I have this task to solve:
Find (if it exists) such a sentence formula α so that the following formulas are simultaneously tautologies of the propositional calculus: $$ (q \rightarrow \alpha) \leftrightarrow (q \rightarrow (p \wedge r))$$ and $$ (\alpha \rightarrow q) \leftrightarrow (\neg(p \vee r) \rightarrow q) $$
Generally I have idea to solve this task in use of table. So I created a table with 7 columns: $p$,$q$,$r$,$(p \wedge r)$, $(q \rightarrow (p \wedge r)$ $(q \rightarrow \alpha)$, $LS \leftrightarrow RS $ and 8 rows with each case. In last column I get these values: $\alpha$, $\neg \alpha$, $1$, $1$, $\neg \alpha$, $\neg \alpha$, $1$, $1$ So if we consider tautology, both $\alpha$ and $\neg \alpha$ should be $1$ so it can't be done. So $\alpha$ like that does not exists and I don't have to consider second potential tautology. Have I right? Eventually, there are other ways to solve tasks like that?