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With $X$ being the set of all finite or cofinite subsets of the power set of $\mathbb{N}$ and $\subseteq$ as a relation.

How do I prove that the union and intersection of some finite subset of $X$, $X_1$, and some cofinite subset of $X$, $X_2$ is, again, in $X$?

I know how to do it when both subsets are either finite or cofinite by using the DeMorgan laws.

But how to do it in this case?

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Hint: The intersection of a finite set with anything is finite. The union of a cofinite set with anything is cofinite.

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    I thought the latter was only true for infinite sets. Can you give a hint how to prove the latter please? – 2017-01-03
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    It is the complement of the first. There are only finitely many things not in the set. Taking the union can only reduce the set of things not in the set. – 2017-01-03