Answer
$a.\,\, \bigcup_{i=0}^{4}C_{i}=\left \{ 0,1,-1,2,-2,3,-3,4,-4 \right \}$
$b.\,\, \bigcap_{i=0}^{4}C_{i}=\varnothing $
$c.\,\,yes\,\, \,\,, C_{0},C_{1},C_{2},C_{3},.....\,\,are\,\,mutually\,\,dijoint$
$d.\,\,\bigcup_{i=0}^{n}C_{i}=\left \{ 0,1,-1,2,-2,3,-3,4,-4,.....,n,-n \right \}$
$e.\,\,\,\, \bigcap_{i=0}^{n}C_{i}=\varnothing$
$f.\,\,\bigcup_{i=0}^{\infty }C_{i}=\left \{ 0,1,-1,2,-2,3,-3,..... \right \}$
$g.\,\, \bigcap_{i=0}^{\infty}C_{i }=\varnothing $
Work Step by Step
$C_{i}=\left \{ i,-i \right \} \,\, for\,all\,nonnegative\,integers\,i . $
$C_{0}=\left \{ 0,0 \right \}=\left \{ 0 \right \} $
$C_{1}=\left \{ 1,-1 \right \} ,
C_{2}=\left \{ 2,-2 \right \} ,
C_{3}=\left \{ 3,-3 \right \} ,
C_{4}=\left \{ 4,-4 \right \}...C_{n}=\left \{ n,-n \right \}$
$a.\,\,\bigcup_{i=0}^{4}C_{i}=C_{0}\cup C_{1}\cup C_{2}\cup C_{3}\cup C_{4}=\left \{ 0,1,-1,2,-2,3,-3,4,-4 \right \}$
$b.\,\, \bigcap_{i=0}^{4}C_{i}=C_{0}\cap C_{1}\cap C_{2}\cap C_{3}\cap C_{4}=\varnothing \,\,,
as\,\,C_{0},C_{1},C_{2},C_{3},C_{4}\,have\,no\,\,element\,\,in\,common
$
$c.\,\,C_{0},C_{1},C_{2},C_{3},C_{4},....\,are\,\,mutually\,\,dis\! joint\,because\, \,\,we\,can\,see\,that\,C_{i}\cap C_{j}=\left \{ i,-i \right \}\cap \left \{ j,-j \right \}= \varnothing \,
for\,integers\,\,i,j\geq 0,i\neq j$
$d.\,\,\bigcup_{i=0}^{n}C_{i}=C_{0}\cup C_{1}\cup C_{2}\cup C_{3}.....\cup C_{n}=\left \{ 0,1,-1,2,-2,.....n,-n \right \}$
$e.\,\,\bigcap_{i=0}^{n}C_{i}=C_{0}\cap C_{1}\cap C_{2}\cap C_{3}\cap.... C_{n}=\left \{ 0 \right \}\cap \left \{ 1,-1 \right \}\cap \left \{ 2,-2 \right \}\cap \left \{ 3,-3 \right \}
\cap .......\left \{ n,-n \right \}=\varnothing$
$f.\,\,\bigcup_{i=0}^{\infty}C_{i}=C_{0}\cup C_{1}\cup C_{2}\cup ....=\left \{ 0 \right \}\cup \left \{ 1,-1 \right \}\cup \left \{ 2,-2 \right \}\cup .....=\left \{ 0,1,-1,2,-2,..... \right \}\,where\,\,the\,\,union\,is\,\,taken\,\,for\,all\,natural\,numbers\,$
$g.\,\,\bigcap_{i=0}^{\infty}C_{i}=C_{0}\cap C_{1}\cap C_{2}\cap C_{3}\cap...=\left \{ 0 \right \}\cap \left \{ 1,-1 \right \}\cap \left \{ 2,-2 \right \}\cap \left \{ 3,-3 \right \}\cap ......=\varnothing\, since\,\,from\,\,c\,\,\, C_{i}'s\,\,are\,mutually\,\,disjoint\,$
