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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 $\forall y Q(y)$?

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    To get $\forall$ you can enclose \forall in dollar signs.2011-05-31

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For your first question I think your first expression, that is $\exists x P(x)$ would suffice. Your second candidate is a bit pointless unless you can express the relation between students and classrooms (that is a student belongs to a classroom). If you can express it, then maybe it will be considered a nice addition but I don't think it's required.

For your second question what you wrote is sufficient.

P.S. The command for "for all" is "\forall".