Question
Determine the validity of the following arguments using the direct method of truth table:
$\sim A\ \&\ \sim B$
$\therefore\ \sim (A\ v\ B)$
$\sim A\ \&\ \sim B$
$\therefore\ \sim (A\ v\ B)$
| Support Statement | The resulting statement | ||||||
| $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | |
| $A$ | $B$ | $\sim A$ | $\sim B$ | $A\ v\ B$ | $\sim A\ \&\ \sim B$ | $\sim (A\ v\ B)$ | |
| $1$ | $T$ | $T$ | $F$ | $F$ | $T$ | $F$ | $F$ |
| $2$ | $T$ | $F$ | $F$ | $T$ | $T$ | $F$ | $F$ |
| $3$ | $F$ | $T$ | $T$ | $F$ | $T$ | $F$ | $F$ |
| $4$ | $F$ | $F$ | $T$ | $T$ | $F$ | $T^*$ | $T^*$ |
| $1(\sim )$ | $2(\sim )$ | $1, 2(v)$ | $3, 4 (\&)$ | $5(\sim )$ | |||
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| $E\ v\ (E\ \&\ Q)$ |
| $E \rightarrow L$ |
| $\sim L$ |
| $Q\ v\ M$ |
| $A\ \&\ B.$ |
| $B\rightarrow (D\ v\ E)$ |
| $\sim E$ |
| $D \rightarrow (P \rightarrow \sim Q)$ |
| $\therefore P\rightarrow \sim Q$ |