The proposition $ \sim \left( {p\,\vee \sim q} \right) \vee  \sim \left( {p\, \vee q} \right)$ is logically equivalent to

Consider the statement : "For an integer $n$, if $n ^{3}-1$ is even, then $n$ is odd." The contrapositive statement of this statement is

Which of the following is logically equivalent to $\sim(\sim p \Rightarrow q)$

Negation of $p \wedge( q \wedge \sim( p \wedge q ))$ is

The compound statement $(\sim( P \wedge Q )) \vee((\sim P ) \wedge Q ) \Rightarrow((\sim P ) \wedge(\sim Q ))$ is equivalent to

Let $p$ and $q$ be two statements.Then $\sim( p \wedge( p \Rightarrow \sim q ))$ is equivalent to