If $p$ and $q$ are simple propositions,then $p \Leftrightarrow \sim q$ is true when

The statement pattern $\sim(p \vee q) \vee(\sim p \wedge q)$ is equivalent to

The dual of the statement pattern $(p \land \sim q) \to (q \land \sim p)$ is equivalent to

Consider the following two statements :
$P :$ If $7$ is an odd number,then $7$ is divisible by $2.$
$Q :$ If $7$ is a prime number,then $7$ is an odd number.
If $V_1$ is the truth value of the contrapositive of $P$ and $V_2$ is the truth value of the contrapositive of $Q,$ then the ordered pair $(V_1, V_2)$ equals

If the truth value of the compound statement $[(p \leftrightarrow q) \land (q \to r) \land \sim r] \to (p \land \sim q)$ is false, then the truth values of the statement patterns $(p \to q) \leftrightarrow (q \to r)$ and $\sim (p \lor r) \to (q \land p)$ are, respectively ...

Let $\Delta, \nabla \in \{\wedge, \vee\}$ be such that $( p \rightarrow q ) \Delta ( p \nabla q )$ is a tautology. Then