« Section 1.2: Problem 15 Solution

# Section 1.2: Problem 16-A 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
(Additional) Show that for any formula we can find a tautologically equivalent formula in which negation (if it occurs at all) is applied only to sentence symbols.
This can be shown by induction. In fact, the only case we need to consider is $\gamma=(\neg(\alpha\square\beta))$ where both $\neg\alpha$ and $\neg\beta$ are tautologically equivalent to some formulas $\alpha'$ and $\beta'$ , respectively, in which negation is applied only to sentence symbols. Then, if $\square=\wedge,\vee,\rightarrow,\leftrightarrow$ , then $\gamma$ is tautologically equivalent to, respectively, $(\alpha'\vee\beta')$ , $(\alpha'\wedge\beta')$ , $(\alpha\wedge\beta')$ , and $(\alpha\wedge\beta')\vee(\alpha'\wedge\beta)$ .