Question
Identify the false statements
A.
$$ \sim \left[ {p \vee \left( { \sim q} \right)} \right] \equiv \left( { \sim p} \right) \vee q$$
B.
$$\left[ {p \vee q} \right] \vee \left( { \sim p} \right)$$ is a tautology
C.
$$\left[ {p \wedge q} \right] \wedge \left( { \sim p} \right)\,$$ is a contradiction
D.
$$ \sim \left[ {p \vee q} \right] \equiv \left( { \sim p} \right) \vee \left( { \sim q} \right)$$
Answer :
$$ \sim \left[ {p \vee q} \right] \equiv \left( { \sim p} \right) \vee \left( { \sim q} \right)$$
Solution :
Since, $$ \sim \left( {p \vee q} \right) \equiv \, \sim p \, \wedge \sim q\,\,$$ (By De-Morgans' law)
$$\therefore \,\, \sim \left( {p \vee q} \right) \ne \,\, \sim p \, \vee \sim q$$
$$\therefore \left( D \right)$$ is the false statement.