Section 1.2: Problem 5 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
Prove or refute each of the following assertions:
(a) If either
or
, then
.
(b) If
, then either
or
.
Note: formally we would have to consider truth assignments that use symbols from the formulas on both sides of
only. You can either assume that this is the case without explicitly saying that, or use Exercise 6(b) to see that this is not so important.
(a) Either for every truth assignment
that satisfies
,
, or for every truth assignment
that satisfies
,
. Hence, for every truth assignment
that satisfies
, either
or
is
, but in either case,
. Therefore,
. But not vice versa, see (b).
(b) For any
, it is true that
. However, of course, in general, neither
nor
holds.