Indexed Unions: Subscripts for Infinitely Many Sets

Define $C_n$, list $C_1$ through $C_4$, and compute $\bigcup_{j=4}^{8}C_j$ and $\bigcap_{j=4}^{8}C_j$, recording that the bounds on the symbol pick which sets are combined.

Write both set-builder definitions, marking that the union uses 'for some $j$' and the intersection 'for all $j$', then write the versions for $j \geq 1$ over infinitely many sets.

Compute $\bigcup_{j=1}^{6}A_j$ and $\bigcap_{j=1}^{6}A_j$, then $\bigcup_{j=1}^{\infty}A_j = \mathbb{N}$ and $\bigcap_{j=1}^{\infty}A_j = \{1\}$, and note that unions grow and intersections shrink as more sets are included.