Trying express,"Someone in your school has visited Japan."
I'm wondering why the given answer is:$\exists x(A(x) \land P(x))$,where A(x) denotes,"x is in your school" and P(x) denotes,"x has visited Japan,domain of x consists of all people in the world;
But not $\exists x(A(x) \rightarrow P(x))$? Since if there's someone not in your school,he's visited Japan could be whether true or false.
Suppose that no one in your school has visited Japan, but somewhere there is some person $a$ who has visited Japan. Then $P(a)$ is true, so $A(a)\to P(a)$ is true, and therefore $\exists x\big(A(x)\to P(x)\big)$ is true, even though no one in your school has visited Japan. Thus, the expression $\exists x\big(A(x)\to P(x)\big)$ definitely does not capture the English sentence. (In fact it really doesn’t matter whether $a$ has visited Japan, since the implication $A(a)\to P(a)$ is automatically true whenever $A(a)$ is false, i.e., whenever $a$ is not in your school.)
The expression $\exists x\big(A(x)\land P(x)\big)$, however, does: it says that there is someone who has visited Japan and is in your school, which is exactly the semantic content of the sentence Someone in your school has visited Japan.