For any two statements $p$ and $q,$ the negation of the expression $p \vee ( \sim p \wedge q)$ is

Write the negation of the following statement:
$r:$ For every real number $x$,either $x > 1$ or $x < 1.$

Given below are two statements:
$p: 25 \text{ is a multiple of } 5.$
$q: 25 \text{ is a multiple of } 8.$
Write the compound statements connecting these two statements with "And" and "Or". In both cases,check the validity of the compound statement.

The statement pattern $(p \lor q) \to \sim r$ is logically equivalent to

The statement $(\sim( p \Leftrightarrow \sim q )) \wedge q$ is :

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