$p$ and $q$ are two logical statements. If $r: p \rightarrow (\sim p \vee q)$ has truth value false,then the truth values of $p$ and $q$ are respectively:

The negation of the contrapositive of the statement $(p \vee \sim q) \to (p \wedge \sim q)$ is

Consider the two statements:
$(S1): (p$ $\rightarrow q) \vee (\sim q$ $\rightarrow p)$ is a tautology.
$(S2): (p \wedge \sim q) \wedge (\sim p \vee q)$ is a fallacy.
Then:

Check whether the following pair of statements are negation of each other. Give reasons for your answer.
$I$: There exists real numbers $x$ and $y$ for which $x+y=y+x.$
$II$: For all real numbers $x$ and $y$,$x+y \neq y+x.$

Show that the statement $p:$ 'If $x$ is a real number such that $x^{3}+4x=0$,then $x$ is $0$' is true by the method of contradiction.

The statement pattern $(p \wedge q) \to (r \vee \sim s)$ is false. Then the truth values of $p, q, r$ and $s$ are respectively