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2.2.8.6 Laws of Absorption
With classical sets, the absorption laws
A
A
,
always hold. With fuzzy sets in standard algebras, the formulas and respective equa-
tions
∩
(
A
∪
B
)
=
A
, and
A
∪
(
A
∩
B
)
=
•
μ
·
(μ
+
˃)
=
μ,
T
(
a
,
S
(
a
,
b
))
=
a
•
μ
+
(μ
·
˃)
=
μ,
S
(
a
,
T
(
a
,
b
))
=
a
must be studied to find for which algebras these laws do hold.
Lemma 2.2.44
If T
and S are, respectively, a t-norm and a t-conorm, it is
T
(
a
,
S
(
a
,
b
))
=
a for all a
,
bin
[
0
,
1
]
if and only if T
=
min
.
Proof
If
T
=
min, since
a
S
(
a
,
b
)
, it follows min
(
a
,
S
(
a
,
b
))
=
a
. With
b
=
0,
the equation gives
T
(
a
,
a
)
=
a
, and
=
min.
Lemma 2.2.45
If T and S are, respectively, a t-norm and a continuous t-conorm, it
is S
(
a
,
T
(
a
,
b
))
=
a for all a
,
bin
[
0
,
1
]
if and only if S
=
max
.
Proof
Since
T
(
a
,
b
)
a
,itfollowsmax
(
a
,
T
(
a
,
b
))
=
a
. With
b
=
1, the equation
gives
S
(
a
,
a
)
=
a
, and
=
max.
Hence,
•
μ
·
(μ
+
˃)
=
μ,
=
The law
holds for all
S
and
T
min
•
μ
+
(μ
·
˃)
=
μ,
=
The law
holds for all
T
and
S
max
•
The two laws hold jointly if and only if
T
=
min and
S
=
max.
2.2.8.7 The Law of von Neumann
With classical sets it always holds the law of von Neumann, or law of the perfect
repartition,
B
c
A
=
(
A
∩
B
)
∪
(
A
∩
),
B
c
B
c
that follows from
A
=
A
∩
X
=
A
∩
(
B
∪
)
=
(
A
∩
B
)
∪
(
A
∩
)
, and generalizes
B
c
B
c
.
that of the excluded-middle since
A
=
X
implies
X
=
(
B
∩
X
)
∪
(
X
∩
)
=
B
∪
From that law, by duality it follows
A
c
c
B
c
c
A
c
B
c
=
(
A
∩
B
)
∩
(
A
∩
)
=
(
∪
)
∩
A
c
(
∪
B
)
, that is
B
c
A
=
(
A
∪
B
)
∩
(
A
∪
),
B
c
.
a law that generalizes that of non-contradiction since
A
= ∅
implies
∅ =
B
∩
With fuzzy sets, the question is the validity of the laws
μ
=
μ
·
˃
+
μ
·
˃
,
μ
=
(μ
+
˃)
·
(μ
+
˃
),
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