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What does set D mean here? Could someone please explain in words what it is a set of?
How does the sentence follow from it and can someone please translate the sentence?
And how do De Morgan's laws follow? I have understood the definition of a power set and all the previous parts of the book but am really confused by this! Any help is much appreciated.

halmos p20

Pie
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1 Answers1

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You have in the background a universal set $E$ and some collection $\mathcal{C}$ of subsets of $E$. The collection $\mathcal{D}$ is the one containing any set whose complement is a member of $\mathcal{C}$ (provided you stay inside the universal set $E$, of course).

Another way to imagine this is to form $\mathcal{D}$ by taking the complement of each set belonging to $\mathcal{C}$ and placing it in $\mathcal{D}$, since $X^{\prime\prime} = X$ for any set $X$.

The collection $\mathcal{D}$ is only related to DeMorgan's laws because they both reference the collection of all set complements of another collection. For example, $$ \bigcup_{X \in \mathcal{C}} X^\prime $$ is the same as $$ \bigcup_{X \in \mathcal{D}} X. $$ The latter hides the complementation in the indexing set.

Austin Mohr
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    @Austin_Mohr Thanks for your explanation, I now understand where D comes from. Would you be able to give a hint as to how I might go about proving the general form of De Morgan's laws? – Pie Jul 29 '14 at 16:35
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    @Pie This question might be helpful: http://math.stackexchange.com/questions/37045/de-morgans-law-on-infinite-unions-and-intersections – Austin Mohr Jul 29 '14 at 21:40