If Las Vegas is the capital of Fiji, then $x^2=4$. If Las Vegas is the capital of Fiji, then $x^2=4$.


I was asked to state either the above claim is true or false. I must give a proof if it is true and counter example if it is false.

I prove its contrapositive: if not $q$ then not $p$ in the truth value table which  is true always and is false only when $q$ or conclusion is false. 

So since $x^2=4$ is false because the value of $x$ is unknown.
I say the claim is false.
Am I correct?
This is a vacuous truth, since the condition "Las vegas is capital of Fiji" is never satisfied, so for some $x$, this is equivalent to
$$ \text{False} \implies \text{True}
$$
and for some $x$, this is
$$ \text{False} \implies \text{False}
$$
Both statements are true, so this is true.
This is vacuously true by definition of material implication:
https://en.wikipedia.org/wiki/Material_implication_(rule_of_inference)

It is true because the law of the excluded middle demands implication to have a truthy or falsy value. The truth table of implication is such that if P is false, then regardless of the validity of Q, the whole statement is true. Vacuous truths, however, are not of much interest beyond this.