The statement $p$ $\Rightarrow (q$ $\Rightarrow p)$ is logically equivalent to .....

The statement $[p \wedge (q \vee r)] \vee [\sim r \wedge \sim q \wedge p]$ is equivalent to

Which of the following statement patterns is a tautology?
$S_{1} \equiv \sim p \rightarrow (q \leftrightarrow p)$
$S_{2} \equiv \sim p \vee \sim q$
$S_{3} \equiv (p$ $\rightarrow q) \wedge (q$ $\rightarrow p)$
$S_{4} \equiv (q \rightarrow p) \vee (\sim p \leftrightarrow q)$

Consider the following statements.
$p$: If $3^4 > 4^3$, then $3^3 > 4^4$
$q$: The roots of the equation $x^2 - 2x + 2 = 0$ are real if and only if Mumbai is in Maharashtra.
$r$: Statement $p$ is true or statement $q$ is false.
Which of the following has truth value $T$ (true)?

The statement pattern $(p \lor q) \to \sim r$ is logically equivalent to

The negation of the contrapositive of the statement pattern $(p \vee \sim q) \rightarrow (p \wedge \sim q)$ is