The simplest form of the following switching circuit is:

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 $x - 17 < 20$.
$S_3 : \forall n \in N, n^2 + 3n - 10 = 0$.
$S_4 : \forall n \in N, n^2 \ge 1$.

If $c$ denotes the contradiction,then the dual of the compound statement $\sim p \wedge (q \vee c)$ is:

For each of the following compound statements,first identify the connecting words and then break it into component statements.
All rational numbers are real and all real numbers are not complex.

The statement $\sim(p \leftrightarrow \sim q)$ is

The logical statement $[\sim(\sim p \vee q) \vee (p \wedge r) \wedge (\sim q \wedge r)]$ is equivalent to