Recall the alternative denition for the star of a language A that we gave just b
ID: 3807863 • Letter: R
Question
Recall the alternative denition for the star of a language A that we gave just before(A^k is concatenation of A^k and A^k-1) Theorem 2.3.1. In Theorems 2.3.1 and 2.6.2, we have shown that the class of regular languages is closed under the union and concatenation operations. Since A* = (definition of A star), why doesn’t this imply that the class of regular languages is closed under the star operation?
What does this mean? Does it mean A* is also closed under the union and concatenation operations?
Please help me what should i prove
Explanation / Answer
We know that a class of regular languages is closed under the union and concatenation operations means if we perform concatenation or union of two regular languages the resultant is also a regular language. And while we state this we confirm that union and concatenation are binary operators and perform operation on two languages. A* on other hand is an unary operator, so we don't state that regular languages are closed under star operator. We state a language is closed under when we are able to perform the operation on two different grammar.
But applying our common sense we can say that regular languages are also closed under star operator. We can say that star operator does not violate regular language property by applying pumping lemma for regular language. Even we can draw an NFA with lambda transmission from final state to start state and also make start state as the final state will represent star operation.
From the definition perspective, 'A* is also closed under the union and concatenation operations' is also very true, since A* means union of Ak, where Ak is nothing but concatenation of k many As. So star operation is nothing but a combination of union and concatenation. Since both union and concatenation are both included in closure property of Regular languages, we don't need to add star operation as a separate operation.
Both the statement given above are correct, you are free to choose either of them. But statement two is more apt for this question.
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