I
nte
rna
t
io
na
l J
o
urna
l o
f
Adv
a
nces in Applie
d Science
s
(
I
J
AAS)
Vo
l.
15
,
No
.
1
,
Ma
r
ch
20
26
,
p
p
.
4
2
7
~
4
3
6
I
SS
N:
2252
-
8
8
1
4
,
DOI
:
1
0
.
1
1
5
9
1
/ijaas
.
v15.
i
1
.
pp
427
-
4
3
6
427
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Dep
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h
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ash
ik
–
4
2
2
2
1
3
,
Ma
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ar
ash
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a,
I
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in
i2
6
1
2
@
g
m
ail
.
co
m
1.
I
NT
RO
D
UCT
I
O
N
Acc
o
r
d
in
g
to
Z
ad
e
h
'
s
f
u
zz
y
s
et
th
eo
r
y
,
elem
en
ts
ca
n
h
av
e
d
e
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r
ee
s
o
f
m
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b
er
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ip
in
[
0
,
1
]
in
s
tead
o
f
o
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ly
"in
"
o
r
"o
u
t"
[
1
]
.
He
also
g
en
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alize
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o
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er
atio
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s
lik
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n
io
n
,
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ter
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,
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c
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p
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m
en
t
f
o
r
f
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zz
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ets,
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d
estab
lis
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ed
p
r
o
p
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ties
o
f
f
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zz
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r
elatio
n
s
an
d
co
n
v
ex
f
u
zz
y
s
ets.
Kr
am
o
s
il
an
d
Mic
h
álek
[
2
]
in
tr
o
d
u
ce
d
f
u
zz
y
m
etr
ic
s
p
ac
es
,
wh
ich
co
m
b
in
e
n
o
tio
n
s
o
f
d
is
tan
ce
f
u
z
zily
with
an
ex
tr
a
p
ar
am
eter
a
n
d
s
atis
f
y
an
alo
g
u
es
o
f
th
e
tr
ian
g
le
in
eq
u
ality
.
Pre
s
en
ted
f
u
n
d
am
e
n
tal
r
esu
lts
in
f
u
zz
y
m
etr
ic
s
p
ac
es
,
in
clu
d
in
g
co
n
v
e
r
g
en
ce
an
d
C
au
ch
y
s
eq
u
en
ce
c
h
ar
ac
ter
iza
tio
n
s
[
3
]
.
Ma
tth
ews
in
tr
o
d
u
ce
d
th
e
p
a
r
tial
m
etr
ic
s
p
ac
es,
w
h
ich
allo
w
a
p
o
in
t
t
o
h
av
e
a
n
o
n
-
ze
r
o
d
is
tan
ce
to
its
elf
[
4
]
.
T
h
is
r
elax
ed
f
o
r
m
o
f
m
etr
ic
is
u
s
ef
u
l
in
c
o
m
p
u
ter
s
cien
ce
an
d
h
as
in
ter
esti
n
g
to
p
o
l
o
g
ical
co
n
s
eq
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en
ce
s
.
I
n
o
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d
er
to
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al
with
u
n
ce
r
tain
,
p
ar
am
eter
iz
ed
d
ata,
Mo
lo
d
ts
o
v
[
5
]
s
u
g
g
e
s
ted
s
o
f
t
s
ets.
T
h
is
co
n
ce
p
t
allo
ws
attr
ib
u
te
-
b
ase
d
p
ar
am
etr
izatio
n
o
f
elem
e
n
t
m
em
b
er
s
h
ip
with
o
u
t
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o
m
e
o
f
th
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d
r
awb
ac
k
s
o
f
f
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zz
y
o
r
r
o
u
g
h
s
ets.
ex
am
in
es a
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d
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ip
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g
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s
o
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s
ets,
an
d
s
o
f
t
s
ets
[
6
]
.
Usef
u
l
f
o
r
s
ee
in
g
h
o
w
d
if
f
er
en
t
u
n
ce
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tain
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el
s
o
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lap
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d
if
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e
r
.
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n
v
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ates
f
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zz
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o
f
t
m
etr
i
c
s
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ac
es
co
m
b
in
in
g
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zz
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et
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em
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er
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ip
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n
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o
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t
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et
p
ar
a
m
e
ter
izatio
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in
a
m
etr
ic
-
lik
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s
tr
u
ctu
r
e;
s
tu
d
ies
b
asic
s
tr
u
ctu
r
e
lik
e
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n
v
er
g
e
n
ce
,
co
n
tin
u
ity
,
s
o
th
at
later
f
ix
ed
p
o
i
n
t o
r
to
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o
lo
g
ical
p
r
o
p
er
ties
ca
n
b
e
d
e
v
elo
p
e
d
[
7
]
.
I
n
tr
o
d
u
ce
s
m
u
lti
-
f
u
zz
y
s
o
f
t
s
e
ts
allo
win
g
m
u
ltip
le
m
em
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er
s
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ip
lev
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r
f
u
zz
in
ess
ty
p
es
u
n
d
er
s
o
f
t
s
et
p
ar
am
eter
s
[
8
]
,
an
d
ap
p
l
ies
th
ese
to
th
e
d
ec
is
io
n
m
ak
in
g
,
d
em
o
n
s
tr
atin
g
th
eir
u
tili
ty
in
m
o
d
ellin
g
u
n
ce
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tain
p
r
ef
er
en
ce
o
r
attr
ib
u
te
-
b
ased
ju
d
g
m
en
ts
.
Am
er
[
9
]
d
ef
in
es
f
u
zz
y
p
ar
tial
m
etr
ic
s
p
ac
es
allo
win
g
non
-
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r
o
s
elf
-
d
is
tan
ce
,
b
u
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with
f
u
zz
in
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.
E
x
p
lo
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e
s
tr
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r
e,
d
ef
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n
itio
n
s
,
an
d
o
f
ten
p
r
o
v
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g
f
ix
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o
in
t,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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I
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Ap
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1
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Def
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s
in
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m
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p
ac
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[
1
2
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C
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t
f
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th
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m
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u
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o
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t
G
-
m
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s
[
1
3
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.
A
d
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itio
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elo
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m
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in
f
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p
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tial m
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class
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o
f
m
ap
p
in
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im
p
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ality
in
f
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p
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t
th
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s
[
1
4
]
,
[
1
5
]
.
P
r
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p
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s
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n
eu
tr
o
s
o
p
h
ic
s
o
f
t
m
etr
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s
p
ac
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[
1
6
]
,
s
tu
d
ies
s
o
f
t
co
m
p
atib
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m
ap
p
in
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s
an
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e
m
s
in
s
o
f
t
S
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m
etr
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s
p
ac
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[
1
7
]
,
a
n
d
ex
p
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v
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g
en
ce
in
p
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ti
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ic
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estab
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in
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f
u
n
d
am
en
tal
f
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r
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e
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al
f
ix
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p
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co
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cl
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s
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n
s
[
1
8
]
.
C
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tr
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ted
to
f
i
x
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p
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in
t
ap
p
licatio
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g
r
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lts
f
o
r
s
o
f
t
B
-
m
etr
ic
s
p
ac
es
[
1
9
]
.
I
m
p
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d
co
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p
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tatio
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n
k
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y
ap
p
l
y
in
g
s
o
f
t
s
et
t
h
eo
r
y
to
d
ec
is
io
n
-
m
ak
in
g
p
r
o
b
lem
s
th
r
o
u
g
h
a
s
o
f
t
AND
-
o
p
er
atio
n
ap
p
r
o
ac
h
[
2
0
]
.
Stu
d
ied
f
ix
e
d
-
p
o
in
t
th
e
o
r
y
ap
p
licatio
n
s
o
f
th
e
m
etr
izatio
n
o
f
s
o
f
t
m
etr
ic
s
p
a
ce
s
[
2
1
]
.
Fix
ed
-
p
o
in
t
th
e
o
r
em
s
in
s
o
f
t
p
ar
am
etr
ic
m
etr
ic
s
p
ac
es
wer
e
p
r
o
v
ed
u
s
in
g
C
-
c
lass
f
u
n
ctio
n
s
[
2
2
]
.
E
x
p
an
d
i
n
g
o
n
f
u
zz
y
c
o
n
tr
ac
ti
o
n
co
n
ce
p
ts
,
f
ix
e
d
-
p
o
i
n
t
f
in
d
i
n
g
s
in
s
o
f
t
B
-
f
u
zz
y
m
etr
ic
s
p
ac
es
wer
e
p
r
esen
ted
[
2
3
]
.
Fix
ed
-
p
o
i
n
t
s
o
lu
tio
n
s
with
p
r
ac
tical
ap
p
licatio
n
s
in
m
o
d
if
ied
in
tu
itio
n
is
tic
f
u
zz
y
s
o
f
t
m
etr
ic
s
p
ac
es
we
r
e
p
r
esen
ted
[
2
4
]
,
p
r
o
v
i
n
g
e
x
is
ten
ce
an
d
u
n
iq
u
e
n
ess
r
esu
lts
b
y
ex
am
i
n
in
g
ϕ
-
co
n
tr
ac
tio
n
m
ap
p
in
g
s
u
n
d
er
s
o
f
t
f
u
zz
y
m
etr
ic
s
p
ac
es
[
2
5
]
.
E
s
tab
lis
h
ed
co
m
m
o
n
f
i
x
ed
-
p
o
in
t
th
eo
r
em
s
u
n
d
er
th
e
eq
u
iv
-
asy
m
p
to
tic
(
E.
A.
)
p
r
o
p
er
ty
co
n
d
itio
n
in
f
u
zz
y
p
ar
t
ial
m
etr
ic
s
p
ac
es
[
2
6
]
.
T
h
e
co
n
ce
p
t
o
f
s
o
f
t
f
u
zz
y
p
ar
t
ial
m
etr
ic
s
p
ac
es
is
p
r
esen
ted
in
th
is
s
tu
d
y
alo
n
g
with
an
ex
am
in
atio
n
o
f
th
eir
f
u
n
d
am
en
tal
ch
a
r
ac
ter
is
tics
.
W
e
ex
p
an
d
tr
ad
itio
n
al
co
n
clu
s
io
n
s
to
th
is
g
e
n
er
alize
d
ca
s
e
by
estab
lis
h
in
g
n
ew
f
ix
ed
-
p
o
in
t
th
eo
r
em
s
u
n
d
er
d
if
f
er
en
t
c
o
n
tr
ac
tiv
e
co
n
d
itio
n
s
.
Ou
r
c
o
n
tr
ib
u
tio
n
s
aim
to
d
ee
p
en
th
e
t
h
eo
r
etica
l
u
n
d
e
r
s
tan
d
in
g
o
f
f
ix
e
d
-
p
o
i
n
t
p
h
en
o
m
en
a
in
s
o
f
t
f
u
zz
y
en
v
ir
o
n
m
en
ts
an
d
to
p
r
o
v
id
e
p
r
ac
tical
to
o
ls
f
o
r
a
p
p
licatio
n
s
wh
er
e
u
n
ce
r
ta
in
ty
,
f
u
zz
in
ess
,
an
d
p
ar
a
m
eter
d
e
p
en
d
e
n
ce
co
ex
is
t.
2.
P
RO
P
O
SE
D
M
E
T
H
O
D
I
n
o
r
d
e
r
to
d
ev
elo
p
a
n
o
v
el
co
n
ce
p
t
o
f
s
o
f
t
f
u
zz
y
p
ar
tial
m
et
r
ic
s
p
ac
e
an
d
r
elate
d
f
ix
ed
-
p
o
i
n
t
th
eo
r
y
,
we
d
escr
ib
e
ce
r
tain
f
u
n
d
am
e
n
t
al
d
ef
in
itio
n
s
an
d
p
r
o
p
er
ties
o
f
m
etr
ic
s
p
ac
e
s
an
d
s
o
f
t sets
in
th
is
p
ar
t.
2
.
1
.
Def
ini
t
io
n
2.
1
A
p
ar
tial m
etr
ic
s
p
ac
e
o
n
‘
’
is
a
p
air
(
‘
’
,
)
s
u
ch
th
at
‘
’
is
a
n
o
n
-
em
p
ty
s
et
an
d
:
‘
’
×
‘
’
→
ℝ
+
is
a
m
ap
p
in
g
p
r
o
v
id
in
g
t
h
e
lis
ted
co
n
d
itio
n
s
∀
,
,
∈
‘
’
s
u
ch
th
at
:
i)
‘
‘
(
,
)
≤
(
,
)
’
ii)
‘
(
,
)
=
(
,
)
=
(
,
)
’
if
‘
=
’
iii)
‘
(
,
)
=
(
,
)
’
iv
)
‘
(
,
)
≤
(
,
)
+
(
,
)
–
(
,
)
′
N
o
t
e
t
h
a
t
a
p
o
i
n
t'
s
s
el
f
-
d
i
s
t
a
n
c
e
d
o
e
s
n
o
t
a
l
w
a
y
s
e
q
u
a
l
0
i
n
p
a
r
t
i
a
l
m
e
t
r
i
c
s
p
a
c
e
.
T
h
e
p
a
r
ti
a
l
m
e
t
r
i
c
‘
’
i
s
a
n
o
r
d
i
n
a
r
y
m
e
t
r
i
c
o
n
‘
′
i
f
‘
(
,
)
=
0′
,
∀
∈
‘
’
.
S
o
,
a
p
a
r
t
i
a
l
m
et
r
i
c
is
a
n
e
x
t
e
n
s
i
o
n
o
f
a
n
o
r
d
i
n
a
r
y
m
e
t
r
i
c
[
4
].
2
.
2
.
Def
ini
t
io
n
2
.2
I
f
th
e
f
o
llo
win
g
cr
iter
ia
a
r
e
s
a
tis
f
ied
,
a
b
in
ar
y
o
p
er
atio
n
"
ʘ
’
"
o
n
[
0
,
1
]
is
r
ef
er
r
e
d
to
as
a
c
o
n
tin
u
o
u
s
t
-
n
o
r
m
:
∀
,
,
,
∈
[
0
,
1
]
:
i)
‘
ʘ
=
ʘ
’
an
d
‘
ʘ
(
ʘ
)
=
(
ʘ
)
ʘ
’
ii)
‘
ʘ
is
co
n
tin
u
o
u
s
o
n
[
0
,
1
]
×[
0
,
1
]
’
iii)
‘
ʘ
1
=
’
iv
)
I
f
‘
≤
’
an
d
‘
≤
’
,
th
en
‘
ʘ
≤
ʘ
’
[
2
]
2
.
3
.
Def
ini
t
io
n
2.
3
C
o
n
s
id
er
in
g
‘
’
b
e
a
n
o
n
-
e
m
p
ty
s
et,
‘
ʘ
’
b
e
a
co
n
tin
u
o
u
s
t
-
n
o
r
m
,
an
d
:
‘
′
×
‘
′
×
(
0
,
∞
)
→
[
0
,
1
]
be
a
m
ap
p
in
g
.
C
o
n
s
id
er
‘
’
b
e
a
f
u
zz
y
s
e
t.
I
f
th
e
s
p
ec
if
ie
d
co
n
d
itio
n
s
ar
e
s
atis
f
ied
∀
,
,
∈
‘
’
an
d
,
>
0
,
th
en
th
e
tr
i
p
let
(
‘
’
,
,
ʘ
)
is
s
aid
to
b
e
a
f
u
zz
y
m
etr
ic
s
p
ac
e,
if
it
s
atis
f
ies
th
e
s
u
b
s
eq
u
e
n
t
p
r
o
p
er
ties
f
o
r
:
i)
‘
(
,
)
>
0
’
,
ii)
‘
(
,
)
=
1
’
,
if
‘
=
iii)
‘
(
,
)
=
(
,
,
)
’
,
iv
)
‘
(
,
+
)
≥
(
,
)
ʘ
(
,
)
’
v)
‘
(
,
ʘ
)
,
is
co
n
tin
u
o
u
s
o
n
(
0
,
∞
)
.
’
If
(
‘
’
,
,
ʘ
)
is
a
“
f
u
zz
y
m
et
r
ic
s
p
ac
e,
th
e
n
‘
’
is
a
f
u
zz
y
m
etr
ic
o
n
‘
’
[
2
]
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Ad
v
Ap
p
l Sci
I
SS
N:
2252
-
8
8
1
4
S
o
ft fu
z
z
y
p
a
r
tia
l m
etri
c
a
n
d
s
o
me
r
esu
lts
o
n
fixed
p
o
in
t th
e
o
r
y
u
n
d
er
…
(
R
o
h
in
i R
.
G
o
r
e)
429
2
.
4
.
De
f
in
it
io
n
2.
4
C
o
n
s
id
er
in
g
‘
’
b
e
a
n
o
n
-
em
p
ty
s
et,
‘
ʘ’
b
e
a
co
n
tin
u
o
u
s
t
-
n
o
r
m
an
d
∶
‘
′
×
‘
′
×
[
0
,
∞
)
→
[
0
,
1
]
b
e
a
m
ap
p
in
g
.
C
o
n
s
id
er
in
g
b
e
a
p
ar
tial
m
etr
ic
s
p
ac
e.
I
f
th
e
s
p
ec
if
ied
co
n
d
itio
n
s
ar
e
s
atis
f
ied
∀
,
,
∈
‘
’
an
d
,
≥
0
,
th
en
th
e
tr
i
p
let
(
‘
’
,
,
ʘ
)
,
is
s
aid
to
b
e
a
f
u
zz
y
p
a
r
tial m
etr
ic
s
p
ac
e:
i)
‘
(
,
0
)
=
0’
ii)
‘
(
,
)
=
(
,
,
)
’
iii)
‘
(
,
,
+
)
≥
(
,
)
ʘ
(
,
)
’
iv
)
‘
(
,
,
)
≤
1
,
>
0
’
&‘
(
,
,
)
=
1’
if
‘
(
,
=
0’
v)
′
(
,
,
ʘ
)
,
:
(
0,
∞
)
→[
0
,
1
]
is
co
n
tin
u
o
u
s
W
h
e
r
e
‘
(
,
,
)
=
+
(
,
)
,
i
f
(
‘
’
,
,
ʘ
)
,
r
ep
r
esen
ts
a
f
u
z
zy
p
ar
tial
m
etr
ic
s
p
ac
e,
th
e
n
‘
′
d
en
o
tes
f
u
zz
y
p
ar
tial m
etr
ic
o
n
‘
’
[
9
]
.
No
te:
i
n
th
is
r
esear
ch
p
ap
er
,
r
ef
er
s
u
n
iv
er
s
al
”
s
et,
℘
,
wh
ich
is
th
e
s
et
o
f
p
ar
am
eter
s
,
(
)
is
th
e
p
o
wer
s
et
o
f
.
W
e
d
ef
in
e
℘
as
th
e
ab
s
o
lu
te
s
o
f
t
s
et
o
v
er
with
p
ar
am
eter
s
et
℘
.
(
ℜ
)
is
a
ll
non
-
v
o
id
b
o
u
n
d
ed
s
u
b
s
et
o
f
ℜ
th
at
is
a
co
llectio
n
o
f
all
r
ea
l n
u
m
b
er
s
.
2
.
5
.
De
f
in
it
io
n
2.
5
A
s
o
f
t
s
et
(
,
℘
)
o
v
er
a
u
n
iv
e
r
s
al
s
et
is
a
p
ai
r
,
wh
e
r
e
℘
is
a
s
et
o
f
p
ar
a
m
eter
s
.
is
a
m
ap
p
i
n
g
g
iv
en
b
y
:
℘
→
(
)
,
w
h
er
e
(
)
r
ep
r
esen
ts
p
o
wer
s
e
t
o
f
.
Pu
t
d
if
f
er
en
tly
,
f
o
r
ev
e
ry
p
ar
a
m
eter
∈
℘
,
(
e)
is
a
s
u
b
s
et
o
f
th
e
u
n
iv
er
s
al
s
et
[
5
]
.
2
.
5
.
1
.
E
x
a
m
ple
2
.
5
.
1
T
h
e
ex
am
p
le
illu
s
tr
ates
th
e
d
ef
in
itio
n
o
f
a
s
o
f
t
s
et
(
,
℘
)
o
v
er
a
u
n
i
v
er
s
al
s
et
.
T
h
e
th
eo
r
etica
l
f
r
am
ewo
r
k
in
v
o
lv
es
a
f
u
n
ctio
n
th
at
m
ap
s
ea
ch
p
ar
am
eter
in
th
e
s
et
o
f
p
ar
am
eter
s
℘
,
wh
ich
is
a
s
u
b
s
et
o
f
th
e
u
n
iv
er
s
al
s
et
.
I
t m
ea
n
s
th
at
p
o
wer
s
et
o
f
u
n
iv
er
s
al
s
et.
C
o
n
s
id
er
u
n
iv
er
s
al
s
et
:
=
{
,
,
,
,
}
(
s
et
o
f
h
o
u
s
es)
Par
am
eter
s
:
℘
=
{e
x
p
en
s
iv
e,
b
ea
u
tifu
l,
m
o
d
e
r
n
}
(
s
et
o
f
p
ar
am
et
er
s
)
Def
in
e
(
,
℘
)
is
a
s
o
f
t set o
v
er
,
wh
er
e:
(
)
=
{
,
}
,
(
)
=
{
,
}
,
(
)
=
{
,
}
T
h
e
s
o
f
t set (
,
℘
)
ca
n
b
e
i
n
ter
p
r
e
ted
as:
‒
{
,
}
ar
e
ex
p
e
n
s
iv
e
h
o
u
s
es
‒
{
,
}
ar
e
b
ea
u
tif
u
l h
o
u
s
es
‒
{
,
}
ar
e
m
o
d
e
r
n
h
o
u
s
es
.
T
h
is
ex
am
p
le
s
h
o
ws h
o
w
c
o
m
p
lex
s
y
s
tem
s
with
m
an
y
p
a
r
a
m
eter
s
ca
n
b
e
m
o
d
elled
u
s
in
g
s
o
f
t sets
.
2
.
6
.
De
f
in
it
io
n
2.
6
I
n
s
o
f
t
s
et
th
eo
r
y
,
an
ab
s
o
lu
te
s
o
f
t
s
et
i
s
th
e
“m
ax
im
u
m
’
p
o
s
s
ib
le
s
o
f
t
s
e
t
o
v
er
a
u
n
iv
er
s
al
,
g
iv
en
a
s
et
o
f
p
ar
am
eter
s
℘
.
I
t
is
ess
e
n
tial
f
o
r
d
e
f
in
in
g
th
e
b
asic
lo
g
ical
o
p
er
atio
n
s
th
at
allo
w
s
o
f
t
s
et
th
eo
r
y
to
f
u
n
ctio
n
as
a
m
ath
em
atica
l
to
o
l.
A
s
o
f
t
s
e
t (
,
℘
)
o
v
er
a
u
n
iv
er
s
al
s
et
is
r
ef
er
r
e
d
to
as
an
ab
s
o
lu
te
s
o
f
t
s
et
if
(
)
=
,
∀
∈
℘
[
5
]
.
2
.
6
.
1
.
E
x
a
m
ple
2
.
6
.
1
T
h
e
ex
am
p
le
illu
s
tr
ates th
e
th
e
o
r
etica
l c
o
n
ce
p
t o
f
an
a
b
s
o
lu
te
s
o
f
t set,
wh
ich
is
a
s
p
ec
if
ic
ty
p
e
o
f
s
o
f
t
s
et
with
p
ar
ticu
lar
p
r
o
p
er
ties
.
I
t
s
h
o
ws
th
at
ea
ch
p
ar
am
eter
m
ap
s
to
a
s
in
g
leto
n
s
et
co
n
tain
in
g
o
n
ly
its
elf
,
an
d
th
e
u
n
io
n
o
f
all
r
esu
ltin
g
s
u
b
s
ets co
v
er
s
th
e
en
tire
u
n
iv
er
s
al
s
et.
C
o
n
s
id
er
u
n
iv
er
s
al
s
et
:
=
{
,
,
,
}
Def
in
e
(
,
℘
)
is
an
ab
s
o
lu
te
s
o
f
t set o
v
er
,
wh
er
e:
℘
=
=
{
,
,
,
}
(
)
=
{
}
,
(
)
=
{
}
,
(
)
=
{
}
,
(
)
=
{
}
T
h
e
ab
s
o
lu
te
s
o
f
t set (
,
℘
)
s
atis
f
ie
s
:
i)
(
)
=
{
}
∈
ii)
∪
(
)
=
L
et's ca
lcu
late
th
e
u
n
io
n
o
f
(
)
an
d
(
)
:
(
)
∪
(
)
=
{
}
∪
{
}
=
{
,
}
T
h
is
ex
am
p
le
illu
s
tr
ates c
o
n
ce
p
t o
f
ab
s
o
lu
te
s
o
f
t sets
as we
ll
as
th
eir
p
r
o
p
e
r
ties
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
8
1
4
I
n
t J Ad
v
Ap
p
l Sci
,
Vo
l.
1
5
,
No
.
1
,
Ma
r
c
h
2
0
2
6
:
427
-
4
3
6
430
2
.
7
.
De
f
in
it
io
n
2.
7
A
n
u
l
l
s
o
f
t
s
et
(
)
is
th
e
f
o
u
n
d
at
i
o
n
al
o
r
em
p
t
y
o
b
je
ct
in
s
o
f
t
s
et
t
h
e
o
r
y
.
I
t
r
ep
r
ese
n
ts
a
s
c
e
n
ar
i
o
wh
e
r
e
n
o
n
e
o
f
t
h
e
ch
o
s
e
n
p
a
r
a
m
et
er
s
ca
n
b
e
ap
p
l
ie
d
t
o
a
n
y
o
f
t
h
e
o
b
je
cts
i
n
t
h
e
u
n
i
v
er
s
e.
Her
e
,
c
o
n
s
id
er
in
g
a
f
u
n
cti
o
n
,
s
et
o
f
p
a
r
am
ete
r
s
℘
w
h
ic
h
is
s
u
b
s
et
o
f
u
n
i
v
e
r
s
al
s
et
,
is
el
em
e
n
t
s
et
o
f
p
a
r
a
m
e
te
r
s
f
o
r
d
e
f
i
n
i
n
g
a
n
u
ll
s
o
f
t
s
e
t.
A
“
s
o
f
t
s
et
(
,
℘
)
o
v
e
r
a
u
n
i
v
e
r
s
al
s
e
t
is
ca
lle
d
a
n
u
l
l
s
o
f
t s
et
(
)
=
{
}
,
∀
∈
℘
[
1
0
]
.
2
.
8
.
De
f
in
it
io
n
2.
8
A
p
a
ir
(
,
℘
)
is
a
s
o
f
t
r
ea
l
s
et
if
:
℘
→
(
ℜ
)
,
wh
er
e
(
ℜ
)
is
all
n
o
n
-
v
o
id
b
o
u
n
d
ed
s
u
b
s
et
s
o
f
ℜ
(
co
llectio
n
o
f
all
r
ea
l
n
u
m
b
e
r
s
)
.
A
s
o
f
t
r
ea
l
s
et
(
F,
℘
)
is
a
s
o
f
t
r
ea
l
n
u
m
b
er
,
if
∀
∈
℘
,
(
)
is
a
s
in
g
leto
n
m
em
b
er
o
f
(
ℜ
)
.
Fo
r
a
s
o
f
t r
ea
l
n
u
m
b
er
,
if
(
)
=
{
}
,
>
0
[
1
0
]
.
2
.
9
.
De
f
in
it
io
n
2.
9
Fo
r
two
s
o
f
t r
ea
l n
u
m
b
er
s
,
,
s
u
b
s
eq
u
en
t
o
p
e
r
atio
n
s
ar
e
as
[
1
0
]
:
i)
(
⊕
)
(
)
=
{
(
)
+
(
)
/
∈
℘
}
ii)
(
⊖
)
(
)
=
{
(
)
−
(
)
/
∈
℘
}
iii)
(
⊗
)
(
)
=
{
(
)
.
(
)
/
∈
℘
}
2
.
1
0
.
De
f
ini
t
i
o
n
2.
10
T
h
e
co
llectio
n
o
f
o
r
d
er
ed
p
air
s
,
=
{
(
0
,
(
0
)
)
/
(
0
∈
℘
,
∈
℘
}
is
a
s
o
f
t
f
u
zz
y
s
et
in
℘
as
is
ca
lled
a
s
o
f
t
m
em
b
er
s
h
ip
f
u
n
ctio
n
d
e
f
in
e
as
:
℘
→
[
0
,
1
]
℘
.
T
h
u
s
,
(
0
)
r
ep
r
esen
ts
th
e
ass
o
ciate
d
s
o
f
t
m
em
b
er
s
h
ip
g
r
a
d
e
o
f
s
o
f
t
p
o
in
t
0
in
[
1
0
]
.
3.
RE
S
E
ARCH
M
E
T
H
O
D
3
.
1
.
De
f
in
it
io
n
3.
1
C
o
n
s
id
er
in
g
‘
’
b
e
a
n
o
n
-
em
p
ty
s
et,
‘
ʘ’
b
e
a
c
o
n
tin
u
o
u
s
t
-
n
o
r
m
an
d
:
‘
′
℘
×
‘
′
℘
×
[
0
,
∞
)
℘
→
[
0
,
1
]
℘
b
e
a
m
ap
p
in
g
.
C
o
n
s
id
er
in
g
b
e
a
f
u
zz
y
p
ar
tial
m
etr
ic
s
p
ac
e
&
‘
′
℘
is
a
s
o
f
t
m
etr
ic
o
v
er
℘
.
I
f
s
p
ec
if
ied
co
n
d
itio
n
s
ar
e
s
atis
f
ied
∀
,
,
∈
℘
an
d
,
≥
0
,
th
en
tr
ip
let
(
℘
,
,
ʘ
)
is
s
a
id
to
be
a
s
o
f
t
f
u
zz
y
p
ar
tial m
etr
i
c
s
p
ac
e:
i)
‘
(
,
,
0
)
=
0’
ii)
‘
(
,
,
)
=
(
,
,
)
’
iii)
‘
(
,
,
+
)
≥
(
,
,
)
ʘ
(
,
,
)
’
iv
)
‘
‘
(
,
,
)
≤
1
,
>
0
’
&‘
(
,
,
)
=
1
’
if
f
‘
(
,
)
=
0’
v)
‘
(
,
,
ʘ
)
:
(
0
,
∞
)
℘
→
[
0
,
1
]
℘
is
co
n
tin
u
o
u
s
,
H
e
r
e
‘
(
,
,
)
=
+
(
,
)
’.
If
(
‘
’℘
,
,
ʘ
)
,
is
a
s
o
f
t
f
u
zz
y
p
ar
tial
m
etr
ic
s
p
ac
e
,
th
e
n
‘
′
is
a
f
u
zz
y
p
a
r
tial m
etr
ic
o
n
℘
.
3
.
1
.
1
.
E
x
a
m
ple
3
.
1
.
1
C
o
n
s
id
er
(
℘
,
)
is
a
s
o
f
t
f
u
zz
y
p
ar
tial
m
etr
ic
s
p
ac
e
&
ʘ
=
min
{
,
}
an
d
ʘ
=
⊗
ar
e
d
ef
in
ed
in
(
℘
,
)
.
Def
in
e
m
a
p
p
in
g
:
‘
′
℘
×
‘
′
℘
×
[
0
,
∞
)
℘
→
[
0
,
1
]
℘
as
:
(
,
,
)
=
⊕
(
,
)
,
,
,
∈
℘
an
d
≥
0
3
.
1
.
2
.
E
x
a
m
ple
3
.
1
.
2
Un
iv
er
s
al
s
et
:
=
{
,
,
}
=
s
et
o
f
o
b
jects
.
Def
in
e
(
,
)
is
a
f
u
zz
y
p
ar
tial m
et
r
ic
o
n
,
h
er
e:
(
,
)
=
0
.
8
,
(
,
)
=
0
.
6
,
(
,
)
=
0
.
7
Def
in
e
(
,
℘
)
is
a
s
o
f
t set o
v
er
”
,
h
e
r
e:
℘
={
p
ar
am
eter
1
,
p
ar
a
m
eter
2
}
(
1
)
=
{
(
0
.
9
)
,
(
0
.
8
)
}
(
2
)
=
{
(
0
.
7
)
,
(
0
.
9
)
}
W
e
ca
n
co
m
b
in
e
th
e
s
o
f
t set (
,
℘
)
an
d
f
u
zz
y
p
a
r
tial m
etr
ic
to
d
e
f
in
e
a
s
o
f
t f
u
zz
y
p
a
r
tial m
etr
ic
s
p
ac
e.
L
et's ca
lcu
late
th
e
s
im
ilar
ity
b
etwe
en
o
b
jects
an
d
u
n
d
e
r
p
ar
am
eter
1
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Ad
v
Ap
p
l Sci
I
SS
N:
2252
-
8
8
1
4
S
o
ft fu
z
z
y
p
a
r
tia
l m
etri
c
a
n
d
s
o
me
r
esu
lts
o
n
fixed
p
o
in
t th
e
o
r
y
u
n
d
er
…
(
R
o
h
in
i R
.
G
o
r
e)
431
(
,
)
=
0
.
8
(
f
u
zz
y
p
ar
tial m
etr
ic)
(
1
)
(
)
=
0
.
9
(
s
o
f
t set)
(
1
)
(
)
=
0
.
8
(
s
o
f
t set)
T
h
e
s
im
ilar
ity
b
etwe
en
an
d
u
n
d
er
p
a
r
am
eter
1
ca
n
b
e
c
alcu
l
ated
as
(
,
)
=
(
,
)
×
{
(
1
)
(
)
,
(
1
)
(
)
}
=
0
.
8
×
{
0
.
9
,
0
.
8
}
=
0
.
8
×
0
.
8
=
0
.
64
T
h
is
ex
am
p
le
d
em
o
n
s
tr
ates
h
o
w
s
o
f
t
s
ets
ca
n
b
e
u
s
ed
to
e
x
ten
d
f
u
zz
y
p
a
r
tial
m
etr
ic
s
p
ac
es
an
d
p
r
o
v
id
e
a
m
o
r
e
f
lex
i
b
le
an
d
r
o
b
u
s
t f
r
am
ewo
r
k
f
o
r
m
o
d
ellin
g
co
m
p
lex
s
y
s
tem
s
.
3
.
2
.
De
f
in
it
io
n
3.
2
E
v
er
y
s
o
f
t
s
e
q
u
e
n
c
e
{
}
in
s
o
f
t
f
u
zz
y
p
ar
tial
m
etr
ic
s
p
ac
e
(
℘
,
,
ʘ
)
,
is
co
n
v
er
g
e
n
t
to
a
s
o
f
t
p
o
in
t
∈
℘
,
if
l
im
℘
→
∞
(
,
,
)
=
1
,
∀
>
0
.
i.e
.
(
,
)
=
0
Similar
ly
,
f
o
r
an
y
>
0
,
>
0
∃
0
∈
+
s
u
ch
th
at
(
,
,
)
>
1
ʘ
,
∀
≥
0
3
.
3
.
De
f
in
it
io
n
3.
3
E
v
er
y
s
o
f
t
s
e
q
u
e
n
ce
{
}
in
s
o
f
t
f
u
z
zy
p
ar
tia
l
m
et
r
ic
s
p
ac
e
(
℘
,
,
ʘ
)
,
is
C
au
ch
y
s
eq
u
en
ce
in
s
o
f
t
f
u
z
zy
p
a
r
ti
al
m
e
tr
ic
s
p
ac
e
,
i
f
l
im
℘
→
∞
(
,
,
)
=
1
,
∀
>
0
.
i.e
.
(
,
)
=
0
.
Similar
ly
,
f
o
r
an
y
>
0
,
>
0
∃
0
∈
+
s
u
ch
th
at
(
,
,
)
>
1
ʘ
,
∀
,
≥
0
R
em
ar
k
,
b
y
d
ef
in
itio
n
3
.
2
,
3
.
3
co
n
clu
d
e
th
at
:
i)
I
f
ev
er
y
C
au
ch
y
s
eq
u
e
n
ce
in
s
o
f
t
f
u
zz
y
p
a
r
ti
al
m
et
r
i
c
s
p
a
ce
is
c
o
n
v
er
g
e
n
t
,
t
h
en
s
o
f
t
f
u
z
zy
p
a
r
t
ial
m
e
tr
ic
s
p
a
ce
(
℘
,
,
ʘ
)
,
is
co
m
p
lete.
ii)
I
f
ev
er
y
s
o
f
t
f
u
zz
y
s
eq
u
en
ce
in
s
o
f
t
f
u
z
zy
p
a
r
ti
al
m
et
r
i
c
s
p
a
ce
a
d
m
it
s
a
t
leas
t
o
n
e
co
n
v
e
r
g
e
n
t
s
o
f
t
s
u
b
s
eq
u
en
ce
,
t
h
e
n
s
o
f
t
f
u
zz
y
p
ar
t
ial
m
et
r
i
c
s
p
a
ce
(
℘
,
,
ʘ
)
is
co
m
p
ac
t.
3
.
4
.
De
f
in
it
io
n
3
.
4
C
o
n
s
id
er
(
℘
,
,
ʘ
)
is
a
s
o
f
t
f
u
zz
y
p
ar
tial
m
etr
ic
s
p
ac
e
.
So
f
t
m
ap
p
in
g
(
,
Ω
)
:
(
℘
,
,
ʘ
)
→
(
℘
,
,
ʘ
)
is
“
a
s
o
f
t
f
u
zz
y
co
n
tr
ac
tio
n
m
ap
p
in
g
o
n
s
o
f
t
f
u
zz
y
p
ar
tial
m
etr
ic
s
p
ac
e,
i
f
∃
a
s
o
f
t
r
ea
l
n
u
m
b
er
∈
[
0
,
1
]
s
atis
f
y
in
g
th
e
co
n
d
itio
n
.
(
(
,
Ω
)
,
(
,
Ω
)
,
)
≥
(
,
,
)
,
∀
,
∈
℘
&
>
0
3
.
5
.
De
f
in
it
io
n
3.
5
T
h
e
m
a
p
Ψ
:
ℛ
(
℘
)
→
[
0
,
∞
)
℘
is
a
–
f
u
n
ctio
n
w
h
ich
is
s
u
b
s
eq
u
en
t
co
n
d
itio
n
s
:
i)
(
)
=
0
⟺
=
0
.
ii)
is
a
n
o
n
-
d
e
cr
ea
s
in
g
f
u
n
ctio
n
.
iii)
is
lef
t c
o
n
tin
u
o
u
s
f
o
r
>
0
.
iv
)
is
co
n
tin
u
o
u
s
at
u
=
0
.
v)
(
)
→
∞
→
∞
.
3
.
6
.
De
f
in
it
io
n
3.
6
C
o
n
s
id
er
(
℘
,
,
ʘ
)
is
a
s
o
f
t
f
u
zz
y
p
ar
tial
m
etr
ic
s
p
ac
e
.
So
f
t
m
ap
p
in
g
(
,
Ω
)
:
(
℘
,
,
ʘ
)
→
(
℘
,
,
ʘ
)
is
s
aid
to
b
e
a
−
co
n
tr
ac
tio
n
m
ap
p
in
g
o
n
a
s
o
f
t
f
u
zz
y
p
ar
tial
m
etr
ic
s
p
ac
e,
if
∃
a
s
o
f
t
r
ea
l
n
u
m
b
er
∈
[
0
,
1
]
s
atis
f
y
in
g
th
e
co
n
d
itio
n
.
(
(
,
Ω
)
,
(
,
Ω
)
,
Ψ
(
)
)
≥
(
,
,
Ψ
(
)
)
,
∀
,
∈
℘
&
>
0
,
h
er
e
is
a
–
f
u
n
ctio
n.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
8
1
4
I
n
t J Ad
v
Ap
p
l Sci
,
Vo
l.
1
5
,
No
.
1
,
Ma
r
c
h
2
0
2
6
:
427
-
4
3
6
432
4.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
4
.
1
.
T
heo
re
m
4.
1
C
o
n
s
id
er
in
g
(
℘
,
,
ʘ
)
is
a
co
m
p
lete
s
o
f
t
f
u
zz
y
p
ar
tial
m
etr
ic
s
p
ac
e
s
u
ch
th
at
l
im
℘
→
∞
(
,
,
)
=
1
,
∀
,
∈
℘
.
T
h
en
s
o
f
t
f
u
zz
y
p
ar
tial
c
o
n
tr
ac
tio
n
m
ap
p
i
n
g
(
,
Ω
)
℘
ad
m
itted
a
co
m
m
o
n
”
s
o
f
t f
ix
e
d
p
o
in
t.
Pro
o
f
:
c
o
n
s
id
er
in
g
a
s
o
f
t
p
o
in
t
0
∈
℘
an
d
c
o
n
s
tr
u
ct
a
s
o
f
t
s
eq
u
e
n
ce
{
}
s
u
ch
th
at
=
(
,
Ω
)
0
.
B
y
u
s
in
g
in
d
u
ctio
n
,
we
g
et
:
(
,
(
+
1
)
,
)
≥
(
0
,
1
,
(
∝
)
)
B
y
ab
o
v
e
co
n
d
itio
n
a
n
d
p
r
o
p
e
r
ty
3
o
f
d
ef
i
n
itio
n
o
f
s
o
f
t f
u
zz
y
p
ar
tial m
etr
ic
s
p
ac
e
,
f
o
r
a
n
y
∈
ℤ
+
,
we
g
et
:
(
,
(
+
)
,
)
≥
(
,
(
+
1
)
,
)
ʘ
…
…
ʘ
⏟
−
(
(
+
−
1
)
,
(
+
)
,
)
≥
(
0
,
1
,
(
∝
)
)
ʘ
…
…
ʘ
⏟
−
(
0
,
1
,
(
∝
+
−
1
)
)
B
u
t g
iv
en
th
at
l
im
℘
→
∞
(
,
,
)
=
1
,
∀
,
∈
℘
.
(
,
(
+
)
,
)
≥
1
ʘ
1
ʘ
…
…
ʘ
1
⏟
−
=
1
Hen
ce
,
th
e
s
o
f
t
f
u
zz
y
p
ar
tial
s
eq
u
en
ce
{
}
is
a
C
au
ch
y
in
(
℘
,
,
ʘ
)
.
T
h
u
s
,
it
is
co
n
v
er
g
en
t
.
T
h
er
ef
o
r
e,
(
℘
,
,
ʘ
)
is
co
m
p
lete.
W
e
o
b
tain
,
{
}
→
,
∀
,
∈
℘
l
im
℘
→
∞
(
,
,
)
=
1
,
∀
,
∈
℘
.
T
h
en
,
(
(
,
Ω
)
,
,
)
≥
(
(
,
Ω
)
,
(
,
Ω
)
,
2
)
ʘ
(
(
,
Ω
)
,
,
2
)
≥
(
,
,
2
∝
)
ʘ
(
(
+
1
)
,
,
2
)
≥
1
ʘ
1=1
(
(
,
Ω
)
,
,
)
=
1
Hen
ce
(
,
Ω
)
=
.
T
h
u
s
,
is
a
s
o
f
t
f
ix
ed
p
o
in
t
o
f
(
,
Ω
)
.
I
t
is
s
im
p
le
to
co
n
f
ir
m
th
at
a
s
o
f
t
f
ix
ed
p
o
i
n
t
o
f
th
e
s
o
f
t f
u
zz
y
p
ar
tial c
o
n
tr
a
ctio
n
m
ap
p
i
n
g
(
,
Ω
)
is
u
n
iq
u
e
an
d
c
o
m
p
lete.
4
.
2
.
T
heo
re
m
4.
2
C
o
n
s
id
er
(
℘
,
,
ʘ
)
is
a
co
m
p
lete
s
o
f
t
f
u
zz
y
p
ar
tial
m
etr
ic
s
p
ac
e
s
u
ch
th
at
l
im
℘
→
∞
(
,
,
)
=
1
,
∀
,
∈
℘
.
T
h
en
Ψ
−
co
n
tr
ac
tio
n
m
ap
p
in
g
(
,
Ω
)
:
(
℘
,
,
ʘ
)
→
(
℘
,
,
ʘ
)
℘
a
d
m
itted
a
co
m
m
o
n
s
o
f
t f
ix
ed
p
o
in
t.
Pro
o
f
:
c
o
n
s
id
er
a
s
o
f
t p
o
in
t
0
∈
℘
an
d
co
n
s
tr
u
ct
a
s
o
f
t seq
u
en
ce
{
}
s
u
ch
th
at
=
(
,
Ω
)
0
.
B
y
u
s
in
g
in
d
u
ctio
n
,
we
g
et
,
(
,
(
+
1
)
,
)
≥
(
0
,
1
,
Ψ
(
∝
)
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Ad
v
Ap
p
l Sci
I
SS
N:
2252
-
8
8
1
4
S
o
ft fu
z
z
y
p
a
r
tia
l m
etri
c
a
n
d
s
o
me
r
esu
lts
o
n
fixed
p
o
in
t th
e
o
r
y
u
n
d
er
…
(
R
o
h
in
i R
.
G
o
r
e)
433
B
y
ab
o
v
e
co
n
d
itio
n
a
n
d
p
r
o
p
e
r
ty
3
o
f
d
ef
i
n
itio
n
o
f
s
o
f
t f
u
zz
y
p
ar
tial m
etr
ic
s
p
ac
e
,
f
o
r
a
n
y
∈
ℤ
+
,
we
g
et
:
(
,
(
+
)
,
)
≥
(
,
(
+
)
,
Ψ
(
v
)
)
≥
(
,
(
+
1
)
,
Ψ
(
)
)
ʘ
…
…
ʘ
⏟
−
(
(
+
−
1
)
,
(
+
)
,
Ψ
(
)
)
≥
(
0
,
1
,
Ψ
(
∝
)
)
ʘ
…
…
ʘ
⏟
−
(
0
,
1
,
Ψ
(
∝
+
−
1
)
)
B
u
t g
iv
en
th
at
l
im
℘
→
∞
(
,
,
)
=
1
,
∀
,
∈
℘
.
(
,
(
+
)
,
)
≥
1
ʘ
1
ʘ
…
…
ʘ
1
⏟
−
=
1
Hen
ce
,
th
e
s
o
f
t
f
u
zz
y
p
ar
tial
s
eq
u
en
ce
{
}
is
a
C
au
ch
y
in
(
℘
,
,
ʘ
)
.
T
h
u
s
,
it
is
co
n
v
er
g
e
n
t.
T
h
er
ef
o
r
e,
(
℘
,
,
ʘ
)
is
co
m
p
lete.
W
e
o
b
tain
,
{
}
→
,
∀
,
∈
℘
l
im
℘
→
∞
(
,
,
)
=
1
,
∀
,
∈
℘
.
T
h
en
,
(
(
,
Ω
)
,
,
)
≥
(
(
,
Ω
)
,
(
,
Ω
)
,
2
)
ʘ
(
(
,
Ω
)
,
,
2
)
≥
(
,
,
Ψ
(
2
∝
)
)
ʘ
(
(
+
1
)
,
,
2
)
≥
1
ʘ
1=1
(
(
,
Ω
)
,
,
)
=
1
Hen
ce
(
,
Ω
)
=
.
T
h
u
s
,
is
a
“so
f
t
f
ix
e
d
p
o
in
t
o
f
(
,
Ω
)
.
T
h
e
u
n
i
q
u
en
ess
o
f
a
s
o
f
t
f
ix
e
d
p
o
in
t
o
f
t
h
e
s
o
f
t f
u
zz
y
p
ar
tial c
o
n
tr
ac
tio
n
m
ap
p
in
g
(
,
Ω
)
is
ea
s
i
ly
v
er
if
ied
;
it is
co
m
p
lete.
4
.
3
.
T
heo
re
m
4.
3
C
o
n
s
id
er
(
℘
,
,
ʘ
)
is
a
co
m
p
lete
s
o
f
t
f
u
zz
y
p
ar
tial
m
etr
ic
s
p
ac
e
s
u
ch
th
at
l
im
℘
→
∞
(
,
,
)
=
1
,
∀
,
∈
℘
.
Def
in
e
co
n
tin
u
o
u
s
t
-
n
o
r
m
ʘ
as
‘
,
ʘ
=
min
(
,
)
’
.
T
h
en
Ψ
−
co
n
tr
ac
tio
n
m
ap
p
in
g
(
,
Ω
)
:
(
℘
,
,
ʘ
)
→
(
℘
,
,
ʘ
)
℘
ad
m
itted
a
co
m
m
o
n
s
o
f
t f
ix
ed
p
o
in
t.
Pro
o
f
:
c
o
n
s
id
er
a
s
o
f
t
p
o
in
t
0
∈
℘
an
d
co
n
s
tr
u
ct
a
s
o
f
t
s
eq
u
e
n
ce
{
}
s
u
ch
th
at
=
(
,
Ω
)
0
.
Ass
u
m
e
th
at
{
}
is
n
o
t
a
C
au
ch
y
s
o
f
t
s
eq
u
en
ce
.
T
h
en
∃
s
o
f
t
r
ea
l
n
u
m
b
e
r
s
>
0
,
>
0
s
atis
f
y
in
g
th
at,
∃
(
0
)
,
(
0
)
≥
0
s
u
ch
th
at
,
(
(
0
)
,
(
0
)
,
)
>
1
⊝
,
∀
0
∈
+
C
h
o
o
s
e
(
0
)
<
(
0
)
s
u
ch
th
at
(
0
)
is
th
e
l
o
west
p
o
s
itiv
e
in
teg
er
w.
r
.
to
(
0
)
th
at
is
s
atis
f
ie
s
ab
o
v
e
co
n
d
itio
n
.
Hen
ce
∃
s
o
f
t
r
ea
l
n
u
m
b
er
s
>
0
,
>
0
f
o
r
wh
ich
two
in
cr
ea
s
in
g
”
s
eq
u
e
n
ce
s
{
(
0
)
}
{
(
0
)
},
(
0
)
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ies
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
8
1
4
I
n
t J Ad
v
Ap
p
l Sci
,
Vo
l.
1
5
,
No
.
1
,
Ma
r
c
h
2
0
2
6
:
427
-
4
3
6
434
C
r
ea
tio
n
o
f
s
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ch
s
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o
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e
n
ce
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ce
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1
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0
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{
/
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,
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1
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⊂
{
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I
t
m
e
a
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t
h
a
t
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c
h
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e
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t
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ai
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l
e
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h
e
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r
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c
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a
t
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i
es
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o
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ti
o
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1
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n
d
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2
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o
r
r
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p
o
n
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i
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o
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n
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>
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h
e
r
e
g
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o
w
,
i
s
a
–
f
u
n
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o
n
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o
r
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h
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t
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i
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l
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f
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t
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o
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3
.
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o
n
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o
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n
d
(
4
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y
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o
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nd
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h
er
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o
r
e,
1
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3
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to
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et
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tr
ad
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n
.
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h
u
s
{
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a
C
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ch
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o
f
t seq
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en
ce
.
I
t
is
ea
s
y
to
co
n
f
ir
m
t
h
at
a
s
o
f
t
f
ix
ed
p
o
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t
o
f
th
e
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o
f
t
f
u
zz
y
p
ar
tial
co
n
t
r
ac
tio
n
m
a
p
p
in
g
(
,
Ω
)
is
u
n
iq
u
e
a
n
d
co
m
p
lete.
I
t
is
ea
s
y
to
co
n
f
ir
m
th
at
a
s
o
f
t
f
ix
ed
p
o
in
t
o
f
th
e
s
o
f
t
f
u
zz
y
p
ar
tial
c
o
n
tr
ac
tio
n
m
a
p
p
in
g
is
u
n
iq
u
e.
No
w,
we
e
x
p
lain
th
e
ex
am
p
le
d
e
p
en
d
s
o
n
T
h
e
o
r
e
m
4
.
3
.
4
.
4
.
E
x
a
m
ple 4
.4
C
o
n
s
id
er
in
g
s
et
=
{
0
.
6
,
0
.
7
,
0
.
8
}
&
p
ar
am
eter
s
et
℘
=
{
1
,
2
}
with
a
co
n
tin
u
o
u
s
t
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n
o
r
m
d
ef
in
ite
as
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=
min
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,
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,
∀
,
∈
[
0
,
1]
℘
.
T
h
en
′
′℘
(
℘
)
=
{
0
.
6
1
,
0
.
6
2
,
0
.
7
1
,
0
.
7
2
,
0
.
8
1
,
0
.
8
2
}
.
W
e
d
ef
in
e
∶
‘
′℘
×
‘
′℘
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(
0
,
∞
)
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→
[
0
,
1
]
℘
as f
o
llo
ws:
f
o
r
all
i,
j
∈
℘
(
0
.
6
,
0
.
7
,
)
=
(
0
.
7
,
0
.
6
,
)
=
{
0
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=
0
0
.
9
,
0
≤
≤
3
1
,
>
3
(
0
.
6
,
0
.
8
,
)
=
(
0
.
8
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6
,
)
=
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8
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0
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7
,
)
=
(
0
.
7
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0
.
8
,
)
=
{
0
,
=
0
0
.
6
,
0
≤
≤
8
1
,
>
8
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Ad
v
Ap
p
l Sci
I
SS
N:
2252
-
8
8
1
4
S
o
ft fu
z
z
y
p
a
r
tia
l m
etri
c
a
n
d
s
o
me
r
esu
lts
o
n
fixed
p
o
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t th
e
o
r
y
u
n
d
er
…
(
R
o
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in
i R
.
G
o
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e)
435
(
,
,
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=
1
⟺
=
,
,
∈
℘
,
>
0
.
T
h
en
(
℘
,
,
ʘ
)
is
a
co
m
p
lete
s
o
f
t
f
u
zz
y
p
ar
tial
m
etr
ic
s
p
ac
e
.
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o
n
s
id
er
(
,
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℘
:
(
℘
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,
ʘ
)
→
(
℘
,
,
ʘ
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℘
as
:
(
,
Ω
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(
0
.
6
1
)
=
0
.
7
1
,
(
,
Ω
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(
0
.
6
2
)
=
0
.
7
1
(
,
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(
0
.
7
1
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.
7
1
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(
,
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(
0
.
7
2
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6
2
(
,
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0
.
8
1
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8
2
,
(
,
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(
0
.
8
2
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=
0
.
8
1
(
)
=
0
.
3
.
T
h
en
−
co
n
tr
ac
tio
n
m
ap
p
in
g
(
,
Ω
)
f
o
llo
ws
th
e
r
e
q
u
ir
em
e
n
ts
f
o
r
th
eo
r
em
3
,
we
g
et
a
co
m
m
o
n
f
ix
ed
p
o
in
t
0
.
7
1
.
5.
CO
NCLU
SI
O
N
W
e
h
av
e
in
v
esti
g
ated
th
e
f
u
n
d
am
en
tal
co
n
ce
p
ts
an
d
p
r
o
p
e
r
ties
o
f
s
o
f
t
f
u
zz
y
p
ar
tial
m
et
r
ic
s
p
ac
e
s
an
d
d
ev
el
o
p
ed
s
ig
n
if
ica
n
t
f
ix
ed
-
p
o
in
t
r
esu
lts
b
y
in
teg
r
atin
g
s
o
f
t
s
et
th
eo
r
y
,
f
u
zz
y
s
ets,
an
d
p
ar
tial
m
etr
ics.
T
h
e
g
iv
en
f
i
x
ed
-
p
o
in
t
th
eo
r
e
m
s
ex
ten
d
class
ical
r
esu
lt
s
t
o
th
e
co
n
tex
t
o
f
s
o
f
t
f
u
zz
y
p
ar
tial
m
etr
ics
b
y
en
s
u
r
in
g
t
h
e
ex
is
ten
ce
a
n
d
u
n
i
q
u
en
ess
o
f
s
o
l
u
tio
n
s
to
t
h
e
s
o
f
t
f
u
zz
y
co
n
tr
ac
tio
n
m
a
p
p
in
g
a
n
d
−
co
n
tr
ac
tio
n
u
n
d
er
c
o
n
d
itio
n
s
ap
p
r
o
p
r
iate
to
th
e
r
elev
an
t
ex
a
m
p
les.
Ap
p
licatio
n
s
o
f
s
o
f
t
f
u
zz
y
p
ar
ti
al
m
e
tr
ic
s
p
ac
es
in
en
g
in
ee
r
in
g
,
p
a
r
ticu
lar
ly
in
im
ag
e
p
r
o
ce
s
s
in
g
an
d
a
n
aly
s
is
,
allo
w
im
ag
es
to
b
e
r
e
p
r
es
en
ted
as
f
u
zz
y
s
ets
wh
er
e
ea
ch
p
ix
el
h
as
a
m
e
m
b
er
s
h
ip
v
al
u
e
in
d
icatin
g
th
e
d
eg
r
ee
o
f
c
o
n
n
ec
ti
o
n
to
a
p
ar
ticu
lar
r
e
g
io
n
o
r
f
ea
tu
r
e.
T
h
u
s
,
s
o
f
t
f
u
zz
y
p
ar
tial
m
etr
ic
s
p
ac
es
ca
n
b
e
u
s
ed
to
m
ea
s
u
r
e
s
im
ilar
ity
b
et
wee
n
im
ag
es;
f
o
r
ex
am
p
le,
a
s
et
o
f
im
ag
es
o
f
d
if
f
er
en
t
r
o
ad
ty
p
es
,
s
u
ch
as
h
ig
h
way
s
,
u
r
b
a
n
ar
ea
s
,
an
d
r
u
r
al
ar
ea
s
,
ca
n
b
e
r
ep
r
esen
ted
as
a
s
o
f
t
f
u
zz
y
s
et,
with
ea
ch
p
ix
el
in
d
icatin
g
th
e
d
eg
r
ee
o
f
m
em
b
er
s
h
i
p
to
a
p
ar
ticu
lar
r
o
a
d
ty
p
e
.
A
s
o
f
t
f
u
zz
y
p
ar
tial
m
etr
ic
s
p
ac
e
ca
n
b
e
d
ef
i
n
ed
o
n
a
s
et
o
f
im
ag
es,
wh
er
e
th
e
d
is
tan
ce
b
etwe
en
im
ag
es
is
a
s
im
ilar
ity
m
ea
s
u
r
e
b
ased
o
n
t
h
e
d
if
f
er
e
n
ce
in
m
em
b
er
s
h
ip
v
alu
es
o
f
co
r
r
esp
o
n
d
in
g
p
ix
el
s
.
I
n
th
e
f
u
t
u
r
e,
th
e
s
co
p
e
o
f
r
e
s
ea
r
c
h
in
clu
d
es
in
teg
r
atio
n
with
o
th
er
m
ath
e
m
atica
l
s
tr
u
ctu
r
es
to
d
ev
elo
p
h
y
b
r
i
d
f
ix
e
d
-
p
o
i
n
t
th
eo
r
em
s
,
ap
p
licatio
n
o
f
f
u
zz
y
f
ix
ed
-
p
o
in
t
th
eo
r
y
in
S
-
m
etr
ic
s
p
ac
es
to
ad
d
r
ess
ch
allen
g
es
o
f
n
av
ig
atio
n
a
n
d
co
n
tr
o
l
s
y
s
tem
s
,
th
er
eb
y
im
p
r
o
v
in
g
s
tab
ilit
y
an
d
p
er
f
o
r
m
an
ce
,
an
d
in
v
esti
g
atio
n
o
f
p
air
ed
f
i
x
ed
-
p
o
in
t
th
eo
r
em
s
in
f
u
zz
y
m
etr
ic
s
p
ac
es
th
at
s
atis
f
y
−
co
n
tr
ac
tiv
e
co
n
d
itio
n
s
to
d
ee
p
en
u
n
d
er
s
tan
d
in
g
o
f
in
ter
ac
tio
n
s
b
etwe
en
m
ap
p
in
g
s
.
F
UNDING
I
NF
O
R
M
A
T
I
O
N
Au
th
o
r
s
s
tate
n
o
f
u
n
d
in
g
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