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.
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Hint Use induction. Another hint for the basis ($n=2$):
Prove that the following theorem holds
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...)