For any two statements $p$ and $q,$ the negation of the expression $p \vee ( \sim p \wedge q)$ is

The symbolic form of the following circuit is (where $p$ and $q$ represent switches $S_{1}$ and $S_{2}$ being closed respectively):

The statement $(P$ $\Rightarrow Q) \wedge (R$ $\Rightarrow Q)$ is logically equivalent to:

Which of the following is a tautology?

Which of the following statement patterns is a contradiction?
$S_{1} \equiv (p \rightarrow q) \wedge (p \wedge \sim q)$
$S_{2} \equiv [p \wedge (p$ $\rightarrow q)]$ $\rightarrow q$
$S_{3} \equiv (p \vee q) \rightarrow \sim p$
$S_{4} \equiv [p \wedge (p \rightarrow q)] \leftrightarrow q$

Find the component statements of the following compound statement and check whether they are true or false.
$100$ is divisible by $3, 11$ and $5$.