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y
1
0.5
0.8
x
1
0.7
y
2
x
2
1
0.8
y
3
4.2 How to Compose Fuzzy Relations?
Given two fuzzy relations
μ
:
X
×
Y
ₒ[
0
,
1
]
, and
˃
:
Y
×
Z
ₒ[
0
,
1
]
, how can
we obtain a relation
ʻ
:
X
×
Z
ₒ[
0
,
1
]
through
μ
and
˃
? To solve this problem,
there is the
Sup
−
T
product of fuzzy relations, given by
ʻ(
x
,
z
)
=
Sup
y
T
(μ(
x
,
y
), ˃(
y
,
z
)),
for all
(
x
,
z
)
∈
X
×
Z
,
∈
Y
a formula that, in the finite case
X
={
x
1
,...,
x
n
}
,
Y
={
y
1
,...,
y
m
}
,
Z
=
{
z
1
,...,
z
p
}
, reduces to,
ʻ(
x
i
,
z
j
)
=
Max
1
T
(μ(
x
i
,
y
k
), ˃(
y
k
,
z
j
)).
k
m
[
μ
]=
(
r
ik
)
[
˃
]=
(
s
kj
)
Provided
,
, then
t
ij
=
ʻ(
x
i
,
z
j
)
=
Max
k
m
T
(
r
i
,
k
,
s
kj
)
,1
i
n
,1
j
p
,
1
giving the matrix
[
ʻ
]=
(
t
ij
)
as the Max-T product, or composition, of the matrices
[
r
ik
]
and
[
s
kj
]
, that was introduced before by:
(
t
ij
)
=
(
r
ik
)
↗
T
(
s
kj
)
.
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