What is the identity element for the conjunction $(AND)$ operation in Boolean algebra of logical statements?

The negation of the statement $p \rightarrow (q \wedge r)$ is equal to .........

Are the following pairs of statements negations of each other:
$1$. The number $x$ is a rational number.
$2$. The number $x$ is an irrational number.

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

Consider the following statements:
$P$: $I$ have fever
$Q$: $I$ will take medicine
$R$: $I$ will take rest
The statement "If $I$ have fever,then $I$ will take medicine and $I$ will take rest" is equivalent to:

The logical statement $[\sim(\sim p \vee q) \vee (p \wedge r) \wedge (\sim q \wedge r)]$ is equivalent to