Answers :
$(A)$ ~$Q →$ ~$P$
$(D)$ ~$ Q → Q$
The meaning of question as I understand :
We want to prove $Q$ when we know $P$ for sure. Meaning we want to say $P→ Q$.
Now question asks from which of the options we can infer $P→ Q$.
So question statement in short
Find $X$ such that $X→ (P→ Q)\equiv True$.
We know contrapositive of a statement is equivalent to that statement.
Means ~$Q →$ ~$P$ $\equiv$ $P→ Q$
$\therefore $ $($~$Q →$ ~$P) → (P→ Q)$ is valid argument. So we don’t even need to check 1st option.
Solution : $P→ Q \equiv$ ~$P \vee Q$
${\color{Green} {Option\space1} }$ : ~$Q →$ ~$P$ $\equiv Q$ $\vee$ ~$P$
Now, $ (Q$ $\vee$ ~$P)→$ $($~$P \vee Q)\equiv True$
${\color{Red} {Option\space2} }$ : $P \wedge Q → Q \equiv $ ~ $P$ $\vee$ ~$Q\vee Q\equiv True$
Now, $ True→ $ $($~$P \vee Q)\equiv\space ($~$P \vee Q) $
${\color{Red} {Option\space3} }$ : ~$P$ $\wedge$ ~$Q → P \equiv $ $P\vee Q\vee P\equiv P \vee Q$
Now, $ ( P \vee Q)→ $ $($~$P \vee Q)\equiv \space($~$P \vee Q) $
${\color{Green} {Option\space4} }$ :~$ Q → Q \equiv Q\vee Q\equiv Q$
Now, $ Q→$ $($~$P \vee Q) \equiv True$
${\color{Red} {Option\space5} }$ : $P$ $\wedge$ ~$ Q \wedge R→ $ ~$R \equiv $ ~$P$ $\vee$ $Q$ $\vee$ ~$R\vee R$ $\equiv True$
Now, $ True→$ $($~$P \vee Q)\equiv\space($~$P \vee Q)$
