Predicate Logic with Nobody

Question

Nobody is a college student and not smart. How does this translate into predicate logic? I am stuck between two:

¬∃x[person’(x) ∧ student’(x) ∧ ¬smart’(x)]

¬∃x[[person’(x) ∧ student’(x)] → ¬smart’(x)]

Thanks!

Answer 1

The first translation is the accurate one of the original English statement.

The second "translation" is actually false (even assuming the English statement is true): consider an $x$ that is not a person, or an $x$ who is not a student, or an $x$ who is a smart student—all of these are examples showing that the negated-existential is false.