If $p \to (q \lor \sim r)$ is false, then the truth values of $(p \leftrightarrow q) \land r$ and $\sim p \to \sim q$ are :

Write the negation of the following statement:
$p:$ For every positive real number $x,$ the number $x-1$ is also positive.

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

The symbolic form of the statement "If it does not rain today or $I$ will not go to school,then $I$ will meet my friend and $I$ will go to watch a movie" is:
$p$: It rains today
$q$: $I$ am going to school
$r$: $I$ will meet my friend
$s$: $I$ will go to watch a movie

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

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