Independence of complement Independence events

Question

I need to show that if $E_1,E_2,\ldots, E_n$ are independent then $E_1^c ,E_2^c,\ldots, E_n^c$ are independent too. Please provide a hint.

Answer 1

Hint Use induction. Another hint for the basis ($n=2$):

Prove that the following theorem holds

Let $E_1,E_2$ independent. Then $E_1$ and $E_2^c$ are independent.

Applying this theorem you can easily prove the independence of $E_1^c$, $E_2^c$

(If you are done with $n=2$, I can add some more hints...)