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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| $(X \rightarrow Y)\ v\ D$ |
| $A \rightarrow [(X \rightarrow Y) \rightarrow R]$ |
| $D \rightarrow E$ |
| $(E\ v\ F) \rightarrow A$ |
| $E\ v\ F$ |
| $\therefore (R\ v\ E)\ \&\ A$ |
| $(F \rightarrow S)\ \&\ (R \rightarrow W)$ |
| $F\ \&\ R$ |
| $\therefore \sim S\ \&\ W$ |