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Be a set $M ⊆ R^2$ given by

$M = ${$x = (x_1,x_2)^T ∈ R^2 | 0 < x_1^2 + x_2^2 < 4\text{ and }x_1 ≤1$}

Determine the boundary of M

Attempt,

i know that $x_1^2 + x_2^2 = 4$ is the circle and $x_1 = 1$ is the line.

How can i combine them, to get the boundary

Patricio
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Vek
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  • Try drawing $M$ – Patricio Jun 09 '22 at 14:39
  • @Patricio i did it and i got a circle who is cut at $x_1=1$ – Vek Jun 09 '22 at 14:42
  • Can you describe those points mathematically? – Patricio Jun 09 '22 at 14:43
  • hm.. i know the inner of M with $I = 0<x_1^2+x_2^2<4, x<1$ and the closure of M with $C= 0 ≤ x_1^2+x_2^2≤4,x≤1$ and i know that the boundary is defined with C\I but i guess i cant describe it mathematically -.- – Vek Jun 09 '22 at 14:49
  • What about enumerating those points? $\partial M={x=(x_1,x_2)^T\in\Bbb R^2\mid -2\leq x_1< 1\dots}$ – Patricio Jun 09 '22 at 14:54
  • so i can say $x_1^2+x_2^2=4 ∩ -2 ≤ x_1<1$ so i have the boundary of the circle, without the line x=1, also i have to do $-1,73 ≤ x_2 ≤ 1,73 ∩ x_1 = 1$ and combine them together? – Vek Jun 09 '22 at 15:34
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    That's what I'd do, except I'd write $\sqrt{3}$ rather than $1.73$ – Patricio Jun 10 '22 at 08:11

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