Families Indexed by Any Set: One Interval per Real Number

State the definition for a nonempty $\Lambda$, and record one example where $\alpha \ne \beta$ but $A_\alpha = A_\beta$, such as $B_1 = B_{-1}$ for $B_x = \{0, x^2, x^4\}$.

Write $\bigcup_{\alpha \in \Lambda} A_\alpha$ and $\bigcap_{\alpha \in \Lambda} A_\alpha$ in set-builder notation, marking 'there exists an $\alpha \in \Lambda$' for the union and 'for all $\alpha \in \Lambda$' for the intersection.

Derive that the union is $(-1, \infty)$ and the intersection is $(-1, 0]$, treating $y \le -1$, $-1 < y \le 0$ and $y > 0$ separately, including the choice $\beta = \frac{y}{2}$.

Write $A_\alpha^c$, compute $\left(\bigcup A_\alpha\right)^c$, $\bigcap A_\alpha^c$, $\left(\bigcap A_\alpha\right)^c$ and $\bigcup A_\alpha^c$ in interval notation, and record which two pairs agree.