Unions And Intersections Of Arbitrarily Many Sets

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Definition (Unions And Intersections Of Arbitrarily Many Sets)

Suppose we have a large collection of sets. Each set has been given a name of the form $A_j$ where $j$ is an element of some nonempty set $J$. We call the collection $\mathrm{A} = \{A_j\mid j\in J\}$ an indexed family of sets with index set $J$. The union and intersection of all the sets in $\mathrm{A}$ are the sets given by $$\bigcup_{j\in J}A_j = \{x\mid x\in A_j \text{ for some } j\in J\}\text{ and } \bigcap_{j\in J}A_j = \{x\mid x\in A_j \text{ for every } j\in J\}.$$ When $J=\mathbb{N}$, then we'll often write the union and intersection using the notation $$\bigcup_{n\in \mathbb{N}}A_n = \bigcup_{n=1}^\infty A_n = A_1\cup A_2\cup A_3\cup\cdots\text{ and } \bigcap_{n\in \mathbb{N}}A_n = \bigcap_{n=1}^\infty A_n = A_1\cap A_2\cap A_3\cap\cdots.$$