Quantification to english?

Is my answer to question 1 correct or is the double ∃ mean each student has taken each CS class?

Let P(x, y) be the statement “Student x has taken class y,” where the domain for x consists of all students in your class and for y consists of all computer science courses at your school. Express each of these quantifications in English.

1) ∃x∃yP(x, y)
2) ∃x∀yP(x, y)

1) There exists a student that has taken one CS class.
2) There exists a student that has taken all CS classes.

Edit: New answers based on suggestions. 1) At least one student has taken at least one CS class. 2) At least one student has taken all CS classes.

• You wrote up there "each student has taken one each CS class. It is either one or each, or each one of the CS classes.
– Pedro
Jul 6, 2012 at 4:36
• Technical warning: "has taken one CS class" could be understood to mean "taken precisely one CS class." Instead you may want to specify e.g. "at least one CS class." Other than that, looks good.
– anon
Jul 6, 2012 at 4:53
• I'd translate $\exists x:$ at least one student (or, some students). Similarly for $\exists y:$ some classes.
– user2468
Jul 6, 2012 at 4:55
• I tried to do something to remember what ∃ meant and that was linking it to exists which probably is not the most correct thing. Is at least one a better term for ∃? Jul 6, 2012 at 4:59
• BTW, I removed the tag (propositional-calculus), because $P()$ is a predict, and this is predicate calculus.
– user2468
Jul 6, 2012 at 4:59