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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,
T
(
a
,
S
(
a
,
b
)) =
a
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,
=
The law
holds for all S and T
min
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 = + ˃) · + ˃ ),
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