Negation of the Boolean expression $p \Leftrightarrow( q \Rightarrow p )$ is.
$(\sim p ) \wedge q$
$p \wedge(\sim q )$
$(\sim p) \vee(\sim q)$
$(\sim p) \wedge(\sim q)$
Which of the following is logically equivalent to $\sim(\sim p \Rightarrow q)$
The converse of the statement "If $p < q$, then $p -x < q -x"$ is -
Statement $-1$ : $ \sim \left( {p \leftrightarrow \, \sim q} \right)$ is equivalent to $p \leftrightarrow q$
Statement $-2$ : $ \sim \left( {p \leftrightarrow \, \sim q} \right)$ is a tautology.
Which of the following statements is a tautology?
$\sim (p \wedge q)$ is equal to .....