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Restriction with Multiple Quantifiers

  1. Oct 7, 2015 #1
    1. The problem statement, all variables and given/known data
    In the questions below suppose the variable x represents students and y represents courses, and: U(y): y is an upper-level course M(y): y is a math course F(x): x is a freshman A(x): x is a part-time student T(x,y): student x is taking course y. Write the statement using these predicates and any needed quantifiers.

    1. There is a part-time student who is not taking any math course. Answer: ∃x∀y[A(x) ∧ (M(y) → ¬T(x,y))].

    2. Every part-time freshman is taking some upper-level course. Answer: ∀x∃y[(F(x) ∧ A(x)) → (U(y) ∧ T(x,y))].

    2. Relevant equations

    3. The attempt at a solution
    I'm having trouble rewriting statements using predicates and quantifiers. I can't seem to find much information on restricted quantifiers especially with multiple quantifiers.

    It seems like the whole purpose of restricted quantifiers is to minimize the domain when rewriting the statements, but I can't seem to translate it well.

    I believe I understand the first one:
    There exists a student x, that for all courses y, if x is a part time student and y is a math course, then x is not taking course y. So the "If" portion minimizes the domain.

    The second one is where I get lost. I thought the answer would be:
    ∀x∃y[(F(x) ∧ A(x)) ∧ U(y) → T(x,y)]
    For all students x, there exists a course y such that if x is a freshmen and x is a part-time student and y is a upper-level course, then x is taking y. I don't understand why this is not correct. The "if" restricts the domain to part-time, freshmen and upper level courses.
  2. jcsd
  3. Oct 7, 2015 #2


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    The statement you wrote would be correct, by vacuous implication, if there is an upper-level course that is not taken by any part-time freshman. To see this, consider the simplest case where the only student is a part-time freshman, the only course is upper-level, and the student doesn't take any courses. That satisfies your expression, but not the original statement or the answer expression.
  4. Oct 8, 2015 #3


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    For the second statement, your qualifier on the domain is referring to x, not y. y should be moved to the right.
    Essentially, you want to say that for all x such that x is a part-time freshman, ... something about y.
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