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