The given circuit is equivalent to:

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

The logical statement $(p \wedge \sim q) \vee q \vee (\sim p \wedge q)$ is equivalent to

The statement pattern $p \wedge (q \vee \sim p)$ is equivalent to

If the inverse of the conditional statement $p \to (\sim q \wedge \sim r)$ is false,then the respective truth values of the statements $p, q,$ and $r$ are:

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