Seminar VI

Exercise I gives a little more practice with semantic tableau. Exercises II and II concern the natural deduction method.

I. Use the method of semantic tableaux to show that the following sentences are tautologies:

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

II. Use the rules of natural deduction to show that the following deductions hold:

1. $\{(p \wedge q)\} \vdash p$
2. $\{p\} \vdash (p \vee q)$

III. Use the rules of natural deduction to show that the following sentences are theorems:

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