The dual of the statement $[p \vee (\sim q)] \wedge (\sim p)$ is

The negation of the statement $q \wedge (\sim p \vee \sim r)$ is:

The simplest form of the following switching circuit is:

The negation of the statement $(( A \wedge ( B \vee C ))$ $\Rightarrow ( A \vee B ))$ $\Rightarrow A$ is

The negation of the proposition "If $2$ is prime,then $3$ is odd" is:

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