Probability and elementary set theory proofs (i) suppose P(A) = $0 $ or $ 1$, prove that A and any subset B of Ω are independent. 
I did:  $P(A)+P(B)+P(C)=1 $ since $ P(Ω)=1.$
If $P(A)=0, $ then $P(B)=1-P(C)=P(B|A)$
Similarly, if $P(A)=1, $ then $P(B)=P(C)=P(B|A)$
Therefore, A and any subset B of Ω are independent. Is this correct and/or sufficient?

This is a bad assumption:
$$P(A)+P(B)+P(C)=1.$$
In the case $P(A) = 1,$ you cannot use this to prove something about
"any subset $B$ of $\Omega$", because the assumption implies that $P(B) = 0$.
We know there are subsets of $\Omega$ that have non-zero probabilities,
and you have discarded all of them.
