# Can you think of a counter example (real analysis)

• robertdeniro
In summary: If you think the first example will work can you show it does? I wouldn't worry about the 'points deleted' example. It's not at all clear to me what that's supposed to mean.
robertdeniro

## Homework Statement

X and Y are two closed non-empty subsets of R (real numbers).

define X+Y to be (x+y | x belongs to X and y belongs to Y)

give an example where X+Y is not closed

## The Attempt at a Solution

i tried X=all integers and Y=[0 1] but that didnt work out.
i know X+Y must be non-finite because all finite subsets are closed.

Well you simply want to define X+Y such that some points are deleted.:-)

╔(σ_σ)╝ said:
Well you simply want to define X+Y such that some points are deleted.:-)

ya i was thinking of that too. haven't found a way yet? can you give 1 more hint?

Well define one point in X to be negative the other point in Y.

╔(σ_σ)╝ said:
Well define one point in X to be negative the other point in Y.

hmm just to make sure you didnt misunderstand the definition of X+Y

if X=[1 3] and Y={-2}

then X+Y = [-1 1]

so I am still confused by your suggestion

My suggestion is a bit vague and not helpful but think about if it is possible to delete stuff from X+Y.

Think about isolated points. Sets all of whose points are isolated are closed, but by manipulating them, you may be able to produce a limit point in a convenient place.

thanks for all the tips, I am going to think about this, hopefully ill discover it soon

i still haven't figure this out yet...but i was wondering if the solution has a "hole" in an interval?

and I am still confused about the "deleting" suggestion. since a negative and a positive would only produce zero and not remove the element from the set. in essence, you can only stretch and shift an interval.

robertdeniro said:
i still haven't figure this out yet...but i was wondering if the solution has a "hole" in an interval?

and I am still confused about the "deleting" suggestion. since a negative and a positive would only produce zero and not remove the element from the set. in essence, you can only stretch and shift an interval.

Take r to be irrational. Did you know you can find whole multiples of r that are arbitrarily close to being integers?

i think if X=the integers and Y=sqrt(2) * the integers then it would work.

i still can't think of an example where points are deleted.

robertdeniro said:
i think if X=the integers and Y=sqrt(2) * the integers then it would work.

i still can't think of an example where points are deleted.

If you think the first example will work can you show it does? I wouldn't worry about the 'points deleted' example. It's not at all clear to me what that's supposed to mean.

## 1. Can you explain what a counter example is in real analysis?

A counter example in real analysis is a specific example that disproves a given statement or theorem. It is used to show that a general statement is not always true by providing a specific case in which it does not hold.

## 2. How do you come up with a counter example in real analysis?

To come up with a counter example, you first need to clearly understand the statement or theorem you are trying to disprove. Then, you can use your knowledge of mathematical concepts and techniques to construct a specific example that goes against the given statement.

## 3. Are counter examples always valid in real analysis?

No, not all counter examples are valid in real analysis. A counter example must be well-defined and follow the rules and principles of mathematics. If a counter example is found to be invalid, it means that the original statement or theorem holds true.

## 4. Can a counter example be used to prove a statement in real analysis?

No, a counter example is used to disprove a statement or theorem. It cannot be used to prove a statement, as a single example does not necessarily prove a general statement to be true.

## 5. Why are counter examples important in real analysis?

Counter examples are important in real analysis because they help to identify the limitations of a statement or theorem. They also encourage critical thinking and can lead to the discovery of new mathematical concepts and theories.

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