The inverse of the proposition $(p \wedge \sim q) \rightarrow r$ is:

The statement pattern $[(p \land q) \to (\sim p \lor r)] \lor [(\sim p \lor r) \to (p \land q)]$ is

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

Consider the following three statements:
$(A)$ If $3+2=7$ then $4+3=8$.
$(B)$ If $5+2=7$ then earth is flat.
$(C)$ If both $(A)$ and $(B)$ are true then $5+6=11$.
Which of the following statements is correct?

If the truth value of $[(p \lor q) \land (q \to r) \land (\sim r)] \to (p \land q)$ is false, then which of the following is $NOT$ true?

Which of the following statements is/are False?
$S_1: \exists n \in N$, such that $n^2 + n + 2$ is divisible by $4$.
$S_2: \exists x \in N$, such that $x - 17 < 20$.
$S_3: \forall n \in N, x^2 + 3x - 10 = 0$.
$S_4: \forall n \in N, n^2 \ge 1$.