Generalized Union and Intersection by Induction Our teacher, told us to prove,
$$\left( \bigcap_{i=1}^n A_i\right)^c = \bigcup_{i=1}^n (A_i^c) $$
By induction. He told us that it has something to do with DeMorgan.
So my question is on knowing what's on the sets. I think that the left one has all the numbers to n except for the number 1, but in the right I get lost. 
Can you explain how I could know the elements on the right-handed set? Also, if you could tell me another hint I would really thank you.

$$P(n)=\left( \bigcap_{i=1}^n A_i\right)^c = \bigcup_{i=1}^n (A_i^c) $$
Now for P(1) $A_1^c=A_1^c$
Let P(k) be true i.e. $\left( \bigcap_{i=1}^k A_i\right)^c = \bigcup_{i=1}^k (A_i^c)$ 
Let $\left( \bigcap_{i=1}^{k} A_i \right)=B$
Now for P(k+1):  
$\left( \bigcap_{i=1}^{k+1} A_i\right)^c=\left( \bigcap_{i=1}^{k} A_i\cap A_{k+1}\right)^c=(B \cap A_{k+1})^c=(B^c \cup A_{k+1}^c)=\left( \bigcap_{i=1}^{k} A_i \right)\cup A_{k+1}^c$
$=\left( \bigcup_{i=1}^{k} A_i^c \right)\cup A_{k+1}^c=\left( \bigcup_{i=1}^{k+1} A_i^c \right)$
