Question

Find  and  if for every positive integer ,

a)   Ai={0,i}           b)  Ai=[-i,i].

Answer 1

a)  if for every positive integer i, $A_i=\{0,i\}$ , then

$\bigcup\limits_{i=1}^{+\infty}A_i=\bigcup\limits_{i=1}^{+\infty}\{0,i\}=\{0\}\cup \bigcup\limits_{i=1}^{+\infty}\{i\}=\{0\}\cup N$

$\bigcap\limits_{i=1}^{+\infty}A_i=\bigcap\limits_{i=1}^{+\infty}\{0,i\}=\{0\}$

 b) if for every positive integer i, $A_i=[-i,i]$ , then

$\bigcup\limits_{i=1}^{+\infty}A_i=\bigcup\limits_{i=1}^{+\infty}[-i,i]=(-\infty,+\infty)$

since for every $x\in R$ there exists $i\in R$ such that $|x|\leq i$, i.g. $x\in [-i,i]$

$\bigcap\limits_{i=1}^{+\infty}A_i=\bigcap\limits_{i=1}^{+\infty}[-i,i]=A_1=[-1,1]$

since $\bigcap\limits_{i=1}^{+\infty}A_i\subset A_1$ and $A_1\subset A_i$ for all i, which implies that $A_1\subset\bigcap\limits_{i=1}^{+\infty}A_i$