Proving: Neighborhoods of x and y Have Empty Intersection

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In summary, the problem is asking to prove that there exists a neighborhood of x and a neighborhood of y such that their intersection is empty. This can be visually represented as two separate brackets, one around x and one around y, without touching each other. The sets (x-r, x+r) and (y-s, y+s) can be used as neighborhoods with a radius of 1/3 the distance between x and y, but further steps are needed to turn this into a rigorous argument.
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Trung
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Homework Statement


Let x and y be real numbers. Prove there is a neighborhood P of x and a neighborhood Q of y such that P intersection Q is the empty set.


Homework Equations





The Attempt at a Solution



Sorry, I know this is elementary to many of you, but I am just starting out in this course and I need some hints on how to get started.
 
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  • #2
It is asking "can you fit a pair of brackets around x and another pair of brackets around y such that the two pairs of brackets do not touch each other?" Remember, you are the one choosing how tight the first pair of brackets (around x) as well as how tight the second pair (around y). (You can make them as tight as you want.)
 
  • #3
Pictorially it would be something like this:

<---------------(---x---)---------(-----y-----)--------------->

...but this diagram does not constitute a proof. I do not know how to make it into a rigorous argument. I have the sets (x-r, x+r) and (y-s, y+s) but I don't know what to do with them.
 
  • #4
If x and y are different then they have some non-zero distance between them. Think about neighborhoods of x and y with radius equal to 1/3 that distance.
 

Related to Proving: Neighborhoods of x and y Have Empty Intersection

1. How do you prove that neighborhoods of x and y have empty intersection?

To prove that neighborhoods of x and y have empty intersection, you can use the definition of a neighborhood and the properties of real numbers. Start by assuming that there is a point that is in both neighborhoods. Then, using the definition of a neighborhood, show that this assumption leads to a contradiction. This contradiction proves that the assumption was false, and therefore, the neighborhoods must have an empty intersection.

2. What is a neighborhood in mathematics?

In mathematics, a neighborhood is a set of points that are close to a given point. It can be visualized as a small region around the point. The exact definition of a neighborhood may vary depending on the context, but it generally includes all points within a certain distance from the given point.

3. Can neighborhoods of x and y have non-empty intersection?

No, if the neighborhoods of x and y have a non-empty intersection, then there must be a point that is in both neighborhoods. This contradicts the definition of a neighborhood, which states that a neighborhood of a point does not include the point itself. Therefore, neighborhoods of x and y cannot have a non-empty intersection.

4. What is the purpose of proving that neighborhoods of x and y have empty intersection?

The purpose of proving that neighborhoods of x and y have empty intersection is to show that the two points are not close to each other. This can be useful in many situations, such as in topology, where the concept of neighborhoods is used to define the continuity of functions. It can also be used in geometry to prove that two objects do not intersect or touch each other.

5. Are there any other methods to prove that neighborhoods of x and y have empty intersection?

Yes, there are other methods to prove that neighborhoods of x and y have empty intersection, such as using proof by contradiction or using the contrapositive of the statement. However, the most common and straightforward method is to use the definition of a neighborhood and properties of real numbers to show that the assumption of a non-empty intersection leads to a contradiction, thus proving that the assumption was false.

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