(0,0) Empty Interval? Basic Set Theory

[teroenza]

Homework Statement

Give an example of an indexed family of sets such that the intersection of any finite subfamily is not empty, but the intersection when the index=infinity, is empty.

The Attempt at a Solution

The family I came up with is the exclusive interval (-1/k , 1/k) where k is the index (natural numbers) which index from 0 to infinity. I believe that the intersection of any finite subfamily is not empty. When I get to infinity, I get (0,0). This is an empty interval correct? Because I have squeezed the intersection to zero, but my interval is exclusive.

Thank you

 

[vela]

Isn't 0 in every interval (-1/k, 1/k)?

 

[alanlu]

I agree with vela. This isn't it. However, it is close to an example...

You don't really need both sides, right? Just consider the positive half of your family...

 

[teroenza]

Ok. I understand that -1/k was redundant, and that 0 is in the successive intervals. But I don't understand what the exclusive interval from 0 to 0 means, even if it not the solution I seek.

 

[alanlu]

The original family of intervals produces an intersection of {0}, as the element 0 is in every set of the form (-1/k, 1/k), but for every real number r > 0, there is a k such that 1/k < r, so r is not contained in the interval (-1/k, 1/k), and a similar argument holds for r < 0.

(a, b) is the set of all reals x such that a < x < b. Therefore, by definition, (0, 0) = {} the empty set (this is a valid real interval). However, {} is not the intersection of the family of intervals (-1/k, 1/k), for the reason vela noted.

 

[teroenza]

If the family is (0, 1/k) that would still produce (0,0), and every interval would not include zero. The finite values of k would produce a non-empty intersection.

 

[alanlu]

Yes.

I would avoid using (0, 0) and use {}. As you can see, (0, 0) = (x, x), so it's not very descriptive... and you might not get taken seriously if you use it later.

Also, if the family is (-1/k, 1/k), it would produce {0}, not {}. These are two different sets.

 

[teroenza]

Thank you. Along those same lines, would (1, 1/k+1) producing (1,1) i.e. {} as k indexes to infinity be another example? It would have the same sort of shrinking interval, just shifted to the the right (positive) side of zero.

 

[vela]

As alanlu noted, the notation (x, x) isn't meaningful. Don't use it.

 

[alanlu]

All your sets in your example are empty. You may be looking for (1, 1 + 1/k).

Anyway, the intersection of the family of (1, 1 + 1/k) is {}.

 

[teroenza]

Thank you both.

 

[teroenza]

My teacher says that I am 1/2 right in providing the example. Now I must justify it. To justify the non-empty intersection part I will say that "as n increases the interval of intersection decreases in length, but still contains infinitely many points of intersection."

He said that is was incorrect, however, to say "at n=infinity, the family_n=infinity is ={}. He said it was like just adding in a foreign piece of information.

I believe that the intersection of an infinite number of sets to be empty, means that there can be no element common to all of them. But if there are an infinite number of points in an interval, no matter how small...

This is a second year class. I just can't seem to justify my answer.

 

[alanlu]

I apologize for my imprecise language. Correction: "Also, if the family is (-1/k, 1/k), the intersection of all elements of the family would be {0}, not {}. These are two different sets."

Try fixing an element r in ℝ. Which sets would r have to belong in if it were to belong in the intersection of a family of sets?

 

[vela]

He said that is was incorrect, however, to say "at n=infinity, the family_n=infinity is ={}. He said it was like just adding in a foreign piece of information.

If you only state that, you're simply asserting what you're trying to prove.

I believe that the intersection of an infinite number of sets to be empty, means that there can be no element common to all of them. But if there are an infinite number of points in an interval, no matter how small...

This is an example of the non-intuitive things that crops up when infinity is involved. If you look at any finite intersection of those intervals, it will always contain an infinite number of points, but the intersection of an infinite number somehow ends up empty.

 

[SteveL27]

My teacher says that I am 1/2 right in providing the example. Now I must justify it. To justify the non-empty intersection part I will say that "as n increases the interval of intersection decreases in length, but still contains infinitely many points of intersection."

He said that is was incorrect, however, to say "at n=infinity, the family_n=infinity is ={}. He said it was like just adding in a foreign piece of information.

I believe that the intersection of an infinite number of sets to be empty, means that there can be no element common to all of them. But if there are an infinite number of points in an interval, no matter how small...

This is a second year class. I just can't seem to justify my answer.

Go back to the definition of intersection. What is the intersection of a family of sets? Can you show that the intersection of your family of sets is empty, but that the intersection of finitely many of the sets in your family is non-empty? Go directly to the definition of intersection. Is that too vague a hint? There's a general technique in doing formal proofs. Whenever you're stuck, go directly to the exact textbook definition and show that the definition is satisfied (or not) by the particular objects in question. Forget about "decreasing lengths" or whatever. This is a question about set intersection and they're trying to get you to think formally about set intersections.

 

[teroenza]

After speaking with him again I have this. A finite intersection is nonempty because for any intersection of arbitrary subfamilies from my indexed family there are common points. However, when I have an infinite number of subfamilies I can always find one which does not contain an arbitrarily chosen element, this making the intersection empty.

 

[Deveno]

After speaking with him again I have this. A finite intersection is nonempty because for any intersection of arbitrary subfamilies from my indexed family there are common points. However, when I have an infinite number of subfamilies I can always find one which does not contain an arbitrarily chosen element, this making the intersection empty.

if A is a subset of R with this property:

for all x in R, x is not in A, then A is the empty set.

that is: ∅ = R - R.

this gives a useful way to show a subset of R is actually the empty set: show that no real number is in it!

 

[SteveL27]

After speaking with him again I have this. A finite intersection is nonempty because for any intersection of arbitrary subfamilies from my indexed family there are common points. However, when I have an infinite number of subfamilies I can always find one which does not contain an arbitrarily chosen element, this making the intersection empty.

Yes but all you've done is give the definition of intersection. In other words, you say, "A finite intersection is nonempty because for any intersection of arbitrary subfamilies from my indexed family there are common points." That's the definition of what it is that you have to prove. (Assuming you meant finite subfamilies, not "arbitrary" subfamilies as you wrote).

So if I was the grader here, I would be looking for specifics. What I'd like to see is:

* A clear statement of exactly what your claimed family of sets is.

* Then prove that the intersection of your family is empty.

* Then prove that the intersection of any finite number of sets in your family is nonempty.