Question
p $\leftrightarrow$ [p v (q & ~ q)]
| 1 | 2 | 3 | 4 | 5 | 6 | |
| p | q | ~ q | q & ~ q | p v (q & ~ q)] | p $\leftrightarrow$ [p v (q & ~ q)] | |
| 1 | T | T | F | F | T | T |
| 2 | T | F | T | F | T | T |
| 3 | F | T | F | F | F | T |
| 4 | F | F | T | F | F | T |
| 2 (~) | 2, 3 (&) | 1,4 (v) | 1, 5 ($\leftrightarrow$) | |||
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| $(P\ v\ R)\ \rightarrow (S\ v\ T)$ |
| $\sim M\ \&\ \sim N$ |
| $N\ v \sim (S\ v\ T)$ |
| $H \rightarrow (P\ v\ R)$ |
| $[\sim H\ \&\ \sim (P\ v\ R)]\ v\ S$ |
| $T \rightarrow W$ |
| $X \rightarrow Y$ |
| $Z \rightarrow W$ |
| $X \rightarrow A$ |
| $W \rightarrow X$ |
| $[(T \rightarrow Y)\ \&\ (Z \rightarrow A)] \rightarrow (T\ v\ Z)$ |
| $M \rightarrow \sim (S \ \&\ T)$ |
| $\sim (A \ \&\ B)\ v\ \sim D$ |
| $\sim D (S \ \&\ T)$ |
| $M$ |
| $[\sim (A \ \&\ B) \ \&\ M)\ v\ D$ |
| $(A\ v\ B)\ \rightarrow\ [D\ \rightarrow\ (P\ \&\ \sim\ Q)]$ |
| $(A\ \&\ J)\ \rightarrow [(P\ \&\ \sim\ Q)\ \rightarrow\ K]$ |
| $(A\ \&\ J)\ \&\ (\sim\ K\ v\ D)$ |
| $\therefore\ (D \rightarrow\ K)\ v\ \sim\ Q$ |
| $A \rightarrow B$ |
| $D\ v\ A$ |
| $E\ \rightarrow \sim\ D$ |
| $F\ v\ E$ |
| $\therefore\ A\ \&\ B$ |