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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.

  • 0
    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*.2012-07-06
  • 1
    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.2012-07-06
  • 1
    I'd translate $\exists x:$ at least one student (or, *some* students). Similarly for $\exists y:$ some classes.2012-07-06
  • 0
    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 ∃?2012-07-06
  • 1
    BTW, I removed the tag `(propositional-calculus)`, because $P()$ is a predict, and this is predicate calculus.2012-07-06
  • 0
    Thanks I'm always confused as what I should tag some of these with. Oh also if you guys want to put your suggestions as an answer then I could accept it.2012-07-06

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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.