# Is the inclusion of a regular language in a context-free language decidable?

Is the following problem decidable in general:

Given a regular language $R$ and context-free $C$, is every string in $R$ also in $C$? That is, is $L(R)\subseteq L(C)$? What about $L(C)\subseteq L(R)$?

I know that this is decidable if $R$ and $C$ are both regular and undecidable if they are both context free.

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This is a correct answer for $R \subseteq C$ for regular $R$ and context-free $C$. The question also mentions the reverse inclusion $C \subseteq R$.
Somewhat surprisingly, that is decidable. We have $C \subseteq R$ iff $C \cap R^c = \emptyset$. Here $R^c$ is the complement of $R$ is regular, and thus the intersection $C \cap R^c$ is context-free. Emptiness of context-free languages is decidable.