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Let $X$ be a set and $\{Y_\alpha\}$ is infinite system of some subsets of $X$. Is it true that: $$\bigcup_\alpha(X\setminus Y_\alpha)=X\setminus\bigcap_\alpha Y_\alpha,$$ $$\bigcap_\alpha(X\setminus Y_\alpha)=X\setminus\bigcup_\alpha Y_\alpha.$$ (infinite DeMorgan laws)

Thanks a lot!

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    Yes, it is, and the proof is basically the same as with finite unions/intersections2012-10-05
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    @DonAntonio: Not exactly, actually. The finite case is done by induction where the step uses the fact we proved these for two sets and associativity. The general case uses a slightly different approach.2012-10-05
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    I suppose induction can be used whenever something countable kicks in, yet I wouldn't do it that way but showing one side is contained in the other one and the other way around, as shown in my answer below. Following this strategy both proofs are practically indistinguishable.2012-10-05
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    @DonAntonio: Yes, this is true.2012-10-05

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