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Show that the set S is Closed but not Compact

  • #1

Homework Statement



Show that the set S of all (x,y) ∈ ℝ2such that 2x2+xy+y2
is closed but not compact.

Homework Equations


set S of all (x,y) ∈ ℝ2such that 2x2+xy+y2

The Attempt at a Solution


I set x = 0 and then y = 0
giving me
[0,±√3] and [±√3,0] which means it is closed

However, for it to be Compact, it needs to be Closed & Bounded.
For it to be bounded, it must have both an upper and lower bound, which to me it appears to have?
The bound for me are when x and y = ±√3

Clearly I am wrong, given how the question is structured. Any ideas?
 

Answers and Replies

  • #2
You'll have to be more clear o. What exactly your set S is. ##{(x,y) \in R^2 : 2x^2+xy+y^2}## is what you've defined. But, I can plug any points x, y into that since you've not told us what the equation must satisfy.
 
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  • #3
Most correct, my omission.
The function equals 3.
 
  • #4
OK, so if the function must equal 3, then x=0 and y=0 is not even in the set. Your definition of compact is closed and bounded. So you need to show that the set is closed but not bounded.
 
  • #5
Ray Vickson
Science Advisor
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Most correct, my omission.
The function equals 3.
Are you saying that your ##(x,y)## must satisfy ##2x^{2+xy+y^2} = 3##? That is the function you wrote. Or, did you mean ##2 x^2 + xy + y^2 = 3##?
 
Last edited:
  • #6
Sorry, I responded on my phone so I'm not sure how the LaTex parts turned out.
 
  • #7
I need to apologise to everyone . . . immediately after I posted, I needed to run to lectures . . . I never noticed the error in my post, and wasn't able to correct it from my mobile phone. Really did not mean to waste your time. Stated correctly, it should read as:

Show that the set S of all (x,y) ∈ ℝ2 such that x2+xy+y2 = 3
is closed but not compact.

(Have been trying to correct the original post, but doesn't seem possible?)
 
  • #8
Are you saying that your ##(x,y)## must satisfy ##2x^{2+xy+y^2} = 3##? That is the function you wrote. Or, did you mean ##2 x^2 + xy + y^2 = 3##?
Please see my apology below.
You are correct, it should read:

Show that the set S of all (x,y) ∈ ℝ2 such that x2+xy+y2 = 3
is closed but not compact.
 
  • #9
OK, so if the function must equal 3, then x=0 and y=0 is not even in the set. Your definition of compact is closed and bounded. So you need to show that the set is closed but not bounded.
Yes, which I think I did.
I tried to show it was closed, but don't know how to show that it is bounded or not, as the case may be?
 
  • #10
My original post contains errors and I cannot see how to correct it. So here is the corrected post:

1. Homework Statement


Show that the set S of all (x,y) ∈ ℝ2such that x2+xy+y2=3
is closed but not compact.

Homework Equations


set S of all (x,y) ∈ ℝ2such that x2+xy+y2=3

The Attempt at a Solution


I set x = 0 and then y = 0
giving me
[0,±√3] and [±√3,0] which means it is closed

However, for it to be Compact, it needs to be Closed & Bounded.
For it to be bounded, it must have both an upper and lower bound, which to me it appears to have?
The bound for me are when x and y = ±√3

Clearly I am wrong, given how the question is structured. Any ideas?
 
  • #11
##x=y=\sqrt(3)## Is not in the set, nor is 0. You'll need to show that they are upper and lower bounds.

Another definition of bounded is that you can find a ball in which the whole set is contained in.
 
  • #12
Dick
Science Advisor
Homework Helper
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  • #13
HallsofIvy
Science Advisor
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Let u= x+ y and v= x- y. Then x= (u+ v)/2 and y= (u- v)/2. Replace x and y in the equation. That will eliminate the "xy" term and it is easy to see what kind of conic you have.
 
  • #14
pasmith
Homework Helper
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I need to apologise to everyone . . . immediately after I posted, I needed to run to lectures . . . I never noticed the error in my post, and wasn't able to correct it from my mobile phone. Really did not mean to waste your time. Stated correctly, it should read as:

Show that the set S of all (x,y) ∈ ℝ2 such that x2+xy+y2 = 3
is closed but not compact.

(Have been trying to correct the original post, but doesn't seem possible?)
I think there's a further error in the problem statement: S as you have defined it is actually compact.
 

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