Let $A$ be a nonempty set and let $P(x)$ and $Q(x)$ be open statements. Consider

the two statements $($iii$)$ EExinA,$[P(x{)}^{{?}^{?}}Q(x)]$ and $($iv$)$

$A,Q(x).$ Argue whether $($iii$)$ and $($iv$)$ are $($logically$)$ equivalent or not. $($Can you

explain your answer mathematically and by giving examples in plain language $?)$

If they are, why? If not, can you say at least which of the two always implies the

other? $($Can you explain your answer mathematically and by giving examples in

plain language $?)$