Find the component statements of the following compound statement and check whether they are true or false.
Number $3$ is prime or it is odd.

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

The proposition $p \rightarrow \sim( p \wedge \sim q )$ is equivalent to

Statement $-1$: The statement $A \to (B \to A)$ is equivalent to $A \to (A \vee B)$.
Statement $-2$: The statement $\sim [(A \wedge B) \to (\sim A \vee B)]$ is a tautology.

Which of the following statements is correct?
$(a)$ $S_1: (p \wedge q) \equiv \sim(p \rightarrow \sim q)$
$(b)$ $S_2: (p \wedge q) \wedge (\sim p \vee \sim q)$ is a tautology
$(c)$ $S_3: [p \wedge (p$ $\rightarrow \sim q)]$ $\rightarrow q$ is a contradiction
$(d)$ $S_4: p$ $\rightarrow (q$ $\rightarrow p)$ is a contingency

$\sim (p \Leftrightarrow q)$ is