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

Negation of $p \wedge (\sim q \vee \sim r)$ is -

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

If the truth value of the statement $(P \wedge (\sim R)) \rightarrow ((\sim R) \wedge Q)$ is $F$,then the truth value of which of the following is $F$?

The maximum number of compound propositions,out of $p \vee r \vee s$,$p \vee \sim r \vee \sim s$,$p \vee \sim q \vee s$,$\sim p \vee \sim r \vee s$,$\sim p \vee \sim r \vee \sim s$,$\sim p \vee q \vee \sim s$,$q \vee r \vee \sim s$,$q \vee \sim r \vee \sim s$,$\sim p \vee \sim q \vee \sim s$ that can be made simultaneously true by an assignment of the truth values to $p, q, r$ and $s$,is equal to

$(p \wedge \sim q \wedge \sim r) \vee (\sim p \wedge q \wedge \sim r) \vee (\sim p \wedge \sim q \wedge r)$ is equivalent to-