# Distance between two sets A and B in Rn

• waternight
In summary: I hope this helps to summarize the main points of the conversation. In summary, we discussed the definition of distance between two sets, defined the largest class of sets for which this definition works, proved that the definition is well-defined, and explored another possible definition of distance which works for any two sets in Rn. We also checked that this second definition gives a reasonable answer for the distance between an open ball and its boundary. I hope this helps to clarify the concept for you. Let me know if you have any further questions.
waternight

## Homework Statement

A is a sequentially compact set, and B is a point ⃗v in Rn−A.
deﬁne the distance between A and B as dist( ⃗u0 , ⃗v), where you showed
that ⃗u0∈ A exists such that dist( ⃗u0 , ⃗v)≤dist(⃗u, ⃗v) for all ⃗u in A.
a) Use this example to state a deﬁnition of the distance between two sets in general,
giving the largest class of sets for which your deﬁnition works.
b) Prove that your deﬁnition is well deﬁned on the class of sets for which you say it
applies in a), and give counterexamples showing you can’t weaken the conditions in your
deﬁnition.
c) think of another deﬁnition of distance which applies to any two sets in Rn . Check that this second deﬁnition gives a reasonable answer for the distance between an open ball and its boundary. If you restrict this second deﬁnition to the class of sets considered in a) you should get the same answer as a).

## The Attempt at a Solution

I don't have any clue of writing the definition and the proof. What does the largest class of sets mean?

Can someone help me with this?

To start off, let's define what we mean by "distance between two sets". If we have two sets, A and B, in Rn, we can define the distance between them as the shortest distance between any point in A and any point in B. In other words, the distance between A and B is the minimum distance between any two points, one in A and one in B.

Now, to answer part a) of your question, the largest class of sets for which this definition works is the class of all non-empty sets in Rn. This is because any two non-empty sets in Rn will have at least one point in common, and therefore the distance between them will be well-defined.

For part b), we need to prove that this definition is well-defined on the class of all non-empty sets in Rn. To do this, we need to show that the distance between two sets, A and B, is the same regardless of which points we choose in A and B. To do this, we can assume that there are two points, ⃗u1 and ⃗u2, in A and two points, ⃗v1 and ⃗v2, in B such that dist(⃗u1, ⃗v1) < dist(⃗u2, ⃗v2). We can then show that this leads to a contradiction, which proves that our definition is well-defined.

For part c), another possible definition of distance between two sets is the Hausdorff distance. This is defined as the maximum distance between any two points, one in each set. This definition also works for any two sets in Rn, and when applied to the class of all non-empty sets, it gives the same answer as the definition in part a).

To check that this second definition gives a reasonable answer for the distance between an open ball and its boundary, we can consider the example of a 2-dimensional open ball with radius r and its boundary, which is a circle with radius r. Using the Hausdorff distance, we can see that the maximum distance between any two points, one in the ball and one on the boundary, is r. This makes sense, as the boundary is at a distance of r

## 1. What does the "distance between two sets" mean?

The distance between two sets A and B in Rn is a measure of the minimum distance between any two points, one from set A and one from set B. It tells us how far apart the two sets are from each other.

## 2. How is the distance between two sets calculated?

The distance between two sets A and B in Rn is calculated by finding the distance between every point in set A and every point in set B, and then taking the minimum of these distances. This can be done using a variety of formulas, such as the Euclidean distance formula or the Manhattan distance formula.

## 3. Can the distance between two sets ever be negative?

No, the distance between two sets A and B in Rn is always a positive value. This is because distance is a measure of how far apart two sets are, and distance cannot be negative. If the two sets are overlapping, the distance between them would be considered 0.

## 4. What does it mean if the distance between two sets is 0?

If the distance between two sets A and B in Rn is 0, it means that the two sets are either identical or they are overlapping. This means that every point in set A is also in set B, and vice versa.

## 5. How is the distance between two sets used in real-world applications?

The distance between two sets in Rn is often used in data analysis and machine learning algorithms. It can also be used in optimization problems, such as finding the shortest distance between two locations or finding the closest match between two sets of data. Additionally, it can be used in image recognition and pattern recognition tasks.

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