Find the sets
$$
\bigcup_{N=1}^\infty\left(\bigcap_{n=N}^\infty A_n\right) \text{ and } \bigcap_{N=1}^\infty\left(\bigcup_{n=N}^\infty A_n\right)
$$
if
(1) $A_1,A_2,\dots$ are pairwise disjoint
(2) if $A_n=\left\{\begin{array}{ll}
B \text{ if } n \text{ is odd}\\
C \text{ if } n \text{ is even}
\end{array}
\right.$
For (1) my argument is that $\bigcup_{N=1}^\infty\left(\bigcap_{n=N}^\infty A_n\right)=\emptyset$ (as I first take the intersection of disjoint sets which is empty) and $\bigcap_{N=1}^\infty\left(\bigcup_{n=N}^\infty A_n\right)=\infty$ (as I first take the union of a infinitely large set, and that the intersection of infinitely large set is infinite) Am I correct?
For (2) $\bigcup_{N=1}^\infty\left(\bigcap_{N=1}^\infty A_n\right)=\bigcup_{N=1}^\infty\left(B_1\cap C_2 \cap B_3 \cap C_4\cap \ldots \right)$ and $\bigcap_{N=1}^\infty\left(\bigcup_{n=N}^\infty A_n\right)=\bigcap_{N=1}^\infty\left(B_1\cup C_2 \cup B_3 \cap C_4\cup \ldots \right)$, but I havve really no idea how to proceed from here.
Best Answer
The following characterization can be helpful:
$$a\in\bigcup_{N=1}^{\infty}\bigcap_{n=N}^{\infty}A_{n}\iff\left\{ n\in\mathbb{N}\mid a\notin A_{n}\right\} \text{ is a finite set}$$
$$a\in\bigcap_{N=1}^{\infty}\bigcup_{n=N}^{\infty}A_{n}\iff\left\{ n\in\mathbb{N}\mid a\in A_{n}\right\} \text{ is not a finite set}$$
(1) Set $\left\{ n\in\mathbb{N}\mid a\notin A_{n}\right\} $ is not finite and $a\in A_{n}$ cannot be true for more than one index $n$. So both sets are empty.
(2) Set $\left\{ n\in\mathbb{N}\mid a\notin A_{n}\right\} $ is finite if and only if $a\in B\cap C$ and set $\left\{ n\in\mathbb{N}\mid a\in A_{n}\right\} $ is infinite if and only if $a\in B\cup C$.