(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

LetIdenote the interval [0,[itex]\infty[/itex]). For each r [itex]\in[/itex]I, define

A_{r}= {(x,y) [itex]\in[/itex]RxR: x^{2}+y^{2}= r^{2}},

B_{r}= {(x,y) [itex]\in[/itex]RxR: x^{2}+y^{2}[itex]\leq[/itex] r^{2}},

C_{r}= {(x,y) [itex]\in[/itex]RxR: x^{2}+y^{2}> r^{2}}

a) Determine [itex]\bigcup[/itex]_{r[itex]\in[/itex]I}A_{r}and [itex]\bigcap[/itex]_{r[itex]\in[/itex]I}A_{r}

b) Determine [itex]\bigcup[/itex]_{r[itex]\in[/itex]I}B_{r}and [itex]\bigcap[/itex]_{r[itex]\in[/itex]I}B_{r}

c) Determine [itex]\bigcup[/itex]_{r[itex]\in[/itex]I}C_{r}and [itex]\bigcap[/itex]_{r[itex]\in[/itex]I}C_{r}

2. Relevant equations

Well, I'm not sure on relevant equations in this situation, other than each one of the equations in a,b,c are of circles with radius r.

3. The attempt at a solution

Well, I know for A the union is all points (x,y) in the x,y plane. This is because infinitely many circles can be drawn from the origin, and their union would include all points. Now, I also know that the intersection for A is nothing, so the empty set.

b) I'm a bit confused as to how to write my answer here. I know B contains all the points on the edge of and within the circle of radius r, drawn from the origin. So, the union, again, is every point (keep drawing circles with an increasing radius starting from the origin, and the union would include every point). Now, for the intersection, I'm confused. I know the only thing they would have in common would be the smallest circle drawn, as it would be contained within each of the larger ones. I just dont know how to write this answer / would it be the circle of radius 0 (i.e. they just have the origin in common)?

c) The union is all points greater than the origin, correct? As, starting from 0, it would be all points greater than (0,0). The intersection, would it be the empty set, or would it be infinity? As the points included in the set are those outside the radius of the circle, so as the circles get larger and infinitely larger, the only area in common is infinity / the area not enclosed by the largest circle's circumference.

Thanks!

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