Question
Determine the validity of the following arguments using the direct method of truth table:
$Pv \sim( Q \& R )$
$\sim P $
$\therefore \sim( Q \& R )$
$Pv \sim( Q \& R )$
$\sim P $
$\therefore \sim( Q \& R )$
| Support Statement | The resulting statement | |||||||
| $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | |
| $A$ | $B$ | $C$ | $\sim C$ | $A v B$ | $(A v B) \rightarrow \sim C$ | $[(A v B) \rightarrow \sim C] \& (A v B)$ | $\sim C$ | |
| $1$ | $T | $T$ | $T$ | $F$ | $T$ | $F$ | $F$ | $F$ |
| $2$ | $T$ | $T$ | $F$ | $T$ | $T$ | $T$ | $T^*$ | $T^*$ |
| $3$ | $T$ | $F$ | $T$ | $F$ | $T$ | $F$ | $F$ | $F$ |
| $4$ | $T$ | $F$ | $F$ | $T$ | $T$ | $T$ | $T^*$ | $T^*$ |
| $5$ | $F$ | $T$ | $T$ | $F$ | $T$ | $F$ | $F$ | $F$ |
| $6$ | $F$ | $T$ | $F$ | $T$ | $T$ | $T$ | $T^*$ | $T^*$ |
| $7$ | $F$ | $F$ | $T$ | $F$ | $F$ | $T$ | $F$ | $F$ |
| $8$ | $F$ | $F$ | $F$ | $T$ | $F$ | $T$ | $F$ | $T$ |
| $3(\sim)$ | $1, 2(v)$ | $5, 4(\rightarrow$) | $6, 5(\&)$ | As $4$ | ||||
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| $(M \leftrightarrow N) \rightarrow O$ |
| $\sim A\ v\ (B\ \&\ D)$ |
| $B \rightarrow (O \rightarrow P)$ |
| $\sim \sim A$ |
| $\therefore (M \leftrightarrow N) \rightarrow P$ |
| $(A\ \rightarrow\ B)\ \&\ (D\ \rightarrow\ E)$ |
| $(B\ \rightarrow\ T)\ \&\ (E\ \rightarrow\ A)$ |
| $\sim\ T$ |
| $\therefore\ \sim\ D\ \&\ \sim\ T$ |