Truth value of a statement I have a problem:
Let P be the statement " x $\in$ A and x $\in$ $\mathbb{Z}$ "
statement: ($\forall$x)P $\Longrightarrow$ ($\exists$x)P
Is there a set A for which the truth value of the above statement is false? Explain.
My approach is:
The statement is only false when the antecedent is true and the consequent is false.
This is not possible because ...
Can someone help, please?
NOTE: Please forgive me : our assumption is that x is in the universal set R
 A: Yes, when $A \cap \mathbb Z$ is the empty set.
Then there are no such $x$ and so $(\forall x) P$ is true vacuously.
But as there exist no such $x$, there can exist no $x$ such that $P$ holds, so $(\exists x)P$ is false.
A: In first-order logic as presented in standard textbooks and standard introductory courses, the universe of discourse (i.e., the set of things that we talk about in our logical language) is non-empty. $\forall x\,P(x)$ means that $P(x)$ holds for every element of the universe and, as that universe is non-empty, $\exists x\,P(x)$ must hold if $\forall x\,P(x)$ does. So $(\forall x\,P(x)) \Rightarrow (\exists x\,P(x))$ is valid, for any predicate $P(x)$. In particular, $(\forall x\,(x \in A \land x \in \Bbb{Z})) \Rightarrow (\exists x\,(x \in A \land x \in \Bbb{Z}))$ is valid for any set $A$. So you correct in concluding that there is no set $A$ that makes this proposition false.
To address Prime Mover's concerns: if we are working in a first-order language that can talk about sets and set membership, then it is convenient to use relative quantifiers: i.e., to write $\forall x \in A\,Q(x)$ as an abbreviation for $\forall x \, (x \in A \Rightarrow Q(x))$ and to write $\exists x \in A, Q(x)$ as an abbreviation  for $\exists x(x \in A \land Q(x))$. Using this short-hand $\forall x \in \emptyset\,Q(x))$ is true and $\exists x \in \emptyset\,Q(x)$ is false, but this does not contradict the statement that $(\forall x\,P(x)) \Rightarrow (\exists x\, P(x))$, since when we expand the abbreviations, the matrix (body) of the two quantified subformulas are not the same.
Finally, I should say that there are logics that do not assume that quantification ranges over non-empty sets. Tennant's Natural Logic is one such system. Martin-Löf type theory is another. Such logics are, in some sense, more precise than traditional first-order logic, but the extra precision has to be paid for by more onerous proof obligations.
