
h
ip
s
b
et
w
ee
n
s
u
c
h
n
o
tio
n
s
a
n
d
a
k
n
o
w
n
o
n
e
ar
e
g
iv
e
n
i
n
T
h
eo
r
e
m
3
.
3
.
,
T
h
e
o
r
e
m
3
.
7
an
d
C
o
r
o
llar
y
3
.
8
.
Mo
r
eo
v
er
,
s
o
m
e
co
n
n
ec
tio
n
s
b
et
wee
n
m
in
i
m
al
it
y
a
n
d
h
o
m
o
g
en
eit
y
ar
e
g
i
v
e
n
in
T
h
eo
r
em
3
.
9
,
T
h
eo
r
em
3
.
1
0
an
d
T
h
eo
r
em
3
.
1
1
.
Sectio
n
4
is
d
e
v
o
ted
to
s
h
o
w
t
h
a
t
cu
t
b
ito
p
o
lo
g
ical
s
p
ac
es
o
f
a
h
o
m
o
g
e
n
eo
u
s
(
r
esp
.
p
air
w
is
e
h
o
m
o
g
e
n
eo
u
s
)
f
u
zz
y
b
ito
p
o
lo
g
ical
s
p
ac
e
ar
e
h
o
m
o
g
en
eo
u
s
(
r
esp
.
p
air
w
i
s
e
h
o
m
o
g
en
eo
u
s
)
b
u
t
n
o
t
co
n
v
er
s
el
y
.
T
h
e
f
o
llo
w
in
g
d
ef
in
i
tio
n
s
an
d
r
es
u
lt
s
w
ill
b
e
u
s
ed
in
t
h
e
s
eq
u
el.
Def
i
n
itio
n
1
.
1
.
L
et
ℑ
₁
an
d
ℑ
₂
b
e
tw
o
f
u
zz
y
to
p
o
lo
g
ies
o
n
a
n
o
n
e
m
p
t
y
s
et
.
T
h
en
ℑ
₁
∪
ℑ
₂
f
o
r
m
s
a
s
u
b
b
ase
f
o
r
s
o
m
e
f
u
zz
y
to
p
o
lo
g
y
o
n
.
T
h
is
f
u
zz
y
to
p
o
lo
g
y
is
ca
lled
th
e
lea
s
t
u
p
p
er
b
o
u
n
d
f
u
zz
y
to
p
o
lo
g
y
o
n
X
an
d
d
en
o
ted
b
y
<
ℑ
₁
,
ℑ
₂
>
.
I
t
is
clea
r
th
at
ea
ch
b
asic
f
u
zz
y
o
p
en
s
et
in
<ℑ
₁,
ℑ₂>
ca
n
b
e
w
r
i
tten
i
n
t
h
e
f
o
r
m
A
∧
B
w
h
er
e
A
∈
ℑ₁
a
n
d
∈
ℑ
₂
.
Def
i
n
itio
n
1
.
2
.
L
et
:
(
,
ℑ
₁
,
ℑ
₂
)
→
(
,
₁
,
₂
)
b
e
a
f
u
n
c
ti
o
n
.
a.
is
ca
lled
f
u
zz
y
co
n
ti
n
u
o
u
s
(
f
u
zz
y
o
p
en
,
f
u
zz
y
h
o
m
eo
m
o
r
p
h
is
m
)
i
f
f
t
h
e
f
u
n
ctio
n
s
:
(
,
ℑ
₁
)
→
(
,
₁
)
an
d
:
(
,
ℑ
₂
)
→
(
,
₂
)
ar
e
f
u
zz
y
co
n
ti
n
u
o
u
s
(
f
u
z
z
y
o
p
en
,
f
u
zz
y
h
o
m
eo
m
o
r
p
h
is
m
r
esp
ec
tiv
e
l
y
)
.
b.
is
ca
lled
f
u
zz
y
p
a
ir
w
is
e
co
n
ti
n
u
o
u
s
i
f
f
f
o
r
ea
ch
∈
₁
∪
₂
,
←
(
)
∈
ℑ
₁
∪
ℑ
₂
.
c.
is
ca
lled
f
u
zz
y
p
air
w
i
s
e
h
o
m
eo
m
o
r
p
h
is
m
i
f
f
is
a
b
ij
e
ctio
n
,
f
u
zz
y
p
air
w
is
e
co
n
t
in
u
o
u
s
a
n
d
⁻
¹
:
(
,
₁
,
₂
)
→
(
,
ℑ
₁
,
ℑ
₂
)
is
f
u
zz
y
p
air
w
i
s
e
co
n
ti
n
u
o
u
s
.
P
r
o
p
o
s
itio
n
1
.
3
.
[
1
2
]
L
et
:
(
,
ℑ
₁
)
→
(
,
ℑ
₂
)
b
e
a
f
u
zz
y
co
n
ti
n
u
o
u
s
a
n
d
f
u
zz
y
o
p
en
f
u
n
c
tio
n
.
I
f
is
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
i
n
(
,
ℑ
₁
)
th
e
n
(
→
(
)
is
a
m
in
i
m
al
f
u
zz
y
o
p
en
s
e
t in
(
,
ℑ
₂
)
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
F
u
z
z
y
Ho
mo
g
en
eo
u
s
B
ito
p
o
lo
g
ica
l S
p
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s
(
S
a
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h
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r
)
4621
P
r
o
p
o
s
itio
n
1
.
4
.
[
1
2
]
L
et
(
,
ℑ
)
b
e
a
h
o
m
o
g
e
n
eo
u
s
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
w
h
ic
h
co
n
tai
n
s
a
m
in
i
m
al
f
u
zz
y
o
p
en
s
et.
T
h
en
w
e
h
av
e
t
h
e
f
o
l
lo
w
i
n
g
.
a.
T
h
e
co
llectio
n
o
f
al
l
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
s
i
n
(
,
ℑ
)
ca
n
b
e
w
r
i
tten
o
f
t
h
e
f
o
r
m
{
:
∈
}
w
h
er
e
∈
(
0
,
1
]
an
d
{
:
∈
}
is
a
p
ar
titi
o
n
o
n
X
an
d
|
|
=
|
|
f
o
r
all
,
∈
.
b.
Fo
r
an
y
f
u
zz
y
o
p
en
s
et
in
an
d
∈
eith
er
|
=
0
o
r
(
)
≥
f
o
r
all
∈
.
Def
i
n
itio
n
1
.
5
.
[
1
5
]
L
et
:
(
,
₁
,
₂
)
→
(
,
₁
,
₂
)
b
et
w
e
en
b
ito
p
o
lo
g
ical
s
p
ac
es.
a.
is
ca
lled
co
n
ti
n
u
o
u
s
(
o
p
en
,
h
o
m
eo
m
o
r
p
h
is
m
)
i
f
f
t
h
e
f
u
n
c
tio
n
s
:
(
,
₁
)
→
(
,
₁
)
an
d
:
(
,
₂
)
→
(
,
₂
)
ar
e
co
n
tin
u
o
u
s
(
o
p
en
,
h
o
m
eo
m
o
r
p
h
is
m
r
esp
ec
ti
v
el
y
)
.
b.
f
is
ca
lled
p
air
w
is
e
co
n
t
in
u
o
u
s
if
f
f
o
r
ea
ch
∈
₁
∪
₂
,
⁻
¹
(
)
∈
₁
∪
₂
.
c.
is
ca
lled
p
air
w
is
e
h
o
m
eo
m
o
r
p
h
is
m
i
f
f
is
a
b
i
j
ec
tio
n
,
p
ai
r
w
i
s
e
co
n
tin
u
o
u
s
an
d
⁻
¹
:
(
,
₁
,
₂
)
→
(
,
₁
,
₂
)
is
p
air
w
i
s
e
co
n
ti
n
u
o
u
s
.
Def
i
n
itio
n
1
.
6
.
[
1
5
]
A
b
ito
p
o
lo
g
ical
s
p
ac
e
(
,
₁
,
₂
)
is
s
aid
to
b
e
h
o
m
o
g
e
n
eo
u
s
(
r
esp
.
p
air
w
is
e
h
o
m
o
g
en
eo
u
s
)
if
f
o
r
an
y
t
w
o
p
o
in
ts
₁
,
₂
∈
th
er
e
ex
is
ts
a
h
o
m
eo
m
o
r
p
h
i
s
m
(
r
esp
.
p
air
w
i
s
e
h
o
m
eo
m
o
r
p
h
is
m
)
ℎ
:
(
,
₁
,
₂
)
→
(
,
₁
,
₂
)
s
u
c
h
t
h
at
ℎ
(
₁
)
=
₂
.
2.
M
I
NI
M
AL
I
T
Y
I
N
F
U
Z
Z
Y
B
I
T
O
P
O
L
O
G
I
CA
L
SPAC
E
S
D
ef
i
n
i
t
i
on 2
.1. Le
t
(
,
ℑ
₁
,
ℑ
₂
)
be a
f
uz
zy
bi
t
opo
l
og
i
ca
l
sp
ac
e.
a.
A
f
u
zz
y
s
e
t
o
f
is
ca
lled
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
i
n
(
,
ℑ
₁
,
ℑ
₂
)
if
is
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
i
n
b
o
th
o
f
(
,
ℑ
₁
)
an
d
(
,
ℑ
₂
)
.
b.
A
n
o
n
ze
r
o
f
u
zz
y
s
et
o
f
is
ca
lled
a
p
air
w
i
s
e
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
in
(
,
ℑ
1
,
ℑ
2
)
if
∈
ℑ
1
∪
ℑ
2
an
d
f
o
r
an
y
f
u
zz
y
s
et
∈
ℑ
1
∪
ℑ
2
w
it
h
≤
,
=
0
or
=
.
As
a
s
i
m
p
le
ex
a
m
p
le
o
f
a
f
u
zz
y
b
ito
p
o
lo
g
ical
s
p
ac
e
th
at
h
av
e
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et,
tak
e
=
{
,
,
}
,
ℑ
₁
=
{
0
,
1
,
{
}
,
{
}
,
{
,
}
}
,
ℑ
₂
=
{
0
,
1
,
{
}
,
{
}
,
{
,
}
}
,
it
is
clea
r
th
at
{
}
is
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
in
(
,
ℑ
₁
,
ℑ
₂
)
.
T
h
eo
r
em
2
.
2
.
Giv
en
a
f
u
zz
y
b
i
to
p
o
lo
g
ical
s
p
ac
e
(
,
ℑ
₁
,
ℑ
₂
)
an
d
b
e
a
f
u
zz
y
s
et
o
f
.
C
o
n
s
id
er
t
h
e
f
o
llo
w
i
n
g
s
ta
te
m
e
n
ts
.
(
)
is
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
in
(
,
ℑ
₁
,
ℑ
₂
)
.
(
)
is
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
in
(
,
〈
ℑ
₁
,
ℑ
₂
〉
)
w
ith
∈
ℑ
₁
∪
ℑ
₂
.
(
)
is
a
p
air
w
i
s
e
m
in
i
m
al
f
u
zz
y
o
p
en
s
et
in
(
,
ℑ
₁
,
ℑ
₂
)
.
T
h
en
(
)
⇒
(
)
⇒
(
)
.
P
r
o
o
f
.
a.
(
)
⇒
(
)
Sin
ce
is
a
m
in
i
m
al
f
u
zz
y
o
p
e
n
s
et
in
(
,
ℑ
₁
,
ℑ
₂
)
,
th
en
∈
ℑ
₁
∩
ℑ
₂
⊆
ℑ
₁
∪
ℑ
₂
⊆
〈
ℑ
₁
,
ℑ
₂
〉
.
Su
p
p
o
s
e
f
o
r
s
o
m
e
n
o
n
ze
r
o
f
u
z
z
y
s
et
∈
〈
ℑ
₁
,
ℑ
₂
〉
w
e
h
av
e
≤
.
C
h
o
o
s
e
₀
∈
s
u
c
h
t
h
at
(
₀
)
>
0
.
C
o
n
s
id
er
th
e
f
u
zz
y
p
o
in
t
w
it
h
s
u
p
p
o
r
t
=
₀
an
d
lev
el
(
)
=
(
₀
)
/
2
.
T
h
en
∈
an
d
s
o
th
er
e
ex
is
t
s
a
f
u
zz
y
s
et
∧
w
h
er
e
∈
ℑ
₁
,
∈
ℑ
₂
,
∈
∧
,
an
d
∧
≤
.
Sin
ce
is
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
in
(
,
ℑ
₁
,
ℑ
₂
)
th
en
≤
an
d
≤
an
d
h
en
ce
≤
∧
≤
.
Th
er
ef
o
r
e,
=
.
b.
(
)
⇒
(
)
Su
p
p
o
s
e
f
o
r
s
o
m
e
n
o
n
ze
r
o
∈
ℑ
1
∪
ℑ
2
,
≤
.
Sin
ce
ℑ
1
∪
ℑ
2
⊆
〈
ℑ
1
,
ℑ
2
〉
,
it
f
o
llo
w
s
th
at
=
.
T
h
e
f
o
llo
w
i
n
g
e
x
a
m
p
le
clar
i
f
i
es in
T
h
eo
r
e
m
2
.
2
th
at
(
)
⇏
(
)
.
E
x
a
m
p
le
2
.
3
.
L
et
=
{
,
}
w
it
h
th
e
f
u
zz
y
to
p
o
lo
g
ies
ℑ
₁
=
{
0
,
1
,
}
an
d
ℑ
₂
=
{
0
,
1
,
}
w
h
er
e
=
{
(
,
0
.
5
)
,
(
,
0
.
25
)
}
an
d
=
{
(
,
0
)
,
(
,
0
.
5
)
}
.
No
te
th
at
∧
∈
〈
ℑ
₁
,
ℑ
₂
〉
,
∧
≠
0
_
{
}
,
an
d
∧
<
.
T
h
er
ef
o
r
e,
A
is
a
p
air
w
i
s
e
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
i
n
(
,
ℑ
₁
,
ℑ
₂
)
b
u
t
n
o
t
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
i
n
(
,
〈
ℑ
₁
,
ℑ
₂
〉
)
.
T
h
e
f
o
llo
w
i
n
g
e
x
a
m
p
le
w
i
ll s
h
o
w
in
T
h
eo
r
e
m
2
.
2
th
at
(
)
⇏
(
)
.
E
x
a
m
p
le
2
.
4
.
L
et
=
{
,
}
w
it
h
th
e
f
u
zz
y
to
p
o
lo
g
ies
ℑ
₁
=
{
0
,
1
,
}
w
h
er
e
=
{
(
,
0
)
,
(
,
0
.
5
)
}
an
d
ℑ
₂
=
{
0
,
1
}
.
T
h
en
is
a
p
air
w
is
e
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
i
n
(
,
ℑ
₁
,
ℑ
₂
)
w
it
h
∈
ℑ
₁
∪
ℑ
₂
b
u
t
n
o
t
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
i
n
(
,
ℑ
₁
,
ℑ
₂
)
.
I
n
T
h
eo
r
e
m
2
.
2
,
if
w
e
ad
d
th
e
co
n
d
itio
n
"
∈
ℑ
₁
∩
ℑ
₂
"
,
th
en
co
n
v
er
s
e
o
f
ea
ch
i
m
p
l
icatio
n
w
ill
b
e
tr
u
e.
T
h
eo
r
em
2
.
5
.
L
e
t
(
,
ℑ
₁
,
ℑ
₂
)
b
e
a
f
u
zz
y
b
ito
p
o
lo
g
ical
s
p
ac
e
an
d
∈
ℑ
₁
∩
ℑ
₂
.
T
h
en
th
e
f
o
llo
w
i
n
g
ar
e
eq
u
iv
ale
n
t.
(
)
is
a
m
i
n
i
m
al
o
p
en
s
et
i
n
(
,
ℑ
₁
,
ℑ
₂
)
.
(
)
is
a
m
i
n
i
m
al
o
p
en
s
et
i
n
(
,
〈
ℑ
₁
,
ℑ
₂
〉
)
w
i
th
∈
ℑ
₁
∪
ℑ
₂
.
(
)
is
a
p
air
w
i
s
e
m
in
i
m
al
o
p
en
s
et
in
(
,
ℑ
₁
,
ℑ
₂
)
.
P
r
o
o
f
.
a.
(
)
⇒
(
)
an
d
(
)
⇒
(
)
f
o
llo
w
f
r
o
m
T
h
eo
r
e
m
2
.
2
.
b.
(
)
⇒
(
)
W
ith
o
u
t lo
s
s
o
f
g
e
n
er
alit
y
w
e
s
h
o
w
th
a
t
is
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
in
(
,
ℑ
₁
)
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
8
,
No
.
6
,
Dec
em
b
er
2
0
1
8
:
4
6
1
9
-
4
6
2
5
4622
L
et
∈
ℑ
₁
b
e
n
o
n
ze
r
o
w
i
th
≤
.
Sin
c
e
∈
ℑ
₁
∪
ℑ
₂
an
d
is
a
p
ai
r
w
is
e
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
i
n
(
,
ℑ
₁
,
ℑ
₂
)
,
th
en
=
.
T
h
eo
r
em
2
.
6
.
I
f
is
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
a
n
d
B
is
a
p
air
w
is
e
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
in
a
f
u
zz
y
b
ito
p
o
lo
g
ical
s
p
ac
e
(
,
ℑ
₁
,
ℑ
₂
)
,
th
en
eit
h
er
∧
=
0
o
r
=
.
P
r
o
o
f
.
Su
p
p
o
s
e
∧
≠
0
.
Fro
m
t
h
e
a
s
s
u
m
p
tio
n
s
,
w
e
co
n
clu
d
e
t
h
at
∧
∈
ℑ
₁
∪
ℑ
₂
.
T
h
en
w
e
h
av
e
t
h
e
f
o
llo
w
in
g
ca
s
e
s
:
a.
C
ase
1
:
I
f
∧
∈
ℑ
₁
,
th
en
w
e
co
n
cl
u
d
e
th
at
=
∧
,
i.e
.
,
≤
,
b
ec
au
s
e
is
m
in
i
m
al
f
u
zz
y
o
p
en
in
(
,
ℑ
₁
)
an
d
∧
≤
.
T
h
er
ef
o
r
e,
as
is
p
air
w
i
s
e
m
in
i
m
al
f
u
zz
y
o
p
en
in
(
,
ℑ
₁
,
ℑ
₂
)
an
d
∈
ℑ
₁
∪
ℑ
₂
,
w
e
co
n
cl
u
d
e
th
at
=
.
b.
C
ase
2
:
I
f
∧
∈
ℑ
₂
,
th
en
w
e
co
n
cl
u
d
e
th
at
=
∧
,
i.e
.
,
≤
,
b
ec
au
s
e
is
m
in
i
m
al
f
u
zz
y
o
p
en
in
(
,
ℑ
₂
)
an
d
∧
≤
.
T
h
er
ef
o
r
e,
as
is
p
air
w
i
s
e
m
in
i
m
al
f
u
zz
y
o
p
en
in
(
,
ℑ
₁
,
ℑ
₂
)
an
d
∈
ℑ
₁
∪
ℑ
₂
,
w
e
co
n
cl
u
d
e
th
at
=
.
T
h
er
ef
o
r
e,
in
b
o
th
ca
s
e
s
w
e
s
h
o
w
t
h
at
=
.
C
o
r
o
llar
y
2
.
7
.
(
i)
L
et
(
,
ℑ
₁
,
ℑ
₂
)
b
e
a
f
u
zz
y
b
ito
p
o
lo
g
ical
s
p
ac
e.
I
f
an
d
ar
e
t
w
o
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
ets
i
n
(
,
ℑ
₁
,
ℑ
₂
)
,
th
en
ei
th
er
∧
=
0
o
r
=
.
(
ii)
[
9
]
L
et
(
,
ℑ
)
b
e
a
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e.
I
f
an
d
ar
e
t
w
o
m
in
i
m
al
f
u
zz
y
o
p
en
s
et
s
i
n
(
,
ℑ
)
,
th
en
eit
h
er
∧
=
0
o
r
=
.
P
r
o
o
f
.
(
i)
(
r
esp
.
(
ii))
is
s
h
o
w
n
b
y
T
h
eo
r
em
s
2
.
2
an
d
2
.
6
(
r
esp
.
(
i)
a
b
o
v
e;
tak
e
ℑ
₁
=
ℑ
₂
).
I
n
E
x
a
m
p
le
2
.
3
,
A
an
d
B
ar
e
p
air
w
is
e
m
in
i
m
al
f
u
zz
y
o
p
en
s
ets
i
n
(
,
ℑ
₁
,
ℑ
₂
)
,
b
u
t
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in
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h
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6
w
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2
.
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m
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o
lo
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ac
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,
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it
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r
s
o
m
e
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{
1
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2
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th
en
is
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m
in
i
m
a
l f
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t in
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,
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r
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ch
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s
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it
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d
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en
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n
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th
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,
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9
.
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m
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a
f
u
zz
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g
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s
p
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th
en
is
a
m
i
n
i
m
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zz
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o
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en
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et
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n
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,
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m
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n
i
m
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zz
y
o
p
en
s
et
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n
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,
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h
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2
.
1
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et
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t
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f
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zz
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lo
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s
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ac
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,
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it
h
≠
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d
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h
en
∧
is
a
m
in
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m
a
l f
u
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o
p
en
s
et
i
n
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,
<
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.
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f
{
,
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h
er
e
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r
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th
en
b
y
T
h
eo
r
em
2
.
8
,
it
f
o
llo
w
s
th
at
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o
th
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d
ar
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m
i
n
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en
s
ets
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n
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d
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C
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r
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2
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7
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ii),
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llo
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s
th
at
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ith
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r
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=
0
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h
er
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o
r
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w
e
m
a
y
a
s
s
u
m
e
th
at
∈
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an
d
∈
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₂
.
Su
p
p
o
s
e
th
er
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ex
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t
s
a
n
o
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ze
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u
zz
y
o
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en
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et
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,
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₂
>
s
u
ch
t
h
at
≤
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h
o
o
s
e
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d
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₂
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u
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h
th
at
0
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h
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r
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2
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1
1
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y
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e
a
f
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zz
y
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o
m
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m
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s
m
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en
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a
m
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et
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l
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)
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a
m
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n
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r
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Pro
p
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1
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3
.
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ii)
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llo
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s
f
r
o
m
(
i)
ab
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v
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h
eo
r
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2
.
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.
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et
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u
zz
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s
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co
n
ti
n
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o
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s
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d
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en
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n
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air
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f
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y
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en
s
et
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ls
o
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en
t
h
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n
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p
p
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e
f
o
r
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o
m
e
n
o
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ze
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zz
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et
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h
en
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ce
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at
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ce
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en
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d
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o
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ts
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r
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s
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en
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.
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ce
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a
p
ai
r
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m
in
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)
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^
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h
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en
d
s
th
e
p
r
o
o
f
.
C
o
r
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llar
y
2
.
1
3
.
L
e
t
:
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,
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₂
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→
(
,
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,
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e
a
f
u
zz
y
p
air
w
is
e
h
o
m
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m
o
r
p
h
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m
.
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h
en
is
a
p
air
w
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e
m
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n
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o
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et
in
(
,
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3.
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₁
)
=
₂
.
L
e
m
m
a
3
.
2
(
)
en
d
s
th
e
p
r
o
o
f
.
(
)
L
et
₁
,
₂
∈
.
As
(
,
ℑ
₁
,
ℑ
₂
)
is
h
o
m
o
g
e
n
eo
u
s
,
th
er
e
e
x
is
t
s
a
f
u
zz
y
h
o
m
eo
m
o
r
p
h
is
m
ℎ
:
(
,
ℑ
₁
,
ℑ
₂
)
→
(
,
ℑ
₁
,
ℑ
₂
)
s
u
c
h
t
h
at
ℎ
(
₁
)
=
₂
.
L
e
m
m
a
3
.
2
(
)
en
d
s
th
e
p
r
o
o
f
.
(
)
L
et
₁
,
₂
∈
.
As
(
,
ℑ
₁
,
ℑ
₂
)
is
h
o
m
o
g
en
eo
u
s
,
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
₁
,
ℑ
₂
)
→
(
,
ℑ
₁
,
ℑ
₂
)
s
u
c
h
t
h
at
ℎ
(
₁
)
=
₂
.
T
h
er
ef
o
r
e,
w
e
h
a
v
e
ℎ
:
(
,
ℑ
₁
)
→
(
,
ℑ
₁
)
an
d
ℎ
:
(
,
ℑ
₂
)
→
(
,
ℑ
₂
)
ar
e
f
u
zz
y
h
o
m
eo
m
o
r
p
h
is
m
s
.
He
n
ce
(
,
ℑ
₁
)
an
d
(
,
ℑ
₂
)
ar
e
h
o
m
o
g
en
eo
u
s
.
I
m
p
licatio
n
(
)
o
f
T
h
eo
r
em
3
.
3
is
n
o
t r
ev
er
s
ib
le
as th
e
f
o
llo
w
in
g
ex
a
m
p
le
s
h
o
w
s
:
E
x
a
m
p
le
3
.
4
.
L
et
=
{
,
,
}
,
ℑ
₁
=
{
0
,
1
,
{
}
}
,
ℑ
₂
=
{
0
,
1
,
{
}
,
{
}
,
{
,
}
}
.
T
h
en
<
ℑ
₁
,
ℑ
₂
>
=
{
∶
⊆
}
an
d
h
en
ce
(
,
<
ℑ
₁
,
ℑ
₂
>
)
is
h
o
m
o
g
en
eo
u
s
.
I
f
:
→
is
a
b
i
j
ec
tio
n
f
o
r
w
h
ich
(
)
=
,
th
en
w
e
h
a
v
e
{
,
}
∈
ℑ
₁
∪
ℑ
₂
b
u
t
→
(
{
,
}
}
)
=
{
,
−
1
(
)
}
∉
ℑ
₁
∪
ℑ
₂
w
h
ic
h
s
h
o
w
s
th
a
t
is
n
o
t
a
f
u
zz
y
p
air
w
is
e
h
o
m
eo
m
o
r
p
h
is
m
.
He
n
ce
(
,
ℑ
₁
,
ℑ
₂
)
is
n
o
t p
air
w
is
e
h
o
m
o
g
en
e
o
u
s
.
I
m
p
licatio
n
(
)
o
f
T
h
eo
r
em
3
.
3
is
n
o
t r
ev
er
s
ib
le
a
s
th
e
f
o
llo
w
i
n
g
e
x
a
m
p
le
s
h
o
w
s
:
E
x
a
m
p
le
3
.
5
.
L
e
t
=
ℝ
,
ℑ
₁
=
,
ℑ
₂
=
{
0
,
1
,
{
2
}
}
.
I
t
is
n
o
t
d
i
f
f
icu
l
t
to
s
ee
t
h
at
(
,
ℑ
₁
)
is
h
o
m
o
g
e
n
eo
u
s
an
d
th
at
(
,
ℑ
₂
)
is
n
o
t
h
o
m
o
g
en
eo
u
s
.
A
s
ℑ
₁
∪
ℑ
₂
=
ℑ
₁
,
th
en
(
,
ℑ
₁
,
ℑ
₂
)
is
p
ai
r
w
i
s
e
h
o
m
o
g
en
eo
u
s
.
On
th
e
o
th
er
h
an
d
,
b
y
T
h
eo
r
e
m
3
.
3
(
)
,
it f
o
llo
w
s
t
h
at
(
,
ℑ
₁
,
ℑ
₂
)
is
n
o
t h
o
m
o
g
e
n
eo
u
s
.
I
m
p
licatio
n
(
)
o
f
T
h
eo
r
em
3
.
3
is
n
o
t r
ev
er
s
ib
le
a
s
th
e
f
o
llo
w
i
n
g
e
x
a
m
p
le
s
h
o
w
s
:
E
x
a
m
p
le
3
.
6
.
L
et
X
=
{
,
,
,
,
,
}
,
ℑ
₁
=
{
∶
∈
{
∅
,
,
{
,
,
}
,
{
,
,
}
}
}
,
an
d
ℑ
₂
=
{
∶
∈
{
∅
,
,
{
,
}
,
{
,
}
}
,
{
,
}
,
{
,
,
,
}
,
{
,
,
,
}
,
{
,
,
,
}
}
.
I
t
is
n
o
t
d
if
f
ic
u
lt
to
s
e
e
th
at
(
,
ℑ
₁
)
is
h
o
m
o
g
en
eo
u
s
,
(
,
ℑ
₂
)
is
h
o
m
o
g
e
n
eo
u
s
,
an
d
(
,
ℑ
₁
,
ℑ
₂
)
is
n
o
t
p
air
w
i
s
e
h
o
m
o
g
e
n
eo
u
s
an
d
h
e
n
c
e
n
o
t h
o
m
o
g
en
eo
u
s
.
T
h
eo
r
em
3
.
7
.
L
et
(
,
ℑ
₁
,
ℑ
₂
)
b
e
a
h
o
m
o
g
en
eo
u
s
f
u
zz
y
b
ito
p
o
lo
g
ical
s
p
ac
e
h
av
i
n
g
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et.
I
f
∈
ℑ
₁
∪
ℑ
₂
,
th
e
n
th
e
f
o
llo
w
in
g
ar
e
eq
u
iv
ale
n
t:
(
)
is
an
m
i
n
m
al
f
u
zz
y
o
p
en
s
e
t in
(
,
ℑ
₁
,
ℑ
₂
)
.
(
)
is
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
in
(
,
ℑ
₁
)
o
r
is
a
m
in
i
m
al
f
u
zz
y
o
p
e
n
s
et
i
n
(
,
ℑ
₂
)
.
P
r
o
o
f
.
(
)
⇒
(
)
Ob
v
io
u
s
.
(
)
⇒
(
)
W
ith
o
u
t
lo
s
s
o
f
g
en
er
al
it
y
w
e
m
a
y
ass
u
m
e
th
at
is
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
in
(
,
ℑ
₁
)
.
A
p
p
l
y
i
n
g
P
r
o
p
o
s
itio
n
1
.
4
,
th
er
e
ex
is
t
s
₁
∈
(
0
,
1
]
an
d
₁
⊆
s
u
ch
th
at
=
₁
1
.
T
ak
e
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
o
f
(
,
ℑ
₁
,
ℑ
₂
)
.
T
h
en
ag
ai
n
b
y
P
r
o
p
o
s
itio
n
1
.
4
,
th
er
e
ex
is
ts
₂
∈
(
0
,
1
]
an
d
₂
⊆
s
u
c
h
t
h
at
=
₂
2
.
T
ak
e
₁
∈
₁
,
₂
∈
₂
,
an
d
a
f
u
zz
y
h
o
m
eo
m
o
r
p
h
is
m
ℎ
:
(
,
ℑ
₁
,
ℑ
₂
)
→
(
,
ℑ
₁
,
ℑ
₂
)
s
u
c
h
th
at
ℎ
(
₂
)
=
₁
.
A
p
p
l
y
in
g
T
h
eo
r
e
m
2
.
1
1
(
ii),
it
f
o
llo
w
s
t
h
at
ℎ
→
(
)
=
ℎ
→
(
₂
2
)
=
₂
ℎ
(
₂
)
is
a
m
in
i
m
al
f
u
zz
y
o
p
en
s
e
t
in
(
,
ℑ
₁
,
ℑ
₂
)
.
Sin
ce
₁
∈
₁
∩
ℎ
(
₂
)
,
th
en
(
₂
ℎ
(
₂
)
}
)
∧
(
₁
1
)
≠
0
.
Sin
ce
₂
ℎ
(
₂
)
an
d
₁
1
ar
e
m
i
n
i
m
al
f
u
zz
y
o
p
e
n
s
e
ts
i
n
(
,
ℑ
₁
)
,
th
en
b
y
C
o
r
o
llar
y
2
.
7
(
ii),
it f
o
llo
w
s
t
h
at
₂
ℎ
(
₂
)
=
₁
1
an
d
h
en
ce
is
a
m
i
n
m
al
f
u
zz
y
o
p
en
s
et
in
(
,
ℑ
₁
,
ℑ
₂
)
.
C
o
r
o
llar
y
3
.
8
.
L
et
(
,
ℑ
₁
,
ℑ
₂
)
b
e
an
h
o
m
o
g
en
eo
u
s
f
u
zz
y
b
ito
p
o
lo
g
ical
s
p
ac
e
h
a
v
i
n
g
a
m
in
i
m
a
l
f
u
zz
y
o
p
en
s
et
a
n
d
∈
ℑ
₁
∪
ℑ
₂
.
T
h
en
th
e
f
o
llo
w
i
n
g
ar
e
eq
u
i
v
ale
n
t:
(
)
is
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
in
(
,
ℑ
₁
,
ℑ
₂
)
.
(
)
is
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
in
(
,
〈
ℑ
₁
,
ℑ
₂
〉
)
.
(
)
is
a
p
air
w
i
s
e
m
in
i
m
al
f
u
zz
y
o
p
en
s
et
in
(
,
ℑ
₁
,
ℑ
₂
)
.
(
)
is
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
in
(
,
ℑ
₁
)
o
r
is
a
m
in
i
m
al
f
u
zz
y
o
p
e
n
s
et
i
n
(
,
ℑ
₂
)
.
(
)
is
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
in
(
,
ℑ
₁
)
.
(
)
is
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
in
(
,
ℑ
₂
)
.
P
r
o
o
f
.
(
)
⇒
(
)
an
d
(
)
⇒
(
)
T
h
eo
r
em
2
.
2
.
(
)
⇒
(
)
C
o
r
o
llar
y
2
.
9
.
(
)
⇒
(
)
T
h
eo
r
em
3
.
7
.
(
)
⇔
(
)
(
)
⇔
(
)
T
h
eo
r
em
3
.
7
.
T
h
eo
r
em
3
.
9
.
L
et
(
,
ℑ
₁
,
ℑ
₂
)
b
e
a
p
ai
r
w
i
s
e
h
o
m
o
g
e
n
eo
u
s
f
u
zz
y
b
i
to
p
o
lo
g
ical
s
p
ac
e.
I
f
is
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
i
n
(
,
ℑ
₁
,
ℑ
₂
)
,
th
e
n
th
er
e
e
x
is
t
∈
(
0
,
1
]
an
d
⊆
s
u
c
h
th
a
t
=
.
P
r
o
o
f
.
Su
p
p
o
s
e
o
n
t
h
e
co
n
tr
ar
y
t
h
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t
h
er
e
e
x
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t
t
w
o
d
i
f
f
er
e
n
t
p
o
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ts
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∈
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u
c
h
t
h
at
0
<
(
₁
)
<
(
₂
)
.
Sin
ce
(
,
ℑ
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,
ℑ
₂
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is
p
air
w
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e
h
o
m
o
g
en
eo
u
s
t
h
er
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i
s
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f
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h
o
m
eo
m
o
r
p
h
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m
ℎ
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(
,
ℑ
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ℑ
₂
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→
(
,
ℑ
₁
,
ℑ
₂
)
s
u
c
h
t
h
at
ℎ
(
₁
)
=
₂.
As
is
p
air
w
is
e
m
i
n
i
m
al,
t
h
en
b
y
C
o
r
o
llar
y
2
.
1
3
ℎ
→
(
)
is
p
air
w
is
e
m
in
i
m
al.
As
(
∧
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
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8
8
-
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I
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→
(
)
)
(
₂
)
=
{
(
₂
)
,
(
₁
)
}
=
(
₁
)
>
0
,
∧
ℎ
→
(
)
≠
0
.
T
h
er
ef
o
r
e,
b
y
T
h
eo
r
e
m
2
.
6
0
(
)
=
an
d
h
en
ce
(
₁
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(
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(
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₁
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(
₂
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co
n
tr
ad
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n
.
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n
th
e
s
eq
u
el,
GFP
H
(
,
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₁
,
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₂
)
w
ill
d
en
o
te
th
e
g
r
o
u
p
o
f
all
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u
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m
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o
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p
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m
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h
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e
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t
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,
ℑ
₂
)
b
e
a
p
air
w
i
s
e
h
o
m
o
g
en
eo
u
s
f
u
zz
y
b
ito
p
o
lo
g
ical
s
p
ac
e.
I
f
is
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
in
(
,
ℑ
₁
,
ℑ
₂
)
,
th
en
th
e
f
a
m
il
y
{
ℎ
(
)
:
ℎ
∈
(
,
ℑ
₁
,
ℑ
₂
)
}
is
th
e
s
e
t
o
f
all
p
air
w
i
s
e
m
in
i
m
a
l
f
u
zz
y
o
p
en
s
ets
i
n
(
X,
ℑ₁,
ℑ₂)
.
P
r
o
o
f
.
L
et
b
e
a
p
air
w
is
e
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
o
f
(
,
ℑ
₁
,
ℑ
₂
)
.
C
h
o
o
s
e
∈
s
u
c
h
t
h
at
(
)
>
0
an
d
∈
.
As
(
,
ℑ
₁
,
ℑ
₂
)
is
p
air
w
is
e
h
o
m
o
g
e
n
eo
u
s
,
th
er
e
ex
is
t
s
a
f
u
zz
y
p
air
w
is
e
h
o
m
eo
m
o
r
p
h
is
m
ℎ
:
(
,
ℑ
₁
,
ℑ
₂
)
→
(
,
ℑ
₁
,
ℑ
₂
)
s
u
c
h
t
h
at
ℎ
(
)
=
.
A
p
p
l
y
i
n
g
co
r
o
llar
y
2
.
1
3
,
it
f
o
llo
w
s
t
h
at
ℎ
←
(
)
is
p
air
w
i
s
e
m
in
i
m
al.
As
(
(
ℎ
(
)
)
∧
ℎ
←
(
)
)
(
)
=
{
(
)
,
}
>
0
,
∧
ℎ
←
(
)
≠
0
.
T
h
er
ef
o
r
e,
b
y
T
h
eo
r
em
2
.
6
ℎ
←
(
)
=
an
d
h
e
n
ce
=
ℎ
→
(
)
=
ℎ
(
)
.
R
ec
all
th
at
a
p
ar
titi
o
n
ℬ
o
n
a
n
o
n
e
m
p
t
y
s
et
is
ca
lled
a
r
eg
u
lar
p
ar
titi
o
n
if
f
o
r
an
y
t
w
o
ele
m
e
n
ts
,
∈
ℬ
,
|
|
=
|
|
.
T
h
eo
r
e
m
3
.
1
1
.
L
et
(
,
ℑ
₁
,
ℑ
₂
)
b
e
a
p
air
w
is
e
h
o
m
o
g
en
eo
u
s
f
u
zz
y
b
ito
p
o
lo
g
ical
s
p
ac
e.
I
f
is
a
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
et
i
n
(
,
ℑ
₁
,
ℑ
₂
)
,
th
en
th
e
f
a
m
il
y
{
ℎ
(
)
:
ℎ
∈
(
,
ℑ
₁
,
ℑ
₂
)
}
f
o
r
m
s
a
r
eg
u
lar
p
ar
titi
o
n
o
n
.
Pro
o
f
.
I
t
is
clea
r
th
at
ℎ
(
)
≠
∅
f
o
r
ev
er
y
ℎ
∈
(
,
ℑ
₁
,
ℑ
₂
)
.
Su
p
p
o
s
e
f
o
r
s
o
m
e
,
∈
(
,
ℑ
₁
,
ℑ
₂
)
,
(
)
∩
(
)
≠
∅
.
C
h
o
o
s
e
∈
(
)
∩
(
)
,
∈
,
an
d
ℎ
∈
(
,
ℑ
₁
,
ℑ
₂
)
s
u
c
h
th
at
ℎ
(
)
=
.
B
y
C
o
r
o
llar
y
2
.
1
3
,
b
o
th
(
ℎ
∘
)
→
(
)
an
d
(
ℎ
∘
)
→
(
)
ar
e
p
air
w
is
e
m
i
n
i
m
al
f
u
zz
y
o
p
en
s
ets.
Sin
ce
(
(
)
∧
(
ℎ
∘
)
→
(
)
∧
(
ℎ
∘
)
→
(
)
)
(
)
=
>
0
th
e
n
b
y
T
h
eo
r
e
m
2
.
6
,
it
f
o
llo
w
th
at
=
(
ℎ
∘
)
→
(
)
an
d
=
(
ℎ
∘
)
→
(
)
an
d
h
en
c
e
(
ℎ
∘
)
→
(
)
=
(
ℎ
∘
)
→
(
)
.
T
h
u
s
,
(
ℎ
∘
)
(
)
=
(
ℎ
∘
)
(
)
an
d
h
en
ce
(
ℎ
∘
)
(
)
=
(
ℎ
∘
)
(
)
.
T
h
er
ef
o
r
e,
(
)
=
(
)
.
T
o
s
ee
th
at
∪
{
ℎ
(
)
:
ℎ
∈
(
,
ℑ
₁
,
ℑ
₂
)
}
=
,
let
∈
.
P
ick
∈
an
d
ℎ
∈
(
,
ℑ
₁
,
ℑ
₂
)
s
u
c
h
th
at
ℎ
(
)
=
.
T
h
u
s
,
∈
ℎ
(
)
w
h
ic
h
co
m
p
letes th
e
p
r
o
o
f
.
4.
CUT F
U
Z
Z
Y
B
I
T
O
P
O
L
O
G
I
CAL
SPAC
E
S
T
h
eo
r
em
4
.
1
.
I
f
:
(
,
ℑ
₁
,
ℑ
₂
)
→
(
,
₁
,
₂
)
is
f
u
zz
y
p
air
w
i
s
e
co
n
ti
n
u
o
u
s
(
h
o
m
eo
m
o
r
p
h
is
m
)
,
th
e
n
f
o
r
ev
er
y
∈
[
0
,
1
)
,
:
(
,
(
ℑ
₁
)
,
(
ℑ
₂
)
)
→
(
,
(
₁
)
,
(
₂
)
)
is
p
air
w
i
s
e
co
n
ti
n
u
o
u
s
(
h
o
m
eo
m
o
r
p
h
is
m
)
.
P
r
o
o
f
.
L
et
∈
[
0
,
1
)
an
d
let
∈
(
₁
)
∪
(
₂
)
)
.
T
h
en
th
e
r
e
is
∈
₁
∪
₂
s
u
ch
t
h
at
=
⁻
¹
(
,
1
]
.
Sin
c
e
:
(
,
ℑ
₁
,
ℑ
₂
)
→
(
,
₁
,
₂
)
is
f
u
zz
y
p
air
w
is
e
co
n
tin
u
o
u
s
,
th
en
←
(
)
∈
ℑ
₁
∪
ℑ
₂
an
d
s
o
(
←
(
)
)
⁻
¹
(
,
1
]
∈
(
ℑ
₁
)
∪
(
ℑ
₂
)
.
Sin
ce
⁻
¹
(
)
=
⁻
¹
(
⁻
¹
(
,
1
]
)
=
(
←
(
)
)
⁻
¹
(
,
1
]
,
th
en
⁻
¹
(
)
∈
(
ℑ
₁
)
∪
(
ℑ
₂
)
.
I
t
f
o
llo
ws
th
at
:
(
,
(
ℑ
₁
)
,
(
ℑ
₂
)
)
→
(
,
(
₁
)
,
(
₂
)
)
is
p
air
w
is
e
co
n
ti
n
u
o
u
s
.
C
o
r
o
llar
y
4
.
2
.
I
f
(
,
ℑ
₁
,
ℑ
₂
)
is
a
p
air
w
i
s
e
h
o
m
o
g
e
n
eo
u
s
f
u
zz
y
b
ito
p
o
lo
g
ical
s
p
ac
e,
th
en
f
o
r
ev
er
y
∈
[
0
,
1
)
th
e
b
ito
p
o
lo
g
ical
s
p
ac
e
(
,
(
ℑ
₁
)
,
(
ℑ
₂
)
)
is
p
air
w
i
s
e
h
o
m
o
g
e
n
eo
u
s
.
T
h
e
f
o
ll
o
w
i
n
g
e
x
a
m
p
le
s
h
o
w
s
t
h
at
t
h
e
co
n
v
er
s
e
o
f
ea
c
h
o
f
T
h
eo
r
e
m
s
4
.
1
an
d
C
o
r
o
llar
y
4
.
2
n
ee
d
n
o
t
to
b
e
tr
u
e
i
n
g
en
er
al
:
E
x
a
m
p
le
4
.
3
.
L
et
=
{
1
,
2
}
,
ℑ
=
{
0
,
1
,
,
,
,
}
w
h
er
e
=
{
(
1
,
0
)
,
(
2
,
1
)
}
,
=
{
(
1
,
1
)
,
(
2
,
0
)
}
,
=
{
(
1
,
(
1
/
2
)
)
,
(
2
,
1
)
}
,
=
{
(
1
,
(
1
/
2
)
)
,
(
2
,
0
)
}
.
Def
i
n
e
ℎ
:
→
by
ℎ
(
1
)
=
2
a
n
d
ℎ
(
2
)
=
1
.
T
h
en
f
o
r
ev
er
y
∈
[
0
,
1
)
,
ℎ
:
(
,
ℑ
,
ℑ
)
→
(
,
ℑ
,
ℑ
)
is
p
air
w
is
e
co
n
ti
n
u
o
u
s
wh
ile
ℎ
:
(
,
ℑ
,
ℑ
)
→
(
,
ℑ
,
ℑ
)
is
n
o
t
f
u
zz
y
p
air
w
is
e
co
n
tin
u
o
u
s
.
T
h
er
ef
o
r
e,
th
e
co
n
v
er
s
e
o
f
T
h
eo
r
em
4
.
1
is
n
o
t
tr
u
e
in
g
en
er
al.
Fo
r
ev
er
y
∈
[
0
,
1
)
,
ℎ
:
(
,
ℑ
,
ℑ
)
→
(
,
ℑ
,
ℑ
)
is
p
air
w
i
s
e
h
o
m
eo
m
o
r
p
h
is
m
w
it
h
ℎ
(
1
)
=
2
an
d
ℎ⁻
¹
(
2
)
=
1
an
d
s
o
(
,
ℑ
,
ℑ
)
is
p
air
w
i
s
e
h
o
m
o
g
en
eo
u
s
.
On
th
e
o
th
er
h
an
d
,
it
is
n
o
t
d
if
f
ic
u
lt
to
ch
ec
k
th
at
(
X,
ℑ,
ℑ)
is
n
o
t
p
air
w
is
e
h
o
m
o
g
en
eo
u
s
.
T
h
is
s
h
o
w
s
th
a
t
t
h
e
co
n
v
er
s
e
o
f
C
o
r
o
llar
y
4
.
2
is
n
o
t
tr
u
e
i
n
g
en
er
al.
L
e
m
m
a
4
.
4
.
[
2
6
]
I
f
:
(
,
ℑ
)
→
(
,
ℑ
)
is
a
f
u
zz
y
co
n
ti
n
u
o
u
s
(
h
o
m
eo
m
o
r
p
h
is
m
)
m
ap
,
t
h
e
n
f
o
r
all
a
∈
[
0
,
1
)
,
:
(
,
ℑ
)
→
(
,
ℑ
)
is
co
n
tin
u
o
u
s
(
h
o
m
eo
m
o
r
p
h
i
s
m
)
.
T
h
eo
r
em
4
.
5
.
I
f
:
(
,
ℑ
₁
,
ℑ
₂
)
→
(
,
₁
,
₂
)
i
s
a
f
u
zz
y
co
n
ti
n
u
o
u
s
(
h
o
m
eo
m
o
r
p
h
is
m
)
f
u
n
ct
io
n
,
th
e
n
f
o
r
all
,
∈
[
0
,
1
)
,
:
(
,
(
ℑ
₁
)
,
(
ℑ
₂
)
)
→
(
,
(
₁
)
,
(
₂
)
)
is
co
n
ti
n
u
o
u
s
(
h
o
m
eo
m
o
r
p
h
i
s
m
)
.
P
r
o
o
f
.
W
e
p
r
o
v
e
o
n
l
y
th
e
co
n
ti
n
u
it
y
p
ar
t.
Su
p
p
o
s
e
th
at
:
(
,
ℑ
₁
,
ℑ
₂
)
→
(
,
₁
,
₂
)
is
a
f
u
zz
y
co
n
ti
n
u
o
u
s
f
u
n
ctio
n
.
T
h
en
b
o
th
:
(
,
ℑ
₁
)
→
(
,
₁
)
an
d
:
(
,
ℑ
₂
)
→
(
,
₂
)
ar
e
f
u
zz
y
co
n
tin
u
o
u
s
.
T
h
er
ef
o
r
e
b
y
L
e
m
m
a
4
.
4
,
:
(
,
ℑ
)
→
(
,
ℑ
)
an
d
:
(
,
ℑ
)
→
(
,
ℑ
)
ar
e
co
n
tin
u
o
u
s
.
I
t
f
o
llo
w
s
t
h
at
:
(
,
(
ℑ
₁
)
,
(
ℑ
₂
)
)
→
(
,
(
₁
)
,
(
₂
)
)
is
co
n
ti
n
u
o
u
s
.
C
o
r
o
llar
y
4
.
6
.
I
f
(
,
ℑ
₁
,
ℑ
₂
)
is
a
h
o
m
o
g
e
n
eo
u
s
f
u
zz
y
b
ito
p
o
lo
g
ical
s
p
ac
e,
th
en
f
o
r
all
,
∈
[
0
,
1
)
th
e
b
ito
p
o
lo
g
ical
s
p
ac
e
(
,
(
ℑ
₁
)
,
(
ℑ
₂
)
)
is
h
o
m
o
g
e
n
eo
u
s
.
T
h
e
f
o
llo
w
i
n
g
ex
a
m
p
le
w
il
l s
h
o
w
t
h
at
t
h
e
co
n
v
er
s
e
o
f
C
o
r
o
llar
y
4
.
6
n
ee
d
n
o
t to
b
e
tr
u
e
in
g
en
er
al
:
E
x
a
m
p
le
4
.
7
.
L
et
=
{
1
,
2
}
an
d
ℑ
=
{
:
(
)
⊆
[
0
,
(
1
/
2
)
]
}
∪
{
:
(
)
⊆
[
(
1
/
2
)
,
1
]
}
∪
{
:
(
1
)
≤
(
2
)
}
.
I
t
is
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RE
F
E
R
E
NC
E
S
[1
]
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
.
[2
]
C.
K.
W
o
n
g
,
"
F
u
z
z
y
P
o
i
n
ts
a
n
d
L
o
c
a
l
P
ro
p
e
rti
e
s
o
f
F
u
z
z
y
T
o
p
o
lo
g
y
,
"
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
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
.
[3
]
M
.
D.
W
e
iss,
"
F
ix
e
d
P
o
in
ts,
S
e
p
a
ra
ti
o
n
,
a
n
d
I
n
d
u
c
e
d
T
o
p
o
lo
g
ies
f
o
r
F
u
z
z
y
S
e
ts,"
J
o
u
rn
a
l
o
f
M
a
th
e
m
a
ti
c
a
l
An
a
lys
is
a
n
d
Ap
p
li
c
a
ti
o
n
s
,
v
o
l.
5
0
,
p
p
.
1
4
2
-
1
5
0
,
1
9
7
5
.
[4
]
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
An
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
.
[5
]
J.
C.
Ke
ll
y
,
"
Bit
o
p
o
lo
g
ica
l
S
p
a
c
e
s,"
Pro
c
e
e
d
in
g
s
o
f
t
h
e
L
o
n
d
o
n
M
a
th
e
ma
t
ica
l
S
o
c
iety
,
v
o
l.
1
3
,
p
p
.
7
1
--
8
9
,
1
9
6
3
.
[6
]
A
.
Ka
n
d
il
,
"
Bip
ro
x
im
it
ies
a
n
d
F
u
z
z
y
T
it
o
p
o
l
o
g
ica
l
S
p
a
c
e
s,"
S
imo
n
S
tev
in
,
v
o
l.
6
3
,
p
p
.
4
5
-
6
6
,
1
9
8
9
.
[7
]
S
.
De
b
n
a
th
,
"
On
F
u
z
z
y
Bi
m
in
i
m
a
l
S
tru
c
tu
re
S
p
a
c
e
s,"
In
ter
n
a
ti
o
n
a
l
J
o
u
r
n
a
l
o
f
M
a
t
h
e
ma
ti
c
a
l
A
n
a
l
y
sis
,
v
o
l.
6
,
p
p
.
841
--
8
4
5
,
2
0
1
2
.
[8
]
B.
C.
T
rip
a
th
y
,
S
.
De
b
n
a
th
,
"
γ
-
O
p
e
n
S
e
ts
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d
γ
-
Co
n
ti
n
u
o
u
s
M
a
p
p
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g
s
in
F
u
z
z
y
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o
p
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lo
g
ica
l
S
p
a
c
e
s,"
J
o
u
rn
a
l
o
f
In
telli
g
e
n
t
&
Fu
zz
y
S
y
ste
ms
,
v
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l.
2
4
,
p
p
.
6
3
1
--
6
3
5
,
2
0
1
3
.
[9
]
B.
C.
T
rip
a
th
y
,
S
.
De
b
n
a
th
,
"
On
F
u
z
z
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-
L
o
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ll
y
Op
e
n
S
e
ts
in
Bit
o
p
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lo
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ica
l
S
p
a
c
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s,"
S
o
n
g
k
la
n
a
k
a
rin
J
o
u
rn
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l
o
f
S
c
ien
c
e
a
n
d
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c
h
n
o
l
o
g
y
,
v
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l.
3
7
,
p
p
.
9
3
-
9
6
,
2
0
1
5
.
[1
0
]
A
.
F
o
ra
,
S
.
A
l
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
stio
n
s
a
n
d
An
swe
rs
in
Ge
n
e
ra
l
T
o
p
o
lo
g
y
,
v
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l.
1
9
,
p
p
.
1
5
9
--
1
6
4
,
2
0
0
1
.
[1
1
]
F
.
Na
k
a
o
k
a
,
N.
Od
a
,
"
S
o
m
e
A
p
p
li
c
a
ti
o
n
s
o
f
M
in
im
a
l
F
u
z
z
y
Op
e
n
S
e
ts,"
In
ter
n
a
t
io
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
,
v
o
l.
2
7
,
p
p
.
4
7
1
--
4
7
6
,
2
0
0
1
.
[1
2
]
S
.
A
l
G
h
o
u
r,
A
.
F
o
ra
,
"
M
in
im
a
li
ty
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
th
e
ma
ti
c
s,
v
o
l.
1
2
,
p
p
.
725
--
7
3
7
,
2
0
0
4
.
[1
3
]
S
.
A
l
G
h
o
u
r,
"
S
o
m
e
G
e
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
t
i
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
4
]
S
.
A
l
G
h
o
u
r,
"
On
M
x
im
a
l
a
n
d
M
in
im
a
l
F
u
z
z
y
S
e
ts
in
I
-
T
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
r
n
a
l
o
f
M
a
th
e
ma
ti
c
s
a
n
d
M
a
th
e
ma
ti
c
a
l
S
c
ien
c
e
s
,
A
rt.
ID 1
8
0
1
9
6
,
1
1
p
p
,
2
0
1
0
.
[1
5
]
S
.
A
l
G
h
o
u
r,
A
.
A
z
a
ize
h
,
"
M
in
im
a
l
Op
e
n
S
e
ts
in
Bit
o
p
o
l
o
g
ica
l
S
p
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
9
,
p
p
.
1
0
5
-
1
1
2
,
2
0
1
1
.
[1
6
]
R.
L
o
we
n
,
"
A
Co
m
p
a
riso
n
o
f
Diffe
re
n
t
Co
m
p
a
c
tn
e
ss
No
ti
o
n
s
in
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
An
a
lys
is a
n
d
it
s A
p
p
li
c
a
ti
o
n
s
,
v
o
l.
6
4
,
p
p
.
4
4
6
--
4
5
4
,
1
9
7
8
.
[1
7
]
S
.
E.
Ro
d
a
b
a
u
g
h
,
"
T
h
e
Ha
u
s
d
o
rf
f
S
e
p
a
ra
ti
o
n
A
x
io
m
f
o
r
F
u
z
z
y
T
o
p
o
lo
g
ica
l
S
p
a
c
e
s,"
T
o
p
o
l
o
g
y
a
n
d
it
s
Ap
p
li
c
a
ti
o
n
s
.
,
v
o
l.
1
1
,
p
p
.
3
1
9
--
3
3
4
,
1
9
8
0
.
[1
8
]
A
.
J.
Kle
in
,
"
α
-
Clo
su
re
in
F
u
z
z
y
T
o
p
o
lo
g
y
,
"
Ro
c
k
y
M
o
u
n
tain
J.
M
a
th
.
,
v
o
l
.
1
1
,
p
p
.
5
5
3
--
5
6
0
,
1
9
8
1
.
[1
9
]
S
.
E
.
Ro
d
a
b
a
u
g
h
,
"
A
Ca
teg
o
rica
l
A
c
o
m
m
o
d
a
ti
o
n
o
f
V
a
rio
u
s
No
t
io
n
s
o
f
F
u
z
z
y
T
o
p
o
lo
g
y
,
"
Fu
zz
y
S
e
ts
a
n
d
S
y
ste
ms
,
v
o
l.
9
,
p
p
.
2
4
1
--
2
6
5
,
1
9
8
3
.
[2
0
]
P
.
W
u
y
ts,
"
On
th
e
De
term
in
a
ti
o
n
o
f
F
u
z
z
y
T
o
p
o
lo
g
ica
l
S
p
a
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e
s
a
n
d
F
u
z
z
y
Ne
ig
h
b
o
u
rh
o
o
d
S
p
a
c
e
s
b
y
T
h
e
ir
L
e
v
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l
-
T
o
p
o
lo
g
ies
,
"
Fu
zz
y
S
e
ts
a
n
d
S
y
st
e
ms
,
v
o
l.
1
2
,
p
p
.
7
1
--
8
5
,
1
9
8
4
.
[2
1
]
S
.
E.
R
o
d
a
b
a
u
g
h
,
R.
L
o
w
e
n
,
"
P
a
ra
-
L
o
w
e
n
,
a
n
d
α
-
L
e
v
e
l
F
u
n
c
to
rs
a
n
d
F
u
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y
T
o
p
o
lo
g
ies
o
n
t
h
e
Cri
sp
Re
a
l
L
in
e
,
"
J
o
u
rn
a
l
o
f
M
a
th
e
m
a
ti
c
a
l
An
a
lys
is a
n
d
A
p
p
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ti
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n
s
,
v
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l
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1
3
1
,
p
p
.
1
5
7
--
1
6
9
,
1
9
8
8
.
[2
2
]
Y.
C.
Ba
i,
Y.
L
.
Zh
e
n
g
,
"
Ty
p
e
II
S
tro
n
g
F
u
z
z
y
P
a
ra
c
o
m
p
a
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tn
e
s
s
a
n
d
th
e
λ
-
Cu
t
T
o
p
o
l
o
g
y
,
"
Pu
r
e
a
n
d
Ap
p
li
e
d
M
a
th
e
ma
ti
c
s
,
v
o
l.
1
1
,
p
p
.
1
3
9
--
1
4
2
,
1
9
9
5
.
[2
3
]
Y.
L
.
Zh
e
n
g
,
X
.
C.
Du
,
"
λ
-
Cu
t
To
p
o
lo
g
ies
o
f
L
-
F
u
z
z
y
T
o
p
o
lo
g
ies
,
"
J
.
Ba
o
ji
Co
l
l
e
g
e
Arts
S
c
i.
Na
t.
S
c
i.
,
v
o
l.
1
6
,
p
p
.
1
--
3
,
1
9
9
6
.
[2
4
]
P
.
W
u
y
ts,
"
On
L
e
v
e
l
-
T
o
p
o
lo
g
ies
a
n
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