Question
$(P \rightarrow q)\ v\ (p\ \&\ q)$
| $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | |
| $p$ | $q$ | $\sim q$ | $p \rightarrow q$ | $p\ \&\ \sim q$ | $(P \rightarrow q)\ v\ (p\ \&\ \sim\ q)$ | |
| $1$ | $T$ | $T$ | $F$ | $T$ | $F$ | $T$ |
| $2$ | $T$ | $F$ | $T$ | $F$ | $T$ | $T$ |
| $3$ | $F$ | $T$ | $F$ | $T$ | $F$ | $T$ |
| $4$ | $F$ | $F$ | $T$ | $T$ | $F$ | $T$ |
| $2(\sim )$ | $1, 2 ( \rightarrow )$ | $1, 3 (\&)$ | $4, 5 (v)$ | |||
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| $A \rightarrow B$ |
| $A\ v\ C$ |
| $E\ \&\ \sim F$ |
| $\sim B$ |
| $\therefore\ C\ \&\ \sim F$ |
| $(H\ \&\ K)\ \rightarrow\ (J\ v\ K)$ |
| $\sim\ E\ \&\ \sim\ F$ |
| $F\ v\ \sim\ (J\ v\ K)$ |
| $\sim\ (H\ \&\ K)\ \rightarrow\ H$ |
| $H\ \&\ \sim\ E$ |
| $(R\ S) \rightarrow (F\ E)$ |
| $\sim (R\ S) \rightarrow J$ |
| $(F\ E) \rightarrow \sim H$ |
| $\sim\ \sim H$ |
| $\therefore \sim (J\ v\ G) \& \sim\ \sim H$ |