Understanding Double Quantifiers and Sets with Epsilon

[eclayj]

Homework Statement

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

Homework Equations

p: for all ε > 0, ∃ x $\in$ S such that x < ε


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

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 $\geq$ 1

A $\cup$ 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

 

[Dick]

Homework Statement

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

Homework Equations

p: for all ε > 0, ∃ x $\in$ S such that x < ε


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

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 $\geq$ 1

A $\cup$ 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

That sounds just fine.