Formalise the following statement in first-order logic (FOL)

Question

The following is the statement of the problem and my approach also. Can any one guide me on whether have I done it correctly. if something is wrong then please correct me. Thank you

Problem: Formalise the following statement in first-order logic (FOL), making clear what your atomic predicate symbols stand for and what domains of any variable are:

1. "Not all the students in the discrete mathematics class are smarter than everybody in the linear algebra class."
2. Is there an equivalent formula to represent the statement? If yes, then what is it?
3. What are free and bound variables in your formula for part(1)

My calculations have led to the following answers:

1. Let $S(x):$ The Students in the discrete mathematics class are smarter than those in linear algebra class then, the answer to first part is: $\neg\forall x\in S(x)$
2. $\neg\forall x\in S(x) \equiv \exists x\in \neg S(x) $
3. $\neg\forall x\in S(x)$ is the only variable and it is bounded.