The Boolean expression $(p$ $\Rightarrow q) \wedge (q$ $\Rightarrow \sim p)$ is equivalent to:

Let $p$ and $q$ denote the following statements:
$p$: The sun is shining
$q$: $I$ shall play tennis in the afternoon
The negation of the statement "If the sun is shining then $I$ shall play tennis in the afternoon" is:

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

Statement-$1$: $\sim (p \Leftrightarrow \sim q)$ is equivalent to $p \Leftrightarrow q$.
Statement-$2$: $\sim (p \Leftrightarrow \sim q)$ is a tautology.

Which of the following statement patterns is a tautology?

Which of the following is not logically equivalent to the proposition: "$A$ real number is either rational or irrational"?