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

The negation of the statement $(( A \wedge ( B \vee C ))$ $\Rightarrow ( A \vee B ))$ $\Rightarrow A$ is

Given the following two statements:
$(S_{1}): (q \vee p) \rightarrow (p \leftrightarrow \sim q)$ is a tautology.
$(S_{2}): \sim q \wedge (\sim p \leftrightarrow q)$ is a fallacy.
Then:

Consider the three statements -
$p: \forall n \in N, 10n-3$ is a prime number,when $n$ is not divisible by $3$.
$q: \frac{2}{\sqrt{3}}, \frac{-2}{\sqrt{3}}, \frac{-1}{\sqrt{3}}$ are the direction cosines of a directed line.
$r: \sin x$ is an increasing function in the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$.
Then which of the following statement patterns has a truth value of true?

The statement pattern $(p \land q) \to (r \lor \sim s)$ is false. Then the truth values of $p, q, r$ and $s$ are respectively

The negation of the statement pattern $p \vee (q \rightarrow \sim r)$ is