Let A, B, and C be sets. The formula

$\displaystyle A-(B-C) = (A-B)\cup C$

is sometimes false. State an additional necessary and sufficient condition for it to be always true.

(The minus sign here means set difference.) It is pretty easy for me to come up with sufficient conditions (i.e., if $\displaystyle C\subseteq A$ and $\displaystyle B\cap C=\varnothing$, then the equation is true), but I'm not sure how to find a necessary condition.

Any element which is in both sets above must be in A. I think we could have three cases:

1. $\displaystyle x\in A$, $\displaystyle x\not\in B$, $\displaystyle x\in C$.

2. $\displaystyle x\in A$, $\displaystyle x\not\in B$, $\displaystyle x\not\in C$.

3. $\displaystyle x\in A$, $\displaystyle x\in B$, $\displaystyle x\in C$.

But I'm not really sure where to go from there.