Can you think of a counter example (real analysis)

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.
  • #1
robertdeniro
39
0

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

Homework Equations


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.
 
Physics news on Phys.org
  • #2
Well you simply want to define X+Y such that some points are deleted.:-)
 
  • #3
╔(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?
 
  • #4
Well define one point in X to be negative the other point in Y.
 
  • #5
╔(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
 
  • #6
My suggestion is a bit vague and not helpful but think about if it is possible to delete stuff from X+Y.
 
  • #7
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.
 
  • #8
thanks for all the tips, I am going to think about this, hopefully ill discover it soon
 
  • #9
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.
 
  • #10
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?
 
  • #11
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.
 
  • #12
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.
 

Related to Can you think of a counter example (real analysis)

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.

Similar threads

  • Calculus and Beyond Homework Help
Replies
15
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
611
  • Calculus and Beyond Homework Help
Replies
3
Views
876
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
596
  • Calculus and Beyond Homework Help
Replies
3
Views
893
  • Calculus and Beyond Homework Help
Replies
3
Views
433
  • Calculus and Beyond Homework Help
Replies
7
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
10K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
Back
Top