Problem $1$: Building Sets $(20$ Points$)$

Suppose that A$1,$ A$2,$ A$3,...$ are a sequence of sets. We can define infinite unions and infinite intersections in the

following way:

$$

n$=1$

An $=\{$x : $$n in N : x in An$\}$

$$

n$=1$

An $=\{$x : $$n in N : x in An$\}.$

$(1)$

Let us also define the $$closed ball$,$

Bclosed$($r$)=$

$\{$

$($x$,$ y$)$ in R$2$ : $$x$2+$ y$2$ r

$\}$

$(2)$

and the $$open ball$,$

Bopen$($r$)=$

$\{$

$($x$,$ y$)$ in R$2$ : $$x$2+$ y$2<$ r

$\}$

$.(3)$

For each of the following, a$)$ give a $$simplified form$$ of the set, and b$)$ prove that your simplified set is equivalent to

the expression below.

$1)$

n$=1$ Bclosed $(1$

n

$)$

$2)$

n$=1$ Bclosed $(1$

n

$)$

$3)$

n$=2$ Bclosed $(11$

n

$)$

$4)$

n$=2$ Bclosed $(11$

n

$)$

$5)$

n$=1$ Bopen $(1$

n

$)$

$6)$

n$=1$ Bopen $(1$

n

$)$

$7)$

n$=2$ Bopen $(11$

n

$)$

$8)$

n$=2$ Bopen $(11$

n

$)$