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,
·
,
+
,
)
(
It is obvious that a De Morgan algebra
verifying the laws of
non-contradiction and excluded-middle is a Boolean algebra. In all De Morgan
algebra there is included the Boolean algebra whose elements are the Boolean
elements of the algebra, that is, that given by the set
L
a
{
a
∈
L
;
a
·
=
0
}=
a
=
{
a
∈
L
;
a
+
1
}
, a set that is never empty since at least contains 0 and 1.
1.6.1 Examples
1. A good example of a Boolean algebra is the power-set
P
(
X
)
={
A
;
A
ↆ
X
}
,
whose elements are the subsets of a set
X
. In this case, the operations are:
the intersection
of subsets is
•
·
the union
of subsets is
•
+
c
,is
,
•
the complement of subsets
(
)
, and 1 is the full set
X
.
Power-sets are typical instances Boolean algebras. In fact, any Boolean algebra
is isomorphic to a power-set.
2. The unit interval
and 0 is the empty set
∅
[
0
,
1
]
, endowed with the operations:
·=
min,
+=
max, and
=
1
−
id
, is a De Morgan algebra, but not a Boolean one, since, for instance,
min
(
a
,
1
−
a
)
=
0
⃔
a
=
0, or
a
=
1. The Boolean elements of this algebra
are just 0 and 1.
3. The set of functions
X
[
,
]
={
μ
;
μ
:
ₒ[
,
]}
0
1
X
0
1
, endowed with the operations
μ
(
given by
(μ
·
˃)(
x
)
=
min
(μ(
x
), ˃(
x
))
,
(μ
+
˃)(
x
)
=
max
(μ(
x
), ˃(
x
))
,
x
)
=
1
, for all
x
in
X
, is a De Morgan algebra, whose Boolean elements are the
functions
−
μ(
x
)
X
. Hence, this algebra is not a Boolean one.
4. The set whose elements are all the vector subspaces of
R
3
μ
∈{
0
,
1
}
constitute an Ortho-
modular lattice, once endowed with the operations:
•
The intersection of two subspaces
(
·
)
,
•
The minimum subspace that contains two subspaces
(
+
)
,
(
)
•
The subspace that is orthogonal to a subspace
.
R
3
, that is, these three vectors are an orthog-
For instance, if it is
<
u
,v,w >
=
onal basis of
R
3
,itis:
•
<
u
>
·
<v>
=
0, the null subspace, with 0
=
(
0
,
0
,
0
)
,
•
<
u
>
+
<w>
=
<
u
,w >
, the plane given by the two orthogonal vectors
,
•
<v>
=
<
u
an
w
u
,w >
, the same plane.
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