# Predicate Logic - Why is this not sufficient?

Let N be the set on Natural numbers. Let S={2,6,14} and T={6,8,11}.

Question: The intersection of S and T is nonempty

$∃ x : x ∈ S. ∃ y : y ∈ T. x=y$

$∃ x : x ∈ S ∧ x ∈ T$

Is my answer a sufficient way of saying the value x is in the intersection of S and T?

If not, why does the correct answer refer to two variables, x and y? Can x only represent the values in one set, even when they intersect?

• Your answer is correct. – MJD Nov 17 '16 at 14:04
• Note that the "correct" answer only quantifies over defined sets, whereas your answer quantifies over the entire universe. Depending on your text, unrestricted quantification may not be allowed. – DanielV Nov 17 '16 at 23:42

$\exists x (x \in S \land x \in T)$
• I thnk the provided answer has no parenthesis since it requires no parentheses. Think of it as $\exists x \in S \: \exists y \in T x = y$. Some syntactical notations are fine with that. But I have not seen any notation that is fine with $\exists x x \in S \land x \in T$ – Bram28 Nov 17 '16 at 14:09