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Double Quantifiers and Sets

  1. Feb 27, 2014 #1
    1. The problem statement, all variables and given/known data

    Determine if the sets A, B, C, and D satisfy the following proposition (p) for the set S:

    2. Relevant equations

    p: for all ε > 0, ∃ x [itex]\in[/itex] S such that x < ε

    A = {1/n : n [itex]\in[/itex] Z+}
    B = {n : n ε Z+}
    C = A [itex]\cup[/itex] B
    D = {-1}

    3. The attempt at a solution

    I am looking for a little help in reading/interpreting the mathematical statements with double quantifiers and "ε," I have always had terrible trouble understanding what these mean (I have never quite understood the formal definition of a limit for example).

    Let me attempt to explain how I understand proposition p, for example, and how it would relate to set A, B, C, and D. Then please tell me if I am off the mark.

    For p, after I read it a few times I interpreted it as a condition requiring the set to contain at least one element, x, that is less than some number ε, where that number ε can be made arbitrarily close to 0. The only set I could visualize that would allow for this is a set that contains elements which get arbitrarily close to 0 (if the set does not contain negative numbers), or a set which contains negative numbers.

    A satisfies this condition because the members of its set approach 0 as n approaches infinity. So, no matter what epsilon you choose, you can always find a smaller x value in set A

    B does not satisfy this condition, because you can choose 0 < ε < 1, but the elements of this set are restricted to whole numbers, and therefore all elements in set B [itex]\geq[/itex] 1

    A [itex]\cup[/itex] B satisfy p b/c it includes set A, the elements of which approach 0 as n approaches infinity (i.e., get infinitely close to 0).

    Finally set D trivially satisfies condition p b/c condition p restricts the choice of ε > 0, and the only element of set D is < 0
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Feb 27, 2014 #2


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    That sounds just fine.
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