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Is the union of a collection of context-free languages always context-free ? Justify your answer .....

I know that the answer is yes, but how can I prove it ?

dz216
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To show that the finite union of context-free languages is context-free you just have to build a context-free grammar for the union language, exactly as you would do to prove that the union of two context-free languages is context-free.

If G1,...,GN are the context-free grammars for the N context-free languages you have, rename all the symbols in the each grammar (add a subscript just to avoid symbol name clashes) and then make a new grammar G with all the productions from the N grammars, plus the production:

S -> S1 | S2 | ... | SN

This grammar generates the union language, and it is context-free.

Alejandro Piad
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