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where
.
Define fuzzy binary relations of similarity over set
i
=
1
k
;
;
j
=
1
k
l
=
1
m
.
The proof
. Let us prove of reflexivity characteristic for
Ξ
k
(
)
, i.e. that
R
p
p
=
1
4
(
)
μ
X
,
X
=
1
k
X
∈
Ξ
R
i
i
;
i
p
1
1
m
~
(
)
(
)
∑
∫
=
()
()
μ
X
,
X
=
κ
=
1
−
d
X
,
X
=
1
−
μ
x
−
μ
x
dx
=
1
R
i
i
ii
i
i
il
il
2
1
l
1
0
1
[
]
()
()
∫
min
μ
x
,
μ
x
dx
il
il
1
m
(
)
∑
μ
X
,
X
=
κ
=
0
=
1
R
i
i
ii
1
m
2
[
()
()
]
l
=
1
∫
max
μ
x
,
μ
x
dx
il
il
0
1
~
(
)
(
)
()
()
l
ii
∫
μ
X
,
X
=
κ
=
1
−
d
μ
,
μ
=
1
−
μ
x
−
μ
x
dx
=
1
R
i
i
il
il
il
il
3
0
1
[
]
()
()
∫
min
μ
x
,
μ
x
dx
il
il
(
)
l
ii
μ
X
,
X
=
κ
=
0
=
1
R
i
i
1
4
[
]
()
()
∫
max
μ
x
,
μ
x
dx
il
il
Thus,
R
is a set with reflective fuzzy relations. Let us prove symmetry
characteristic for this set, i.e. that
0
(
)
(
)
k
μ
X
,
X
=
μ
X
,
X
;
X
,
X
∈
Ξ
R
i
j
R
j
i
i
j
p
p
1
m
1
(
)
(
)
()
()
∑
∫
=
μ
X
,
X
=
1
−
d
X
,
X
=
1
−
μ
x
−
μ
x
dx
=
R
i
j
i
j
il
jl
2
1
l
1
0
1
1
m
(
)
(
)
()
()
∑
∫
=
=
1
−
μ
x
−
μ
x
dx
=
1
−
d
X
,
X
=
μ
X
,
X
;
jl
il
j
i
R
j
i
2
1
l
1
0
1
[
]
()
()
∫
min
μ
x
,
μ
x
dx
il
jl
1
m
(
)
∑
0
μ
X
,
X
=
=
R
i
j
m
1
[
]
2
()
()
l
=
1
∫
max
μ
x
,
μ
x
dx
il
jl
0
1
[
]
()
()
∫
min
μ
x
,
μ
x
dx
jl
il
m
1
(
)
∑
=
0
=
μ
X
,
X
;
R
j
i
1
m
[
]
2
()
()
l
=
1
∫
max
μ
x
,
μ
x
dx
jl
il
0
∫
1
0
(
)
(
)
()
()
μ
X
,
X
=
1
−
d
μ
,
μ
=
1
−
μ
x
−
μ
x
dx
=
R
i
j
il
jl
il
jl
3
