Question
$P \rightarrow \sim Q$
$\therefore Q \rightarrow (P\ \&\ Q)$
$\therefore Q \rightarrow (P\ \&\ Q)$
| Support Statement | The resulting statement | ||||||
| $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | ||
| $P$ | $Q$ | $\sim Q$ | $P\ \&\ Q$ | $P\rightarrow\sim Q$ | $Q\rightarrow(P\ \&\ Q)$ | ||
| $1$ | $T$ | $T$ | $F$ | $T$ | $F$ | $T$ | |
| $2$ | $T$ | $F$ | $T$ | $F$ | $T$ | $T$ | |
| $3$ | $F$ | $T$ | $F$ | $F$ | $T^*$ | $F^*$ | |
| $4$ | $F$ | $F$ | $T$ | $F$ | $T$ | $T$ | |
| $2 (\sim )$ | $1, 2(\&)$ | $1, 3 (\rightarrow)$ | $2, 4 (\rightarrow)$ | ||||
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| $(R\ \&\ S)\ v\ P$ |
| $P\ \rightarrow\ Q$ |
| $T\ v\ (R\ \&\ S)\ \rightarrow\ W ]$ |
| $S\ \&\ \sim\ T$ |
| $[S\ \&\ (W\ v\ Q)]\ v\ \sim\ P$ |
| $(A\ v\ B) \rightarrow D$ |
| $A$ |
| $\therefore (D\ \&\ A)\ v\ P$ |
| $(A\ v\ B)\ \rightarrow\ [D\ \rightarrow\ (P\ \&\ \sim\ Q)]$ |
| $(A\ \&\ J)\ \rightarrow [(P\ \&\ \sim\ Q)\ \rightarrow\ K]$ |
| $(A\ \&\ J)\ \&\ (\sim\ K\ v\ D)$ |
| $\therefore\ (D \rightarrow\ K)\ v\ \sim\ Q$ |