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