Question

The negation of $p \to (\sim p \vee q)$ is

• A. $p \vee (p \vee \sim q)$
• B. $p \rightarrow \sim (p \vee q)$
• C. $p \rightarrow q$
• D. $p \wedge \sim q$

Answer 1

Answer: (D)

$p \wedge \sim q$

Answer 2

Truth table

p
q
$\sim p$
$\sim q$
$(\sim p \vee q)$
$p \rightarrow(\sim p \vee q )$
$( - p \vee- q )$
$p \vee(\sim p \vee q )$
$\sim( p \vee q )$
$p \rightarrow\sim( p \vee q )$
$p \rightarrow q$
$p \wedge \sim q$
$\sim[p \rightarrow(\sim p \vee q)]$

F
F
T
T
T
T
T
T
T
T
T
F
F

F
T
T
F
T
T
F
F
F
T
T
F
F

T
F
F
T
F
F
T
T
F
F
F
T
T

T
T
F
F
T
T
T
T
F
F
T
F
F

Here, we observe that the truth value of column $\sim[p \rightarrow(\sim p \vee q)]$ and $(p \wedge \sim q)$ are same.
Hence,$\sim[p \rightarrow(\sim p \vee q)] \equiv(p \wedge \sim q)$