Let $S\subset 2^{\Sigma^*}$ be some family of formal languages over some alphabet $\Sigma$.
Consider the the following statement:
$A,A\cap B, A\cup B\in S$ implies $B\in S$
For which (preferably known) families $S$ does the statement hold ( regular / CFG / recursively enumerable / poly-time solvable/ etc. ) ? for which it doesn't?
It seems that since $$B = ((A\cup B)\setminus A) \cup (A\cap B)$$ Then if $S$ is closed for union and subtraction, then the property holds (which proves it for regular languages or $P$ for example).
Also, if $\emptyset,\Sigma^*\in S$, and $S$ is not closed for complement, then the statement is false (take $B=\bar A=\Sigma^*\setminus A$ (so $RE$/$CFG$ are out).