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