Question
Determine the validity of the following arguments using the direct method of truth table:
$\sim P\ v \sim Q$
$\therefore P\ \&\ Q$
$\sim P\ v \sim Q$
$\therefore P\ \&\ Q$
| Support Statement | The resulting statement | ||||||
| $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | ||
| $P$ | $Q$ | $\sim P$ | $\sim Q$ | $\sim P\ v \sim Q$ | $P\ \&\ Q$ | ||
| $1$ | $T$ | $T$ | $F$ | $F$ | $F$ | $T$ | |
| $2$ | $T$ | $F$ | $F$ | $T$ | $T^*$ | $F^*$ | |
| $3$ | $F$ | $T$ | $T$ | $F$ | $T^*$ | $F^*$ | |
| $4$ | $F$ | $F$ | $T$ | $T$ | $T^*$ | $F^*$ | |
| $1 (\sim )$ | $1, 2(v)$ | $3, 4(v)$ | $1, 2(\&)$ | ||||
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| $M\ \rightarrow\ N$ |
| $D\ v\ (N\ \rightarrow\ P)$ |
| $R\ v\ \sim\ D$ |
| $(A\ \&\ B)\ \rightarrow\ \sim\ R$ |
| $A\ \&\ B$ |
| $(M \rightarrow P)\ v\ Z$ |
| $(R\ \rightarrow\ S)\ \&\ (A\ v\ C)$ |
| $B\ \&\ (P\ \rightarrow\ W)$ |
| $(W\ \rightarrow\ T)\ \&\ (S\ \rightarrow\ Y)$ |
| $P\ v\ R$ |
| $T\ v\ Y$ |
| $J \rightarrow K$ |
| $J\ v\ (K\ v\ \sim\ L)$ |
| $\sim K$ |
| $\therefore \sim L\ \&\ \sim K$ |
| $(T\ \&\ P) \rightarrow \sim Q$ |
| $\sim (T\ \&\ P) \rightarrow (R \rightarrow \sim Q)$ |
| $(\sim S\ v\ R)\rightarrow\ \sim\ \sim Q$ |
| $\sim S$ |
| $\therefore \sim R\ \&\ \sim S$ |
| $(P\ \&\ R)\ v\ (S\ \rightarrow\ T)$ |
| $Q \rightarrow \sim\ (P\ \&\ R)$ |
| $P\ v\ Q$ |
| $\sim\ P$ |
| $(S\ \rightarrow\ T)\ \&\ Q$ |