Union and Intersection of Multiple Sets

Sometimes, we want to find the union or intersection of more than 2 sets. Instead of writing the taunting long notation $A_1 \cup A_2 \cup A_3 \,$$\cup A_4 \cup \,$$\dots \cup A_n$ for the union of all these sets, we can instead write: $$\bigcup_{1 \le i \le n} A_i = \bigcup A_i$$ (

$\bigcup​_{1 \le i \le n} A_i$

and

$\bigcup A_i$

). Similarly, instead of writing $A_1 \cap A_2 \cap A_3 \cap \,$$A_4 \cap \,$$\dots \cap A_n$ for the intersection of all these sets, we can instead write: $$\bigcap_{1 \le i \le n} A_i = \bigcap A_i$$ (

$\bigcap_​{1 \le i \le n} A_i$

and

$\bigcap A_i$

). You can replace $i$ with any other variable of your choice.