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Intersection and partial quantity decidability

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Problem Detail: 

I'm still insecure in the section decidability (no proof needed, I want to divine it):

X is decidable and Y is undecidable. Is the intersection of X and Y decidable or undecidable?

X is decidable and Y is a partial quantity of X. Is Y decidable, too or not?


Asked By : user22709
Answered By : Rick Decker

For your first question, the answer is that $X\cap Y$ will not necessarily be decidable. Let $Y$ be an undecidable language over an alphabet $\Sigma$ and let $X=\Sigma^*$. Then obviously $X$ will be decidable and $X\cap Y=Y$ will be undecidable. On the other hand, the intersection may be decidable, as we could show by letting $X=\emptyset$.

For your second question, I assume that by "$Y$ is a partial quantity of $X$" you mean that $Y$ is a subset of $X$ (i.e., $Y\subseteq X$). Then we can do a similar construction, letting $X=\Sigma^*$, and $Y$ be undecidable, so again, $Y$ may or may not be decidable.

It's worth mentioning that this latter result frequently trips up students. Very few properties of languages are closed under subset/superset: there's no guarantee that a subset of a regular language will be regular, for example.

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