$($a$)$ Let $p,q$ and $r$ be three propositions.

i$.$ Construct a truth table for the compound proposition

$((pq{)}^{{?}^{?}}(r$notp${)}^{{?}^{?}}r)$notq

ii$.$ Is the compound proposition in question $($i$)$ a tautology? Explain your answer.

$($b$)$ Let $p,q,r$ and $s$ be four propositions. Assuming that $p$ and $r$ are false and that $q$ and $s$ are true, find the truth value of the following proposition

$((pvvq{)}^{{?}^{?}}(qvvs))((notrvvp{)}^{{?}^{?}}(qvvs))$

$($c$)$ Define a set of at most three atomic propositions. Then use them to translate all of these sentences.

i$.$ It's time for bed only if it is after $9$ PM and l$\text{'}$m tired

ii$.$ I$\text{'}$m not tired unless it is after $9$ PM$.$

iii. It's time for bed and after $9$ PM if and only if I am tired

$($d$)$ Give the contrapositive, the converse and the inverse of the following statement:

AAxinR, $ifx(x+2)>0$ then

$($e$)$ A tautology is a proposition that is always true. Let $p,q$ and $r$ be three propositions. Using the laws of propositions or the truth table, show that $(p(qvvr))>((p^{{?}^{?}}$notq$)r)$ is a tautology.