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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| $H \rightarrow I$ |
| $H\ \&\ J$ |
| $I \rightarrow G$ |
| $G\ \&\ J$ |
| $\sim P\ \&\ (Q\ v\ R)$ |
| $(Q \rightarrow A)\ \&\ (R \rightarrow B)$ |
| $(A\ v\ B) \rightarrow (P\ v\ R)$ |
| $R$ |
| $(H\ \&\ K)\ \rightarrow\ (J\ v\ K)$ |
| $\sim\ E\ \&\ \sim\ F$ |
| $F\ v\ \sim\ (J\ v\ K)$ |
| $\sim\ (H\ \&\ K)\ \rightarrow\ H$ |
| $H\ \&\ \sim\ E$ |
| $(R\ S) \rightarrow (F\ E)$ |
| $\sim (R\ S) \rightarrow J$ |
| $(F\ E) \rightarrow \sim H$ |
| $\sim\ \sim H$ |
| $\therefore \sim (J\ v\ G) \& \sim\ \sim H$ |
| $P\ v\ Q$ |
| $R \rightarrow \sim P$ |
| $R\ \&\ S$ |
| $Q \rightarrow (R\ \&\ P)$ |
| $\therefore P\ v\ R$ |