Seminar V

The following exercises are concerned with Semantic Tableau and their use in order to determince the (in)consistency of sets of sentences and to decided on the validity of entailment relations.

I. Use the method of Semantic Tableau to decide whether or not the following sets of sentences are consistent or inconsistent.

1. $\{p \rightarrow q, \neg q \}$
2. $\{p \rightarrow q, \neg q \vee r, p \wedge \neg r \}$
3. $\{(p \vee q) \rightarrow r, \neg ( ( \neg p \wedge \neg q) \vee r)\}$

II. Use Semantic Tableaux to decide whether nor not the following entailment statements are valid or not:

1. $\{p\} \models p \vee q$
2. $\{p \rightarrow q, q \} \models p$
3. $\{p \rightarrow q, \neg q\} \models \neg p$

III. Use Semantic Tableaux to decide whether or not the following sentences are tautologies or not:

1. $p \leftrightarrow (p \vee q)$
2. $p \rightarrow (p \vee q)$
3. $\neg (p \wedge q) \leftrightarrow (\neg p \vee \neg q)$