Question
Determine the validity of the following arguments using the direct method of truth table:
$A \rightarrow B$
$\therefore B \rightarrow A$
$A \rightarrow B$
$\therefore B \rightarrow A$
| Support Statement | The resulting statement | |||
| $1$ | $2$ | $3$ | $4$ | |
| $A$ | $B$ | $A \rightarrow B$ | $B \rightarrow A$ | |
| $1$ | $T$ | $T$ | $T^*$ | $T$ |
| $2$ | $T$ | $F$ | $F$ | $T$ |
| $3$ | $F$ | $T$ | $T^*$ | $F^*$ |
| $4$ | $F$ | $F$ | $T$ | $T$ |
| $1, 2(\rightarrow)$ | $2, 1(\rightarrow)$ | |||
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| (P$\rightarrow$Q) & (R v S) |
| (R v S) $\rightarrow$ ~ L |
| L v (M & N) |
| $\therefore$ [(P $\rightarrow$ Q) & M] & ~ L |
| $(A \leftrightarrow B) \rightarrow (D \leftrightarrow E)$ |
| $(D \leftrightarrow E) \rightarrow\ \sim H$ |
| $\sim\ \sim H$ |
| $\sim (A \leftrightarrow B) \rightarrow F$ |
| $(F\ v\ G)\ \&\ \sim\ \sim H$ |
| $(Q\ \&\ B)\ v\ \sim D$ |
| $(Q\ \&\ B) \rightarrow \sim E$ |
| $F \rightarrow \sim\ \sim E$ |
| $\sim D \rightarrow (L\ \&\ N)$ |
| $F$ |
| $L\ v\ (B\ \&\ D)$ |
| $(M \leftrightarrow N) \rightarrow O$ |
| $\sim A\ v\ (B\ \&\ D)$ |
| $B \rightarrow (O \rightarrow P)$ |
| $\sim \sim A$ |
| $\therefore (M \leftrightarrow N) \rightarrow P$ |