The contrapositive of $\sim q \to p$ is equivalent to

Which of the following is not logically equivalent to the proposition: "$A$ real number is either rational or irrational"?

The statement $(\sim( p \Leftrightarrow \sim q )) \wedge q$ is :

For each of the following statements,determine whether an inclusive "Or" or exclusive "Or" is used. Give reasons for your answer.
Students can take French or Sanskrit as their third language.

The negation of the converse of the statement $p \lor q$ is:

The correct logical equivalences from the following are: $(I)$ $p \to (q \to r) \equiv (p \land q) \to r$ $(II)$ $(p \to q) \to r \equiv p \to (q \lor r)$ $(III)$ $(p \to q) \to r \equiv (p \to r) \land (\sim q \to r)$ $(IV)$ $p \to (q \to r) \equiv q \to (p \to r)$