probability set theory proof Denote by A, B and C subsets of the sample space Ω, 
Please advice me on my solutions below as I am unsure and help me answer the part iii.
(ii) if P(A) > 0,P(B) > 0, and A,B are mutually exclusive, then are A and B independent?
I did: If A and B are mutually exclusive then $P(A\cap B)=0$ as $(A\cap B)=\phi$
Since, $P(A\cap B)=0\ne P(A)P(B)> 0$  as $ P(A),P(B) > 0.$ It implies A and B are dependent.
(iii) if A, B, C are (mutually) independent, prove that $A^c, B^c $ and $ C^c$ are (mutually) independent.
I did: $P(A\cap B\cap C)=P(A)P(B)P(C).$
$((A∩B)∩C)^c=(A\cap B)^c\cup C^c=A^c\cup B^c\cup C^c - $ DeMorgan's Laws
$P((A∩B)∩C)^c=1-P(A)P(B)P(C)=P(A^c\cup B^c\cup C^c)=P(A^c)+P(B^c)+P(C^c)$
I don't know what to do from now on. Please help. Also advice on the above solution.

Start by proving that if $A$ and $B$ are independent, then $A^c$ and $B^c$ are independent. I will supply a proof for you; you can then easily extend the proof for the three sets $A, B, C$.  
Proof
$P(A^c \cap B^c) = P(A^c) + P(B^c) - P(A^c \cup B^c)$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 1 - P(A) + 1 - P(B) - P((A \cap B)^c)$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 2 - P(A) - P(B) - (1 - P(A)P(B))$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 1 + P(A)P(B) - P(A) - P(B)$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = (1 - P(A))(1 - P(B))$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = P(A^c)P(B^c)$
