Solution to Find and if for every positive integer , a) Ai={0,i} b) Ai=[-i,i]. - Sikademy
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Find and if for every positive integer , a) Ai={0,i} b) Ai=[-i,i].

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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


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