Question
$P\ v\ Q$
$\therefore \sim P\ \&\ \sim Q$
$\therefore \sim P\ \&\ \sim Q$
| Support Statement | The resulting statement | ||||||
| $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | ||
| $P$ | $Q$ | $\sim P$ | $\sim Q$ | $P\ v\ Q$ | $\sim P\ \&\ \sim Q$ | ||
| $1$ | $T$ | $T$ | $F$ | $F$ | $T^*$ | $F^*$ | |
| $2$ | $T$ | $F$ | $F$ | $T$ | $T^*$ | $F^*$ | |
| $3$ | $F$ | $T$ | $T$ | $F$ | $T^*$ | $F^*$ | |
| $4$ | $F$ | $F$ | $T$ | $T$ | $F$ | $T$ | |
| $1 (\sim )$ | $2 (\sim )$ | $1, 2 (v)$ | $3, 4(\&)$ | ||||
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| $(P\ \&\ Q) \rightarrow R$ |
| $R \rightarrow (H \rightarrow G)$ |
| $H\ \&\ K$ |
| $P\ \&\ Q$ |
| $\therefore (G\ v\ I)\ \&\ H$ |
| $(P\ \&\ Q) \rightarrow [(P \rightarrow R)\ \&\ S]$ |
| $(P\ \&\ Q)\ \&\ T$ |
| $R\ v\ W$ |
| $A \rightarrow B$ |
| $D\ v\ A$ |
| $E\ \rightarrow \sim\ D$ |
| $F\ v\ E$ |
| $\therefore\ A\ \&\ B$ |
| $(H\ \&\ K)\ \rightarrow\ (J\ v\ K)$ |
| $\sim\ E\ \&\ \sim\ F$ |
| $F\ v\ \sim\ (J\ v\ K)$ |
| $\sim\ (H\ \&\ K)\ \rightarrow\ H$ |
| $H\ \&\ \sim\ E$ |
| $P\ v\ Q$ |
| $R \rightarrow\sim P$ |
| $R\ \&\ S$ |
| $Q \rightarrow (R\ \&\ P)$ |
| $\therefore P\ v\ H$ |