Let $A$, $B$ and $C$ be 3 discrete random variables. If $A$ and $B$ are independent ($I(A;B) = 0$, where $I$ represents the mutual information), how can we prove that $I(A;B|C)=0$? When I draw a Venn diagram this seems trivially true, but I just cannot find a way to prove it.
$I(A;B\mid C)$ indicates "the reduction in the uncertainty of $A$ due to knowledge of $B$ when $C$ is given".
To find an example where $I(A;B\mid C)\not= 0$, we can see if the combination of $B$ and $C$ can help determine $A$ even though knowing $B$ alone does not help at all.
So let us try to create $A$ from the combination of $B$ and $C$. Let $B$ and $C$ be two simplest non-trivial independent random variables, $$P(B=0, C=0)=P(B=0, C=1)=P(B=1, C=0)=P(B=1, C=1)=1/4.$$ Let $A$ be 0 if $B$ and $C$ turns out the same and 1 otherwise.
We can check that $A$, $B$ and $C$ are pairwise independent. However, given the values of any two of them, the third one is determined. (In modulo arithmetic, $B+C=A(\bmod 2)$). When $C$ is given, we will not know anything more about $A$; if $B$ is known additionally, we will know $A$ completely. The above shows how we can find and understand an example. Beginners are encouraged to verify various "obvious" (or "obscure") claims above by using the corresponding definitions.
Exercise. Verify that $I(A;B)=0$ and $I(A;B\mid C)=1$ in the example above.