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1
1
N
(
)
()
()
=
=
1
μ
n
j
x
μ
n
i
x
dx
=
μ
M
,
M
;
R
j
i
N
1
n
1
0
1
[
]
()
()
n
i
n
j
min
μ
x
,
μ
x
dx
N
(
)
1
μ
M
,
M
=
0
=
R
i
j
1
N
[
]
2
()
()
n
=
1
n
i
n
j
max
μ
x
,
μ
x
dx
0
1
[
]
()
()
n
j
n
i
min
μ
x
,
μ
x
dx
1
N
(
)
=
0
=
μ
M
,
M
.
R
j
i
1
N
[
]
2
()
()
n
=
1
max
μ
n
j
x
,
μ
n
i
x
dx
0
Thus, R and R are symmetric. The proposition 3.2 is proved.
If the constructed fuzzy relations of similarity
R
1 , R
are not transitive, let us
2
denote by
, accordingly, fuzzy relations of similarity which are transitive
closures of fuzzy relations
R
1 , R
2
R
1 , R
. Since in accordance with the proposition 3.2
2
R
1 , R
R
=
R
k
p
1
p
=
1
2
are reflective fuzzy relations, then
,
.
2
p
(
)
Let us denote elements
M
,
M
,...,
M
q
k
of the set
as similar (versus
Θ
k
1
2
q
()
α
0
similarity relation R
) with confidence level
, if for all
M ,
i M
j
(
)
(
)
i
=
1
q
;
j
=
1
q
μ
M ,
M
α
is satisfied.
Let us consider fuzzy clusterization of a set
the relation
R
i
j
Θ
k
under fuzzy relations of
similarity
R
1 , R
.
2
~
~
(
)
(
)
μ
M
,
M
=
k
μ
M
,
M
=
k
I n th e pr opo sition 3.2 it is proved that
,
R
i
j
ij
R
i
j
ij
1
2
(
)
i
=
1
k
;
j
=
1
k
are values of membership functions of fuzzy relations of
R
1 , R
k
similarity
defined over set
Θ
. Let us make a matrix of fuzzy relations of
2
similarity for these relations:
(
)
(
)
1
μ
M
,
M
.
.
.
μ
M
,
M
R
1
2
R
1
k
p
p
(
)
(
)
μ
M
,
M
1
.
.
.
μ
M
,
M
R
1
2
R
2
k
R
=
,
p
=
1
2
p
p
p
.
.
.
.
.
.
(
)
(
)
μ
M
,
M
μ
2
M
,
M
.
.
.
1
R
1
k
R
2
k
p
p
Generally, constructed fuzzy relations of similarity
R
1 , R
are not transitive,
2
therefore let us construct their transitive closures
R
1 , R
, which are fuzzy relations
2
of similarity. Transitive closures for
1 , R are constructed on the basis of
compositions of each considered relation with themselves. Let us write out
matrixes of fuzzy relations of similarity
2
 
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