Effect of difference operator on sampling Suppose I have a dataset $D$ and let it be split into 2 smaller datasets $D_1$ and $D_2$ such that $D = D_1 + D_2 $. Thus we can also say that $D_1=D - D_2$. 
Will we get a simple random sample of $D_1$ if we took a simple random sample of $D$ and removed from it those data points obtained by taking a random sample of $D_2$?
 A: I don't see how you'd be guaranteed a simple sample of $D_1$ by subtracting points in $D_2$ from different points in $D$. For example, consider these distributions:
$$D=\{1,2,...100\}\\D_1=\{1,1,...1\}\\D_2=\{0,1,...99\}$$
If you happen to randomly sample the first five values in a row (it could happen!) from $D$ and subtract five randomly sampled values from $D_2$, you're pretty unlikely to end up with a sample of $D_1$; you'll probably get a bunch of negative values, none of which belong to $D_1$ of course.
I.e., I think you'd only get to sample randomly from $D$ or $D_2$, not both. Same goes for stratified sampling: it'll only work if you sample systematically within strata, not randomly – at least, not randomly with both samples. E.g., you could sample randomly from $D$, with or without stratification, but if you happen to sample the 8th, 48th, and 88th values from $D$, the only way I can see to guarantee that you'll get values from $D_1$ by subtracting values sampled from $D_2$ is to systematically sample the same 8th, 48th, and 88th values from $D_2$.
