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let l1 be a finite language and l2 a context-free language. show that the language l2 − l1 is context-free.

Answer :

To show that the language L2 - L1 is context-free, we need to show that it can be generated by context-free grammar.

First, let's define the languages L1 and L2. The language L1 is a finite language, which means that it consists of a finite number of strings. A language is context-free if it can be generated by a context-free grammar. Therefore, L2 is a language that can be generated by a context-free grammar.

Now, let's consider the language L2 - L1. This language consists of all strings that can be generated by a context-free grammar that are not in L1. In other words, it is the set of all strings in L2 that are not in L1.

Since L1 is a finite language, we can generate a regular grammar for it. This is a context-free grammar in which each production rule has the form A → a or A → aB, where A and B are non-terminal symbols and a is a terminal symbol. We can then use this regular grammar to generate a finite automaton that recognizes L1.

To generate a context-free grammar for L2 - L1, we can modify the context-free grammar for L2 by adding a new non-terminal symbol S' and a new production rule S' → S, where S is the start symbol of the grammar for L2. This ensures that all strings in L2 that are not in L1 can be generated by this modified grammar.

To see why this works, consider that any string in L2 that is also in L1 can be generated by the grammar for L1. By adding a new start symbol and production rule, we can generate all strings that are not in L1 without affecting the generation of strings that are in L1.

Therefore, we have shown that the language L2 - L1 is context-free by constructing a context-free grammar for it.

To learn more about context-free grammar: https://brainly.com/question/29453027

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