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In-Depth Information
Remark 4.2.2
1. As it is easy to prove, it is
t
t
t
(
[
μ
]↗
T
[
˃
]
)
=[
˃
]
↗
T
[
μ
]
,
t
,
t
, defined by
t
t
with the matrices
[
˃
]
[
μ
]
˃
(
x
,
y
)
=
˃(
y
,
x
)
,
μ
(
x
,
y
)
=
μ(
y
,
x
)
,
giving
t
t
(
[
μ
]
)
=[
μ
]
.
t
.
2. In general, the max-T composition is associative, but not commutative. That is,
if
the compositions
The matrix
[
μ
]
is the
transposed
of
[
μ
]
[
μ
]↗
T
(
[
˃
]↗
T
[
ʻ
]
)
, and
(
[
μ
]↗
T
[
˃
]
)
↗
T
[
ʻ
]
,
are possible
it is,
(
[
μ
]↗
T
[
˃
]
)
↗
T
[
ʻ
]=[
μ
]↗
T
(
[
˃
]↗
T
[
ʻ
]
),
but, in general,
[
μ
]↗
T
[
˃
] =[
˃
]↗
T
[
μ
]
.
4.3 Which Relevant Properties Do Have a Fuzzy Binary
Relation?
The most relevant properties of a fuzzy relation
μ
:
X
×
X
ₒ[
0
,
1
]
are the following,
1. Reflexive property,
μ(
x
,
x
)
=
1,
for all
x
∈
X
.
2. Symmetric property,
μ(
x
,
y
)
=
μ(
y
,
x
)
,
for all
x
,
y
∈
X
, implies
x
=
y
.
3. Antisymmetric property,
μ(
x
,
y
)>
0,
μ(
y
,
x
)>
0 implies
x
=
y
.
4. T-transitive property
T
(μ(
x
,
y
), μ(
y
,
z
))
μ(
x
,
z
)
,
for all
x
,
y
,
z
∈
X
, and
some continuous t-norm
T
.
In the finite case, for what concerns properties reflexive and symmetric, the matrix
[
μ
]=
(
shows the respective properties,
1
.Itis
t
ii
=
t
ij
)
1,
for all 1
i
n
, that is, the main diagonal of
[
μ
]
is constituted
by
n
numbers equal to 1.
2
.Itis
t
ij
=
t
ji
,
for all 1
i
,
j
n
, that is, the elements of
[
μ
]
are placed
symmetrically with respect to the main diagonal.
For example, the matrix
10
.
7
is reflexive, but not symmetric,
0
.
61
and the matrix
⊛
⊞
10
.
60
.
7
⊝
⊠
0
.
600
.
9
is symmetric, but not reflexive.
0
.
70
.
90
.
5
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