I have to equations: P(x)=student x knows calculus, and Q(y)=class y contains a student who knows calculus.
For something like "Some students know calculus" would it be enough to write $\exists x P(x)$, or would a better way be $\exists y \exists x Q(y) \wedge P(x)$?
Similarly, to express "every class has a student in it who knows calculus", would it be sufficient to write $\forall y Q(y)$?