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In-Depth Information
·
(
ₒ
)
=
·
(
ₒ
)
For this goal, it should be
a
a
b
0, and
a
a
b
b
. In a boolean
algebra it is
a
+
a
·
z
b
⃔
z
b
,
because of:
1.
a
a
+
a
+
a
+
a
+
a
+
·
z
b
⃒
a
·
z
=
z
b
, and since
z
z
, follows
z
b
a
+
a
+
2.
z
b
⃒
a
·
z
a
·
(
b
)
=
a
·
b
b
.
a
+
Then,
a
·
(
a
ₒ
b
)
b
⃔
a
ₒ
b
b
.
A conditional function is a mapping
ₒ:
B
×
B
ₒ
B
, such that
a
·
(
a
ₒ
b
)
b
B
. Hence, in a Boolean algebra, the biggest conditional is
a
+
for all
a
b
,the
so-called material conditional, and any smaller function is also a conditional. For
example, from
,
b
∈
a
+
a
a
+
a
·
b
b
b
,
b
a
, are conditionals.
it follows that
a
ₒ
b
=
a
·
b
,
a
ₒ
b
=
b
,
a
ₒ
b
=
Analogously, from
a
·
b
+
a
+
a
·
b
b
,
a
·
b
+
it also follows that
a
ₒ
b
=
a
·
b
is a conditional. Different ways of writing
a
+
b
in a boolean algebra, are
a
·
(
b
)
+
a
+
a
·
b
b
+
a
·
b
,
a
·
b
,
b
+
since
a
·
(
b
)
+
a
+
a
)
·
(
a
+
a
+
b
+
a
·
b
=
a
·
b
=
(
a
+
b
)
=
b
, and
a
·
b
=
(
a
)
·
(
b
)
=
a
+
b
+
b
+
b
+
b
.
a
+
a
+
a
+
a
+
Notice that from
z
1
b
,
z
2
b
, follows
z
1
·
z
2
(
b
)
·
(
b
)
=
a
+
a
+
a
+
a
+
,
z
1
+
z
2
(
)
+
(
)
=
,
hence, the union and the intersection
of conditionals is also a conditional. For example,
a
b
b
b
b
a
+
a
+
·
+
=
b
b
b
is obviously
a conditional.
There are two-variable functions
a
a
+
ₒ
b
such that
a
ₒ
b
b
, but are not
expressible as a single formula with the connectives
,
·
,
+
as the before considered
cases. For example,
a
·
b
,
if
a
·
b
=
0
a
ₒ
b
=
a
+
b
,
if
a
·
b
=
0
,
a
·
b
,
if
a
·
b
=
0
verifies
(
a
ₒ
b
)
·
a
=
=
a
·
b
b
,
that is, is
a
+
a
·
(
b
)
=
a
·
b
,
if
a
·
b
=
0
,
a conditional. Analogously,
1
,
if
a
b
a
ₒ
b
=
b
,
otherwise
,
a
,
if
a
b
verifies
(
a
ₒ
b
)
·
a
=
b
,
that is, is a conditional.
a
·
b
,
otherwise
,
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