Determine Union of Sets Belonging to Interval

[knowLittle]

Let $I$ denote the interval $[0, \infty )$ . For each r $\in I$ define:

$A_{r} = \{ (x,y), \in$R x R : $x^{2} +y^{2} = r^{2} \}$
$B_{r} = \{ (x,y), \in$R x R : $x^{2} +y^{2} \leq r^{2} \}$
$C_{r} = \{$ ... $: ... > r^{2} \}$

a.) Determine $\bigcup_{r\in I} A_{r}$ and $\bigcap_{r \in I} A_{r}$

For case, $A_{3}$
Is this right?
For, $A_{3} = \{ (3,0), (0,3), (\sqrt(4.5), \sqrt(4.5)) , (\sqrt(4.6), \sqrt(4.4)), \dots \}$

Can I just list partitions of square roots that would give me 9?

 

[Office_Shredder]

Yes, that is a partial list of elements in A₃. Obviously there are infinitely many of them.

For the purposes of solving the problem it would probably be instructive to think about what the set A_r is geometrically as well.

 

[knowLittle]

Are they points that map the radius of a circle for $A_{r}$?

 

[haruspex]

Are they points that map the radius of a circle for $A_{r}$?

That's a slightly odd way of saying it, but yes, A_r consists of the points of a circle of radius r, centred at the origin.