# Determine Union of Sets Belonging to Interval

1. Feb 3, 2014

### 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?

Last edited: Feb 3, 2014
2. Feb 4, 2014

### Office_Shredder

Staff Emeritus
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 Ar is geometrically as well.

3. Feb 4, 2014

### knowLittle

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

4. Feb 5, 2014

### haruspex

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