Question
Which of the following is true ?
A.
$$p \Rightarrow q \equiv \,\, \sim p \Rightarrow \, \sim q$$
B.
$$ \sim \left( { p \Rightarrow \, \sim q} \right) \equiv \,\, \sim p \wedge q$$
C.
$$ \sim \left( { \sim p \Rightarrow \, \sim q} \right) \equiv \,\, \sim p \wedge q$$
D.
$$ \sim \left( { \sim p \Leftrightarrow q} \right) \equiv \left[ { \sim \left( {p \Rightarrow q} \right) \wedge \sim \left( {q \Rightarrow p} \right)} \right]$$
Answer :
$$ \sim \left( { \sim p \Rightarrow \, \sim q} \right) \equiv \,\, \sim p \wedge q$$
Solution :
$$\eqalign{
& \sim \left( {p \Rightarrow q} \right) \equiv p \wedge \sim q \cr
& \therefore \,\, \sim \left( { \sim p \Rightarrow \, \sim q} \right) \equiv \,\, \sim p \wedge \sim \left( { \sim q} \right) \equiv \,\, \sim p \wedge q \cr
& {\text{Thus, }} \sim \left( { \sim p \Rightarrow \, \sim q} \right) \equiv \,\, \sim p \wedge q \cr} $$