Question
Which of the following is always true ?
A.
$$\left( { \sim p \, \vee \sim q} \right) \equiv \left( {p \wedge q} \right)$$
B.
$$\left( {p \to q} \right) \equiv \left( { \sim q \to \, \sim p} \right)$$
C.
$$ \sim \left( {p \to \, \sim q} \right) \equiv \left( {p \, \wedge \sim q} \right)$$
D.
$$ \sim \left( {p \leftrightarrow q} \right) \equiv \left( {p \to q} \right) \to \left( {q \to p} \right)$$
Answer :
$$\left( {p \to q} \right) \equiv \left( { \sim q \to \, \sim p} \right)$$
Solution :
Since, $$ \sim \left( {p \vee q} \right) \equiv \left( { \sim p \, \wedge \sim q} \right){\text{and }} \sim \left( {p \wedge q} \right) \equiv \left( { \sim p \wedge q} \right)$$
So option $$\left( B \right)$$ and $$\left( D \right)$$ are not true.
$$\left( {p \to q} \right) \equiv p \wedge \sim q),$$ so option $$\left( C \right)$$ are not true.
$$\eqalign{
& {\text{Now, }}p \to q \sim p \vee q \cr
& \sim q \to \sim p \equiv \left[ { \sim \left( { \sim q} \right) \vee \sim p} \right] \equiv q \, \vee \sim p \equiv \, \sim p \vee q \cr
& p \to q \equiv \sim q \to \sim p \cr} $$