The minimum number of switches in the simplified form of the following switching circuit is

The contrapositive of $(\sim p \wedge q) \rightarrow (q \wedge \sim r)$ is

Let $A, B, C$ and $D$ be four nonempty sets. The contrapositive of 'if $A \subseteq B$ and $B \subseteq D$ then $A \subseteq C$' is

The Boolean expression $(p \wedge \sim q) \vee q \vee (\sim p \wedge q)$ is equivalent to:

Consider the following two statements :
$P :$ If $7$ is an odd number,then $7$ is divisible by $2.$
$Q :$ If $7$ is a prime number,then $7$ is an odd number.
If $V_1$ is the truth value of the contrapositive of $P$ and $V_2$ is the truth value of the contrapositive of $Q,$ then the ordered pair $(V_1, V_2)$ equals

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.