Intersection/Union of Language of PDA, CFG and NFA
$L(P)$ and $L(G)$ are context-free; $L(N)$ is regular.  By standard closure properties, the intersection of a context-free language with a regular language is context-free, and the union of two context-free languages is context-free. Therefore, $L(P)\cap L(N)$ is some context-free language&nbsp;$C$ and $C\cup L(G)$ is context free.

The closure properties can be proven without reference to the specific language by using the fact that every context-free language is accepted by a PDA or defined by a context-free grammar.  For example, to show that the union of two context-free languages is context free, let those languages be defined by grammars $G_1$ and&nbsp;$G_2$ with start symbols $S_1$ and&nbsp;$S_2$. We can rename the non-terminals in $G_1$ and&nbsp;$G_2$ so that no non-terminal appears in both grammars. Now form the a grammar&nbsp;$G$ with start symbol&nbsp;$S$ consisting of the rule $S\to S_1|S_2$, all the rules of $G_1$ and all the rules of&amp;nbsp:$G_2$.  You should be able to convince yourself that $L(G) = L(G_1)\cup L(G_2)$. But proving this didn't require us to know anything about what rules are contained in the grammars.