[opend 10/20/09=MM/DD/YY]
“We note that by the principle of duality the condition (4a) holds
if and only if (4b) holds”
(lines 5–6, page 349)
is an inappropriate expression.
It may cause the misunderstanding that
two conditions dual to each other are equivalent to each other.
For example,
- (a) there exists the greatest element
- (b) there exists the least element
are conditions dual to each other,
and
(a) and (b) are not equivalent.
Indeed,
the set N of positive integers
with the ordinary ordering has the least element 1,
and it does not have the greatest element
(note that the infinity ∞ is not an integer).
The conditions
- (4a) ∀a, b, c ;
a∧(b∨c)
=(a∧b)∨(a∧c)
- (4b) ∀a, b, c ;
a∨(b∧c)
=(a∨b)∧(a∨c)
are dual to each other
and they are equivalent,
but the equivalence follows
not directly
from the principle of duality.
The proof of the equivalence is as follows:
(4a) ⇒ (4b):
|
a∨(b∧c)
|
=
a∨(a∧c)
∨(b∧c)
|
∵
absorption law
|
|
|
=
a∨((a∨b)
∧c)
|
∵
(4a)
|
|
|
=
((a∨b)
∧a)
∨((a∨b)
∧c)
|
∵
absorption law
|
|
|
=
(a∨b)
∧(a∨c)
|
∵
(4a)
|
(4b) ⇒ (4a)
follows from
the above
by the principle of duality.
q.e.d.
