$\sim p \wedge q$ is logically equivalent to

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

Consider the following statements:
$(A)$ If $4+3=8$,then $5+3=9$
$(B)$ If $6+4=10$,then the moon is flat
$(C)$ If both $(A)$ and $(B)$ are true,then $5+6=17$
Which of the following statements is correct?

The compound statement $(P \vee Q) \wedge (\sim P) \Rightarrow Q$ is equivalent to:

Let $a: \sim(p \wedge \sim r) \vee(\sim q \vee s)$ and $b: (p \vee s) \leftrightarrow(q \wedge r)$. If the truth values of $p$ and $q$ are $T$ and that of $r$ and $s$ are $F$,then the truth values of $a$ and $b$ are respectively...

Identify the type of "Or" used in the following statement and check whether the statement is true or false:
To enter into a public library,children need an identity card from the school or a letter from the school authorities.