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In-Depth Information
Fuzzy
n
-ary relation
R
between sets
X
,
X
,....,
X
is referred to as such fuzzy
1
2
n
(
)
(
)
[]
∀
x
,
x
,...,
x
∈
X
×
X
×
....
×
X
μ
x
,
x
,...,
x
∈
0
1
set
R
when
;
,
1
2
n
1
2
n
R
1
2
n
{}
n
{}
1
{}
2
where
are ordinary sets.
Fuzzy binary relation
R
between sets
X
,
Y
is referred to as such fuzzy set
R
when
;
, …,
X
=
x
X
=
x
X
=
x
n
1
2
(
)
) []
(
{}
{}
∀
x
,
y
∈
X
×
Y
μ
x
,
y
∈
0
1
;
, where
X
=
x
;
Y
=
y
are ordinary
R
sets.
If sets
{
}
{
}
X
x
,
x
,...,
x
Y
y
,
y
,...,
y
are finite, the fuzzy binary
relation
R
can be specified by means of a matrix, which rows and columns are
put in correspondence with elements of sets
X
,
Y
; and an element
=
;
=
1
2
n
1
2
n
(
)
μ
x
,
y
is
R
i
j
put as cross-section the of
i
-th row and
j
-th column. Thus
(
)
(
)
(
)
μ
x
,
y
μ
x
,
y
...
μ
x
,
y
⎛
⎞
R
1
1
R
1
2
R
1
m
⎜
⎟
(
)
(
)
(
)
μ
x
,
y
μ
x
,
y
...
μ
x
,
y
⎜
⎟
R
2
1
R
2
2
R
2
m
R
=
.
⎜
⎟
...
...
...
...
⎜
⎜
⎟
⎟
(
)
(
)
(
)
μ
x
,
y
μ
x
,
y
μ
x
,
y
⎝
⎠
Fuzzy binary relation
R
over set
X
is referred to as such fuzzy set
R
, that
(
R
n
1
R
n
2
R
n
m
[
]
(
)
μ
x
,
y
∈
0
1
)
;
.
∀
x
,
y
∈
X
×
Y
R
Let
R
,
R
are fuzzy binary relations between sets
and
Y
,
Z
,
X
,
Y
accordingly.
Composition of fuzzy relations
R
,
R
is referred to as such fuzzy set
R
1
R
2
when for
∀
x
∈
X
,
∀
y
∈
Y
,
∀
z
∈
Z
[
]
(
)
(
)
(
)
μ
x
,
z
=
∨
μ
x
,
y
∧
μ
y
,
z
,
R
R
R
R
1
2
y
1
2
where
are operators of triangular norm and conorm class, accordingly.
For example, (
max
—
min
)-composition
and
∧
∨
R
is defined by the expression
1
R
2
[
{
}
]
(
)
(
)
(
)
μ
x
,
z
=
max
min
μ
x
,
y
,
μ
y
,
z
,
R
R
R
R
1
2
1
2
y
where
.
The fuzzy binary relation
R
is referred to as reflective, if
x
∈
X
;
y
∈
Y
;
x
∈
X
(
)
μ
x
,
x
=
1
;
R
(
)
(
)
∀ ,
.
One of the important properties of fuzzy binary relations is that they can be
presented as a population of ordinary binary relations ordered by inclusion and
representing a hierarchical population of relations [15]. Expansion of fuzzy binary
relations to a population of ordinary binary relations is based on the concept of
α
-level of a fuzzy binary relation which is represented in the form
(
μ
x
,
y
=
μ
y
,
x
x
y
∈
X
∀
x
∈
X
, and symmetric, if
;
R
R
{
)
(
)
}
R
=
x
,
y
∈
X
×
Y
:
μ
x
,
y
≥
α
.
α
R
