Question
Determine the validity of the following arguments using the direct method of truth table:
$P \rightarrow Q$
$Q \rightarrow P$
$\therefore P \leftrightarrow Q$
$P \rightarrow Q$
$Q \rightarrow P$
$\therefore P \leftrightarrow Q$
| Support Statement | The resulting statement | |||||
| $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | |
| $P$ | $Q$ | $P \rightarrow Q$ | $Q \rightarrow P$ | $(P \rightarrow Q)\ \&\ (Q \rightarrow P)$ | $P \leftrightarrow Q$ | |
| $1$ | $T$ | $T$ | $T$ | $T$ | $T^*$ | $T^*$ |
| $2$ | $T$ | $F$ | $F$ | $T$ | $F$ | $F$ |
| $3$ | $F$ | $T$ | $T$ | $F$ | $F$ | $F$ |
| $4$ | $F$ | $F$ | $T$ | $T$ | $T^*$ | $T^*$ |
| $1,2 (\rightarrow)$ | $2, 1(\rightarrow)$ | $3, 4(\&)$ | $1, 2(\leftrightarrow)$ | |||
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| $G \rightarrow J$ |
| $J \rightarrow K$ |
| $(G \rightarrow K) v (J \rightarrow L)$ |
| $L \rightarrow M$ |
| $\therefore (J \rightarrow M) v\ Q$ |
| $P \rightarrow Q$ |
| $\sim Q\ v\ R$ |
| $\sim R$ |
| $\therefore (\sim P \& \sim R)\ v\ S$ |
| $(A\ v\ B)\ \rightarrow\ [D\ \rightarrow\ (P\ \&\ \sim\ Q)]$ |
| $(A\ \&\ J)\ \rightarrow [(P\ \&\ \sim\ Q)\ \rightarrow\ K]$ |
| $(A\ \&\ J)\ \&\ (\sim\ K\ v\ D)$ |
| $\therefore\ (D \rightarrow\ K)\ v\ \sim\ Q$ |