Is the following correct? (concerns sets and convergence)

  • Thread starter RVP91
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  • #1
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Let A = {(x,y) in R^2 | x^2 + y^2 <= 81}
Let B = {((x,y) in R^2 | (x-10)^2 + (y-10)^2 <= 1}

then here "A intersection B" is the empty set.
Then let x_n be the sequence (0,10-(2/n)) which is a sequence in A and y_n be the sequence (10/n,10) which is a sequence in B.
would |x_n - y_n| tend to 0 in this case?
 

Answers and Replies

  • #2
jgens
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Why do you think xn is a sequence in A? It is not.
 
  • #3
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Is it not? Isn't a sequence in A that converges to a point in A compliment?
 
  • #4
jgens
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A sequence in A is a sequence all of whose values are in A. Almost all of the values of xn are not in A.
 
  • #5
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Is it not? Isn't a sequence in A that converges to a point in A compliment?
A is a disk with radius 9. The sequence [itex]x_n[/itex] converges to (0,10). So the sequence is not in A.
 
  • #6
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But can't a sequence x_n [itex]\subseteq[/itex] S converge to a point in S compliment?
 
  • #7
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But can't a sequence x_n [itex]\subseteq[/itex] S converge to a point in S compliment?
Not if S is closed, no.
 
  • #8
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Oh right I see. Thank you.

Can there exist a pair of non-empty closed sets A,B [itex]\subseteq[/itex] R^2 with A [itex]\cap[/itex] B equals the empty set with a pair of sequences x_n [itex]\subseteq[/itex] A and y_n [itex]\subseteq[/itex] B such that ||x_n - y_n|| -> 0.

I was trying to find a specific example and hence my original thought was my first attempt.
 
  • #9
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Yes, such a thing exists. Take

[tex]A=\{(x,0)~\vert~x\in \mathbb{R}\}[/tex]

and

[tex]B=\{(x,1/x)~\vert~x\in \mathbb{R}\}[/tex]

(Note the distance between these two sets is 0). Can you find suitable sequences in these sets?
 
  • #10
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Could you explain why the distance between the two sets is 0 please. I thought it was because d|x-y| = (x,0) - (x,1/x) = (0, -1/x) but wasn't sure how this means the distance is 0.

Also I'm finding it hard to grasp the concept of sequences in 2 dimension in the sense of how to represent them but my guess would be,
x_n = (3n, 0) and y_n = (3n, 1/3n) ?
 
  • #11
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The distance between two sets is usually defined as the greater lower bound of the set of distances between 2 points in the sets. In symbols:
[tex]
d(A,B) = \inf \{d(a,b) : a \in A, b \in B\} \, .
[/tex]
Take a look at this: http://en.wikipedia.org/wiki/Distance#Distances_between_sets_and_between_a_point_and_a_set

So you were right that [itex] d(a,b) = |a-b| = |(x,0) - (x, 1/x)| = |(0,-1/x)| = 1/|x| [/itex]. But now you need to find the greatest lower bound of the set [itex] \{d(a,b) : a \in A, b \in B\} [/itex]. Do you see why it will be zero?
 
  • #12
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Is it because as x gets smaller and tends to -infinity but you take the modulus and so u get limn-> infinity 1/x = 0?

Also were my choices of sequences okay? That is the main issue I am stuck with? I need x_n in A and y_n in B such that ||x_n - y_n|| tends to 0.
I think x_n=(3n,0) and y_n=(3n/1/3n) would work right?
 
  • #13
22,089
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Yes, that choice of sequence is a good one!
 

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