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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| $(A\ \rightarrow\ B)\ \&\ (D\ \rightarrow\ E)$ |
| $(B\ \rightarrow\ T)\ \&\ (E\ \rightarrow\ A)$ |
| $\sim\ T$ |
| $\therefore\ \sim\ D\ \&\ \sim\ T$ |
| $R \rightarrow S$ |
| $P \rightarrow Q$ |
| $P\ v\ Q$ |
| $Q \rightarrow R$ |
| $\therefore (P \rightarrow R) \& (R\ v\ S)$ |
| $E\rightarrow (F\ \&\ \sim G)$ |
| $( F\ v\ G)\rightarrow H$ |
| $E$ |
| $\therefore H$ |
| $(T\ \&\ P) \rightarrow \sim Q$ |
| $\sim (T\ \&\ P) \rightarrow (R \rightarrow \sim Q)$ |
| $(\sim S\ v\ R)\rightarrow\ \sim\ \sim Q$ |
| $\sim S$ |
| $\therefore \sim R\ \&\ \sim S$ |
| $A \rightarrow B$ |
| $A\ v\ C$ |
| $E\ \&\ \sim F$ |
| $\sim B$ |
| $\therefore\ C\ \&\ \sim F$ |
| $\sim (A \ \&\ B) \rightarrow H$ |
| $F\ v \sim (H \ \&\ F)$ |
| $(A \ \&\ B) \rightarrow (H \ \&\ F)$ |
| $\sim F \ \&\ (D \ \&\ E)$ |
| $(D \ \&\ E) \ \&\ H$ |