Question
Determine the validity of the following arguments using the direct method of truth table:
$\sim A \leftrightarrow \sim B$
$\therefore\ \sim B \rightarrow \sim A$
$\sim A \leftrightarrow \sim B$
$\therefore\ \sim B \rightarrow \sim A$
| Support Statement | The resulting statement | ||||||
| $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | ||
| $A$ | $B$ | $\sim A$ | $\sim B$ | $\sim A \leftrightarrow \sim B$ | $\sim B \rightarrow \sim A$ | ||
| $1$ | $T$ | $T$ | $F$ | $F$ | $T^*$ | $T^*$ | |
| $2$ | $T$ | $F$ | $F$ | $T$ | $F$ | $F$ | |
| $3$ | $F$ | $T$ | $T$ | $F$ | $F$ | $T$ | |
| $4$ | $F$ | $F$ | $T$ | $T$ | $T^*$ | $T^*$ | |
| $1 (\sim )$ | $2(\sim )$ | $3, 4(\leftrightarrow)$ | $4, 3(\leftrightarrow)$ | ||||
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
| $P\ v\ Q$ |
| $R \rightarrow \sim P$ |
| $R\ \&\ S$ |
| $Q \rightarrow (R\ \&\ P)$ |
| $\therefore P\ v\ R$ |
| $(A\ \rightarrow\ B)\ v\ D$ |
| $H\ \rightarrow\ [(A \ \rightarrow\ B)\ \rightarrow\ R]$ |
| $D\ \rightarrow\ E$ |
| $(E\ v\ F)\ \rightarrow\ H$ |
| $E\ v\ F$ |
| $(R\ v\ E)\ \&\ H$ |
| $(X \rightarrow Y)\ v\ D$ |
| $A \rightarrow [(X \rightarrow Y) \rightarrow R]$ |
| $D \rightarrow E$ |
| $(E\ v\ F) \rightarrow A$ |
| $E\ v\ F$ |
| $\therefore (R\ v\ E)\ \&\ A$ |
| $(A \rightarrow B) \rightarrow R$ |
| $R \rightarrow S$ |
| $(A \rightarrow B)\ \&\ T$ |
| $S\ \&\ T$ |