The statement pattern $[p$ $\rightarrow(q$ $\rightarrow p)]$ $\rightarrow[p$ $\rightarrow(p \vee q)]$ is

The false statement in the following is

Consider statements $p: S_1$ is closed; $q: S_2$ is closed; $r: S_3$ is closed. The simplified equivalent circuit diagram and its logical statement for the switching circuit is respectively.

The logically equivalent statement of $(\sim p \wedge q) \vee (\sim p \wedge \sim q) \vee (p \wedge \sim q)$ is

Which one of the following is the pair of equivalent circuits?
$i. (p \land q) \lor (p \land r)$
$ii. p \lor (q \land r)$
$iii. p \land (q \lor r)$
$iv. p \land q \land r$
$v. (p \land q) \lor r$

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