Section 1.2: Problem 11 Solution
Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
Show that a truth assignment
satisfies the wff
iff
for an even number of
’s,
. (By the associative law for
, the placement of the parentheses is not crucial.)
It is true for
. Then, by induction, if it is true for
, then by denoting
, we have
, which is true iff
is true and, therefore, has even number of false assignments, whereas
is true, or
is false and, hence, has odd number of false assignments, whereas
is false as well, that is again an even number of false assignments in
.