Using the truth table techniques employed above, test the following argument forms for validity. (For your own entertainment, guess whether the argument form is valid or invalid before working it out.)
$$
\begin{aligned}
& \text { 1. } p \supset q \\
& \therefore q \supset p
\end{aligned}
$$
$$
\begin{aligned}
& \text { 2. } p \supset q \\
& \therefore \sim q \supset \sim p \\
& \text { 3. } \sim q \supset \sim p \\
& \therefore p \supset q
\end{aligned}
$$
$$
\begin{aligned}
& \text { 3. } \sim q \supset \sim p \\
& \therefore p \supset q
\end{aligned}
$$
4.
$$
\begin{array}{rl}
\text { 4. } p & p q \\
q & \supset r \\
\therefore & p \supset(q \& r)
\end{array}
$$
5.
$$
\text { 5. } \begin{aligned}
& p \supset q \\
& q \supset r \\
& \sim r \\
\therefore & \sim p
\end{aligned}
$$
$$
\begin{aligned}
& \text { 6. } p \supset q \\
& \quad q \supset r \\
& \therefore \sim r \supset \sim p
\end{aligned}
$$
$$
\text { 7. } \begin{aligned}
p \vee q \\
p \supset q \\
q \supset r
\end{aligned} \quad \therefore r
$$
8. $p \supset(q \vee r)$
$$
\begin{aligned}
& \sim q \\
& \sim r \\
\therefore & \sim p
\end{aligned}
$$
$$
\text { 9. }(p \vee q) \supset r
$$
10
$$
\text { 10. }(p \& q) \supset r
$$
$$
\begin{aligned}
& \text { 11. } p \supset(q \supset r) \\
& \therefore(p \& q) \supset r \\
&
\end{aligned}
$$
12
12. $(p \& q) \supset r$
$$
\therefore p \supset(q \supset r)
$$
13.
$$
\begin{aligned}
& \text { 3. } p \supset(q \supset r) \\
& q \\
& \therefore \sim r \\
&
\end{aligned}
$$
14.
$$
\begin{aligned}
& \text { 14. } p \supset(q \supset r) \\
& \therefore \supset r \\
& \therefore r
\end{aligned}
$$
$\begin{aligned} & \text { 15. }(p \vee q) \&(p \vee r) \\ & \quad \sim r \\ & \therefore \sim q \\ & \text { 16. }(p \supset q) \&(p \supset \sim r) \\ & \quad q \& r \\ & \therefore \sim p \\ & \text { 17. }(p \vee q) \supset p \\ & \therefore \sim q\end{aligned}$
18. $(p \vee q) \supset(p \& q)$
$$
\therefore(p \supset q) \&(q \supset p)
$$
19. $(p \& q) \supset(p \cap q)$
$\therefore(p \supset q) \mathrm{n}(q \supset p)$
20. $r$
$$
\therefore \overline{(p \supset q) \vee(q \supset p)}
$$