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