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