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