# Prove the diameter of a union of sets is finite

• chipotleaway
In summary: If x and y are elements of the union then ##x \in S_i## and ##y \in S_j## for some ##i \in I## and ##j \in I##. Start like that. Now say why you can find a z that is in both sets. Your inequality is fine. Just fill in some more words about what belongs to what.If x and y are elements of the union, then z must be an element of both sets.
chipotleaway

## Homework Statement

Prove that the union of a collection of indexed sets has finite diameter if the intersection of the collection is non-empty, and every set in the collection is bounded by a constant A.

## The Attempt at a Solution

The picture I have is if they all intersect (and assuming there are infinitely many sets), then all the sets must be 'localized' in some way so that there's some overlap. I'm not yet sure how to make this rigorous though.

chipotleaway said:

## Homework Statement

Prove that the union of a collection of indexed sets has finite diameter if the intersection of the collection is non-empty, and every set in the collection is bounded by a constant A.

## The Attempt at a Solution

The picture I have is if they all intersect (and assuming there are infinitely many sets), then all the sets must be 'localized' in some way so that there's some overlap. I'm not yet sure how to make this rigorous though.

Use the triangle inequality to make a rigorous proof.

1 person
I have the diameter of the union of sets is less than or equal to the sum of the diameters of each set, and that's less than the sum of M (summed over each alpha). How might one go about working in the fact they all intersect?

(If two sets have a maximum diameter of M, and intersect, then I would expect the diameter of the union to be strictly less than the 2M, but if there are infinite sets...)

EDIT: Oh hang on I think I have an idea - x and y in the diameter of the union have to be in either one set or two sets. If in two sets, then since they intersect, then it's less than 2M

Prove these things you 'expect' to be true using the triangle inequality. And there is at least one point, z, that is contained in every set of the collection.

1 person
Letting d(x,y) be the diameter of the union of the sets, if x,y are in different sets, then d(x,y)≤d(x,z)+d(z,y)≤2M since every set contains z and the diameter of every set is bounded by M. Therefore d(x,y) is finite since 2M is finite.

If x,y are in the same set, then it would be bounded by the diameter of just one set, which is bounded by M.

--

I'm a little unsure about using d(x,y) as the diameter of the union of sets and assuming that they're in the sets - should I let the diameter be d(z,w) where z,w are just points in the containing metric space X (which may or may not be in the subsets we're looking)?

chipotleaway said:
Letting d(x,y) be the diameter of the union of the sets, if x,y are in different sets, then d(x,y)≤d(x,z)+d(z,y)≤2M since every set contains z and the diameter of every set is bounded by M. Therefore d(x,y) is finite since 2M is finite.

If x,y are in the same set, then it would be bounded by the diameter of just one set, which is bounded by M.

--

I'm a little unsure about using d(x,y) as the diameter of the union of sets and assuming that they're in the sets - should I let the diameter be d(z,w) where z,w are just points in the containing metric space X (which may or may not be in the subsets we're looking)?

Call your indexed sets ##S_i## where ##i \in I##. Then if x and y are elements of the union then ##x \in S_i## and ##y \in S_j## for some ##i \in I## and ##j \in I##. Start like that. Now say why you can find a z that is in both sets. Your inequality is fine. Just fill in some more words about what belongs to what.

1 person

## 1. What does it mean for a diameter of a union of sets to be finite?

A diameter of a union of sets refers to the largest distance between any two points in the union of sets. When this distance is finite, it means that there is a maximum distance between any two points in the union of sets.

## 2. How is the diameter of a union of sets calculated?

The diameter of a union of sets can be calculated by finding the maximum distance between any two points in the union of sets. This can be done by finding the distance between each pair of points and then selecting the largest value.

## 3. Why is it important to prove that the diameter of a union of sets is finite?

Proving that the diameter of a union of sets is finite is important because it helps us understand the size and structure of the union of sets. It also allows us to make predictions and draw conclusions about the behavior of the union of sets.

## 4. What are some applications of proving the diameter of a union of sets is finite?

Proving the diameter of a union of sets is finite has various applications in fields such as mathematics, physics, and engineering. For example, it can be used to analyze the convergence of sequences, study the behavior of physical systems, and optimize the design of structures.

## 5. Can the diameter of a union of sets ever be infinite?

No, the diameter of a union of sets cannot be infinite. By definition, the diameter is the largest distance between any two points in the union of sets, and infinite distance is not a meaningful concept. Therefore, the diameter of a union of sets is always finite.

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