Let $p$ and $q$ be any two logical statements and $r: p \to (\sim p \vee q)$. If $r$ has a truth value $F$,then the truth values of $p$ and $q$ are respectively

The dual of the statement $[p \vee (\sim q)] \wedge (\sim p)$ is

The correct logical equivalences from the following are:
$(I)$ $p \to (q \to r) \equiv (p \land q) \to r$
$(II)$ $(p \to q) \to r \equiv p \to (q \lor r)$
$(III)$ $(p \to q) \to r \equiv (p \to r) \land (\sim q \to r)$
$(IV)$ $p \to (q \to r) \equiv q \to (p \to r)$

Check the validity of the statement given below by the method specified against it.
$p:$ The sum of an irrational number and a rational number is irrational (by contradiction method).

If $p$: It is raining and $q$: It is pleasant,then the symbolic form of "It is neither raining nor pleasant" is

If $p$ and $q$ each have truth value $F$,then the truth values of the statement patterns $(\sim p \vee q) \leftrightarrow \sim(p \wedge q)$ and $\sim p \leftrightarrow (p \rightarrow \sim q)$ respectively are