Let \( A\), \(B\), and \(C_i\) be events.

1. Show that \( (A \cup B )^c= A^c \cap B^c\).
2. Show that \( (A \cap B)^c = A^c \cup B^c \).
3. \[ \Big( \bigcup_{i=1}^n C_i \Big)^c = \bigcap_{i=1}^n C_i^c\]
4. \[ \Big( \bigcap_{i=1}^n C_i \Big)^c = \bigcup_{i=1}^n C_i^c\]
5. (**) Argue that the above previous two statements hold even when \(n=\infty\).