## Discrete Mathematics with Applications 4th Edition

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