Question
Determine the validity of the following arguments using the direct method of truth table:
$\sim A\ v \sim B$
$\sim A$
$\therefore\ \sim B$
$\sim A\ v \sim B$
$\sim A$
$\therefore\ \sim B$
| Support Statement | The resulting statement | ||||||
| $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | |
| $A$ | $B$ | $\sim A$ | $\sim B$ | $\sim A v \sim B$ | $(\sim A\ v \sim B)\ \&\ \sim A$ | $\sim B$ | |
| $1$ | $T$ | $T$ | $F$ | $F$ | $F$ | $F$ | $F$ |
| $2$ | $T$ | $F$ | $F$ | $T$ | $T$ | $F$ | $T$ |
| $3$ | $F$ | $T$ | $T$ | $F$ | $T$ | $T^*$ | $F^*$ |
| $4$ | $F$ | $F$ | $T$ | $T$ | $T$ | $T$ | $T$ |
| $1(\sim )$ | $2(\sim )$ | $3, 4(v)$ | $5, 3(\&)$ | As $4$ | |||
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| $(A\ v\ B)\ \rightarrow\ D$ |
| $E\ \rightarrow\ \sim\ D$ |
| $(G\ \leftrightarrow\ F)\ \rightarrow\ (E\ \&\ H)$ |
| $G\ \leftrightarrow\ F$ |
| $\sim\ (A\ v\ B)\ \&\ H$ |
| $(X \rightarrow Y)\ v\ D$ |
| $A \rightarrow [(X \rightarrow Y) \rightarrow R]$ |
| $D \rightarrow E$ |
| $(E\ v\ F) \rightarrow A$ |
| $E\ v\ F$ |
| $\therefore (R\ v\ E)\ \&\ A$ |
| $R\rightarrow (S\ \&\ T)$ |
| $P\ v\ \sim (S\ \&\ T)$ |
| $Q\ \&\ \sim P$ |
| $\sim R\rightarrow (X\ \&\ Y)$ |
| $Q\ \&\ X$ |
| $(R\ \rightarrow\ S)\ \&\ (A\ v\ C)$ |
| $B\ \&\ (P\ \rightarrow\ W)$ |
| $(W\ \rightarrow\ T)\ \&\ (S\ \rightarrow\ Y)$ |
| $P\ v\ R$ |
| $T\ v\ Y$ |
| $(H\ \&\ K)\ \rightarrow\ (J\ v\ K)$ |
| $\sim\ E\ \&\ \sim\ F$ |
| $F\ v\ \sim\ (J\ v\ K)$ |
| $\sim\ (H\ \&\ K)\ \rightarrow\ H$ |
| $H\ \&\ \sim\ E$ |