If $L_{1} a n d L_{2}$ are context free languages and R a regular set. one of the languages below is not necessarily a context free language. Which one?

L_{1} \cup L_{2}

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L_{1} L_{2}

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L_{1} \cap L_{2}

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L_{1} \cap R

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Solution

The correct option is C $L_{1} \cap L_{2}$
(a) If $L_{1} a n d L_{2}$ are context free language then $L_{1} L_{2}$ is also context free language because context free languages are closed under concatenation.
(b) Since context free languages are not closed under intersection, $L_{1} \cap L_{2}$ is not necessarily a context free language.
(c) If $L_{1}$ is context free language and R is regular set then $L_{1} \cap R$ is surely context free language because context free languages are closed under regular intersection.
(d) If $L_{1} a n d L_{2}$ are context free languages then $L_{1} \cup L_{2}$ is surely context free language because context free languages are closed under union.

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