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μ
x
1
=
/
x
1
+
/
/
x
2
+
/
/
It is
1
1
8
2
8
x
3
, and it results,
μ
x
1
(
x
1
)μ(
x
1
,
y
)
=
μ(
x
1
,
y
)
=
μ
x
1
(
y
).
It is also
μ
x
2
=
1
/
8
/
x
1
+
1
/
x
2
+
2
/
8
/
x
3
, and
1
8
μ(
μ
x
2
(
x
1
)μ(
x
1
,
)
=
x
1
,
),
y
y
showing,
•
1
1
8
μ(
x
1
,
x
1
)
=
8
=
μ
x
2
(
x
1
)
1
1
8
2
1
•
8
μ(
x
1
,
x
2
)
=
<
8
=
μ
x
2
(
x
2
)
1
2
8
2
3
•
8
μ(
x
1
,
x
3
)
=
<
8
=
μ
x
2
(
x
3
)
,
etc. That is, the three fuzzy sets
μ
x
1
, μ
x
2
, μ
x
3
are
pr od
-states of
μ
.
Remark 4.4.1
When the fuzzy relation
μ
represents a conditional statement
Q
/
P
(a fuzzy rule 'If
x
is
P
, then
y
is
Q
'), the
T
-states of
are among the fuzzy sets
verifying the
Modus Ponens
with respect to the continuous t-norm
T
.
μ
4.5 Fuzzy relations and
α
-cuts
Given a fuzzy relation
μ
:
X
×
X
ₒ[
0
,
1
]
,the
ʱ
-cuts of
μ
are the classical (crisp)
relations
μ
(ʱ)
defined by,
μ
(ʱ)
={
(
x
,
y
)
∈
X
×
X
;
μ(
x
,
y
)
≥
ʱ
}
,
ʱ
∈[
,
]
μ
(
0
)
=
×
ʱ
1
≥
ʱ
2
,itis
μ
(ʱ
1
)
μ
(ʱ
2
)
for all
0
1
. Obviously,
X
X
, and if
.
•
μ
is symmetric, if and only if all its
ʱ
-cuts are symmetric (classical) relations.
μ(
x
,
y
)
=
μ(
y
,
x
)
⃔
(
x
,
y
)
∈
μ
(ʱ)
and
(
y
,
x
)
∈
μ
(ʱ)
.
•
μ
is reflexive, if and only if all
ʱ
-cuts are reflexive (classical) relations.
μ(
x
,
x
)
=
1
⃔
(
x
,
x
)
∈
μ
(ʱ)
, since
ʱ
1
.
•
If
μ
is antisymmetric, all the
ʱ
-cuts are antisymmetric (crisp) relations,
If
μ(
x
,
y
)
≥
ʱ >
0, and
μ(
y
,
x
)
≥
ʱ >
0, it is
x
=
y
.
•
If
μ
is a
T
-transitive fuzzy relation,
(
,
)
∈
μ
(ʱ)
, and
(
,
)
∈
μ
(ʱ)
, implies
(
,
)
∈
μ
(
T
(ʱ,ʱ))
,
x
y
y
z
x
z
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