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## Something like De Morgan’s laws to apply to set union/difference? [closed]

I have two series of subsets (say $x$ and $y$) of some set $S$, so $x_n$, $y_n$ $\subseteq$ $S$.

Can I expand $(\cup x) \setminus (\cup y)$ into anything that groups the $x_n$ and $y_n$ together? So instead of:

$(x_0 \cup x_1 \cup ... \cup x_n)\setminus(y_0 \cup y_1 \cup ... \cup y_n)$

I'm looking for some operations $\circ$ and $\times$ s.t. the above can be written in the form:

$(x_0 \circ y_0) \times (x_1 \circ y_1) \times ... \times (x_n \circ y_n)$

A similar, but separate question: can I do the same for an intersection of differences? ex:

$\cap (x_n \setminus y_n \ \forall \ n \in 1...N)$

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Try writing $A \setminus B$ as $A \cap B^c$ where $B^c$ denotes the complement of $B$. – Yemon Choi Apr 2 2010 at 20:44
Thanks Yemon. Trying that, I get $(x_0 \cup ... \cup x_n) \cap (y_0^c \cap ... \cap y_n^c)$. But union and intersection aren't associative, are they? – Andrey Fedorov Apr 2 2010 at 21:05
Please see en.wikipedia.org/wiki/Algebra_of_sets – François G. Dorais Apr 2 2010 at 21:06
I'm afraid this is not research level mathematics, so this question is not appropriate for MO. See the FAQ for suggestions of other places to ask this - mathoverflow.net/faq#whatquestions – François G. Dorais Apr 2 2010 at 21:13
Sorry about that, François. Thanks for the reference to the FAQ. – Andrey Fedorov Apr 2 2010 at 21:35