Intersection of Disjoint Unions and deMorgan's Laws for Disjoint Sets Let $\sqcup$ denote the disjoint union.  Then is there nice deMorgan relation we can obtain for:
$$\left(\bigsqcup_{i=1}^n E_i\right)^c$$
On the other hand, suppose we have the following
$$\bigcap_{i=1}^n \left(\bigsqcup_{j=1}^{m_i} B_{i_j}\right)$$
Is there a particularly pleasant distribution of unions and intersections that yields something like $\sqcup\cap$?  So that ultimately I have a disjoint union (with respect to the outside set operation)?
 A: You can't define disjoint union inside a single set, but DeMorgan's law requires that. For example, if $S=\{1,2,3\}$ and $E_1=E_2=E_3=E_4=\{1\}$, what set are you using to perform the complement of $\bigsqcup E_i$? Is it different if you change $E_4=\emptyset$?
Disjoint unions are also problematic if you are dealing with intersections, since the set you get from a disjoint union is actually order-dependent - a different indexing of the sets gives a different order.
Basically, unlike unions, disjoint unions are really only unique up to isomorphism, and is not necessarily contained in any particular set.
A: Firstly, I apologise for not adding this as a comment but I have under 50 reputation so I cannot at the moment. This is directed @Thomas Andrews.
You are correct in saying that for your given sets $E_1$, $E_2$, $E_3$ and $E_4$ the disjoint union is not defined. But if $(E_i)_i$ is a sequence of pairwise-disjoint sets (i.e., $E_i \cap E_j = \emptyset$ for $i \neq j$), then the disjoint union of the $E_i$'s is $\bigsqcup_i E_i = \bigcup_i E_i$. Please see https://en.wikipedia.org/wiki/Disjoint_union for more details.
